[STATEMENT]
lemma (in Corps) ApproximationTr3:"\<lbrakk>vals_nonequiv K (Suc n) vv;
      \<forall>l \<le> (Suc n). x l \<in> carrier K; j \<le> (Suc n)\<rbrakk> \<Longrightarrow>
     \<exists>L.(\<forall>N. L < N \<longrightarrow> (an m) \<le> (vv j ((\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> (Suc n)}.
        (x k) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a ((1\<^sub>r \<plusminus> -\<^sub>a (((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) k)^\<^bsup>K N\<^esup>))^\<^bsup>K N\<^esup>)))
        (Suc n)) \<plusminus> -\<^sub>a (x j))))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> Suc n}. x k \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) k^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)
[PROOF STEP]
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> Suc n}. x k \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) k^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)
[PROOF STEP]
apply (frule_tac vals_nonequiv_valuation[of "Suc n" "vv" j], assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> Suc n}. x k \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) k^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)
[PROOF STEP]
apply (subgoal_tac "\<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> (Suc n)}. (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus>
 -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) (Suc n) \<plusminus> -\<^sub>a (x j) =
 \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h\<le> (Suc n)}. (if l \<noteq> j then (x l) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a
 ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus>
        -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>  \<plusminus> -\<^sub>a 1\<^sub>r))) (Suc n)")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> Suc n}. x k \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) k^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n
[PROOF STEP]
apply (simp del:nsum_suc)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n
[PROOF STEP]
apply (thin_tac "\<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> (Suc n)}. (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus>
 -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) (Suc n) \<plusminus> -\<^sub>a (x j) = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> (Suc n)}. if l \<noteq> j then (x l) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) (Suc n)")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n
[PROOF STEP]
prefer 2
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N. \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule allI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j = \<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule eSum_minus_x, assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>j\<le>Suc n. (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> j \<le> Suc n
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>ja\<in>{h. h \<le> Suc n} - {j}. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja = x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule allI, rule impI)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); ja \<le> Suc n\<rbrakk> \<Longrightarrow> (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> j \<le> Suc n
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>ja\<in>{h. h \<le> Suc n} - {j}. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja = x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule ApproximationTr0)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); ja \<le> Suc n\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> j \<le> Suc n
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>ja\<in>{h. h \<le> Suc n} - {j}. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja = x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> j \<le> Suc n
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>ja\<in>{h. h \<le> Suc n} - {j}. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja = x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>ja\<in>{h. h \<le> Suc n} - {j}. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja = x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule ballI, simp)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) j = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (frule Ring.ring_one[of "K"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (cut_tac aa = "(\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j" and N = N in
                                               ApproximationTr0)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (subst aGroup.ag_pOp_assoc, assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)+
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (subst aGroup.ag_pOp_commute[of "K" _ "-\<^sub>a 1\<^sub>r"], assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> (-\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)+
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> (-\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (subst aGroup.ag_pOp_assoc[THEN sym], assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 3. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)+
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (simp add:aGroup.ag_r_inv1)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (\<zero> \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (subst aGroup.ag_l_zero, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (simp add:aGroup.ag_mOp_closed)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); 1\<^sub>r \<in> carrier K; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) = x j \<cdot>\<^sub>r (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
(* subgoal 2 done **)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
[PROOF STEP]
apply (subgoal_tac "\<exists>L. \<forall>N. L < N \<longrightarrow>
  (\<forall>ja \<le> (Suc n). (an m) \<le> ((vv j \<circ> (\<lambda>l\<in>{h. h \<le> (Suc n)}. if l \<noteq> j then (x l) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja))")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
(*
 apply (subgoal_tac "\<exists>L. \<forall>N. L < N \<longrightarrow> (an m) \<le> Amin (Suc n) (vv j \<circ> (\<lambda>l\<in>Nset (Suc n). if l\<noteq>j then (x l) \<cdot>\<^sub>K (1\<^sub>K +\<^sub>K -\<^sub>K (1\<^sub>K +\<^sub>K -\<^sub>K ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^sup>K\<^sup> \<^sup>N)^\<^sup>K\<^sup> \<^sup>N)
 else (x j) \<cdot>\<^sub>K (1\<^sub>K +\<^sub>K  -\<^sub>K (1\<^sub>K +\<^sub>K -\<^sub>K ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^sup>K\<^sup> \<^sup>N)^\<^sup>K\<^sup> \<^sup>N
+\<^sub>K -\<^sub>K 1\<^sub>K)))")
 apply (erule exE)
 apply (subgoal_tac "\<forall>N. L < N \<longrightarrow>
  ((an m) \<le> ((vv j) (e\<Sigma> K (\<lambda>l\<in>Nset (Suc n). (if l \<noteq> j then (x l) \<cdot>\<^sub>K
(1\<^sub>K +\<^sub>K -\<^sub>K (1\<^sub>K +\<^sub>K -\<^sub>K ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^sup>K\<^sup> \<^sup>N)^\<^sup>K\<^sup> \<^sup>N) else (x j) \<cdot>\<^sub>K (1\<^sub>K +\<^sub>K -\<^sub>K
(1\<^sub>K +\<^sub>K -\<^sub>K ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^sup>K\<^sup> \<^sup>N)^\<^sup>K\<^sup> \<^sup>N +\<^sub>K -\<^sub>K 1\<^sub>K))) (Suc n))))")
apply blast
*)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (erule exE)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>L. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rename_tac M)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>M. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>M. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (subgoal_tac "\<forall>N. M < (N::nat) \<longrightarrow>
   (an m) \<le> (vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> (Suc n)}. (if l \<noteq> j then
   (x l) \<cdot>\<^sub>r  (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
   else (x j) \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) l)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>
   \<plusminus> -\<^sub>a 1\<^sub>r))) (Suc n)))")
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>M. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; \<forall>N>M. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<And>M. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>M. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>N>M. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>M. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>M. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>N>M. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule allI, rule impI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N>M. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; M < N\<rbrakk> \<Longrightarrow> an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (drule_tac a = N in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) Suc n)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule value_ge_add[of "vv j" "Suc n" _ "an m"], assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule allI, rule impI)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (frule Ring.ring_one[of "K"])
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (case_tac "ja = j", simp)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r) \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 3. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.ring_tOp_closed, assumption+, simp)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 3. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)+
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.npClose, assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.npClose, assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 4. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); j \<le> Suc n; 1\<^sub>r \<in> carrier K; ja = j\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 3. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja; ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)) ja \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.ring_tOp_closed, assumption+, simp)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)+
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.npClose, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (rule Ring.npClose, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>M N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> vv j (if ja \<noteq> j then x ja \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)); ja \<le> Suc n; 1\<^sub>r \<in> carrier K; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja \<in> carrier K
 2. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>M N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); M < N; \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
[PROOF STEP]
apply (subgoal_tac "\<forall>N. \<forall>ja \<le> (Suc n). (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a
 ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) \<in> carrier K")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (subgoal_tac "\<forall>N. (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>
                         \<plusminus> -\<^sub>a 1\<^sub>r) \<in> carrier K")
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja\<le>Suc n. an m \<le> (vv j \<circ> (\<lambda>l\<in>{h. h \<le> Suc n}. if l \<noteq> j then x l \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) else x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))) ja
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:val_t2p)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja. (ja \<noteq> j \<longrightarrow> ja \<le> Suc n \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (cut_tac multi_inequalityTr0[of "Suc n" "(vv j) \<circ> x" "m"])
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja. (ja \<noteq> j \<longrightarrow> ja \<le> Suc n \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (subgoal_tac "\<forall>ja \<le> (Suc n). (vv j \<circ> x) ja \<noteq> - \<infinity>", simp)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja. (ja \<noteq> j \<longrightarrow> ja \<le> Suc n \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (erule exE)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja. (ja \<noteq> j \<longrightarrow> ja \<le> Suc n \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (subgoal_tac "\<forall>N. L < N \<longrightarrow> (\<forall>ja \<le> (Suc n). (ja \<noteq> j \<longrightarrow>
an m \<le> vv j (x ja) + (vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)))
 \<and> (ja = j \<longrightarrow> (an m) \<le> vv j (x j) +  (vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus>
  -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a (1\<^sub>r)))))")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; \<forall>N>L. \<forall>ja\<le>Suc n. (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))\<rbrakk> \<Longrightarrow> \<exists>L. \<forall>N>L. \<forall>ja. (ja \<noteq> j \<longrightarrow> ja \<le> Suc n \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<And>L. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N\<rbrakk> \<Longrightarrow> \<forall>N>L. \<forall>ja\<le>Suc n. (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N\<rbrakk> \<Longrightarrow> \<forall>N>L. \<forall>ja\<le>Suc n. (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule allI, rule impI)+
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (case_tac "ja = j", simp)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "\<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in>
        carrier K")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "\<forall>l\<le>Suc n. x l \<in> carrier K")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (drule_tac x = N in spec)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; \<forall>ja\<le>Suc n. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (drule_tac a = j in forall_spec, assumption,
        thin_tac "\<forall>ja\<le>Suc n. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>
        \<in> carrier K")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (cut_tac N = N in ApproximationTr0 [of "(\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j"])
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule Ring.ring_one[of "K"], frule aGroup.ag_mOp_closed[of "K" "1\<^sub>r"],
         assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (
        frule_tac x = "(1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>" in
        aGroup.ag_mOp_closed[of "K"], assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp only:aGroup.ag_pOp_assoc)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp only:aGroup.ag_pOp_commute[of "K" _ "-\<^sub>a 1\<^sub>r"])
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> (-\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>))
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp only:aGroup.ag_pOp_assoc[THEN sym])
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:aGroup.ag_r_inv1)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (\<zero> \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:aGroup.ag_l_zero)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j (-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp only:val_minus_eq)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j ((1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "(1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K",
         thin_tac "-\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + vv j ((1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (subst val_exp_ring[THEN sym, of "vv j"], assumption+)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed[of "K"], assumption+)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed[of "K"], assumption)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule Ring.npClose, assumption+)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule Ostrowski_base_npow_not_one)
[PROOF STATE]
proof (prove)
goal (8 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> 0 < N
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> j \<le> Suc n
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> vals_nonequiv K (Suc n) vv
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 5. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 8. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> j \<le> Suc n
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> vals_nonequiv K (Suc n) vv
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply assumption+
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (drule_tac a = N in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>l\<le>Suc n. an m < vv j (x l) + an N\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (drule_tac a = j in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule Ostrowski_baseTr1[of "n" "vv" "j"], assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac n = "N - Suc 0" in val_1mx_pow[of "vv j" "(\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j"])
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)\<rbrakk> \<Longrightarrow> 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j); 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K Suc (N - Suc 0)\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)\<rbrakk> \<Longrightarrow> 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j); 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K Suc (N - Suc 0)\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; vv j (x j) \<noteq> - \<infinity>; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j); 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K Suc (N - Suc 0)\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "vv j (x j) \<noteq> - \<infinity>")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j); 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K Suc (N - Suc 0)\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp only:Suc_pred)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j); 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j))")
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (cut_tac b = "vv j (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)" and N = N in
        asprod_ge)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> N \<noteq> 0
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>); an N \<le> int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> N \<noteq> 0
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>); an N \<le> int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>); an N \<le> int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (cut_tac x = "an N" and y = "int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) j)^\<^bsup>K N\<^esup>)" in aadd_le_mono[of _ _ "vv j (x j)"], assumption)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; Ring K; aGroup K; valuation K (vv j); L < N; j \<le> Suc n; ja = j; 1\<^sub>r \<in> carrier K; -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x j) + an N; 0 < vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>); an N \<le> int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>); an N + vv j (x j) \<le> int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>) + vv j (x j)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x j) + int N *\<^sub>a vv j (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:aadd_commute)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (ja \<noteq> j \<longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) \<and> (ja = j \<longrightarrow> an m \<le> vv j (x j) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r))
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac aa = "(\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja" and N = N in
       ApproximationTr2[of "vv j"])
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja \<in> carrier K
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja \<noteq> \<zero>
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 4. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 7. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja \<noteq> \<zero>
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule Ostrowski_base_nonzero, assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac l = ja in Ostrowski_baseTr0[of "n" "vv"], assumption+,
       erule conjE)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); \<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rotate_tac -1, frule_tac a = j in forall_spec)
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> j \<le> Suc n
 2. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 3. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac x = j in bspec, simp)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; 0 < vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> 0 \<le> vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule aless_imp_le)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; 0 < vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> 0 < vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>N>L. \<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rotate_tac -5,
        drule_tac a = N in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; \<forall>ja\<le>Suc n. vv j (x ja) \<noteq> - \<infinity>; \<forall>l\<le>Suc n. an m < vv j (x l) + an N\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rotate_tac -2,
        drule_tac a = ja in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>l\<le>Suc n. an m < vv j (x l) + an N; L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; vv j (x ja) \<noteq> - \<infinity>\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (
        drule_tac a = ja in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; vv j (x ja) \<noteq> - \<infinity>; an m < vv j (x ja) + an N\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac l = ja in Ostrowski_baseTr0[of  "n" "vv"], assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; vv j (x ja) \<noteq> - \<infinity>; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<and> (\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja))\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (erule conjE, rotate_tac -1,
        frule_tac a = j in forall_spec, assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; vv j (x ja) \<noteq> - \<infinity>; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (thin_tac "vv j (x ja) \<noteq> - \<infinity>")
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (cut_tac b = "vv j ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)" and N = N in asprod_ge)
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> 0 < vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)
 2. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> N \<noteq> 0
 3. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 6. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K\<rbrakk> \<Longrightarrow> N \<noteq> 0
 2. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 5. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac x = "an N" and y = "int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)" and
        z = "vv j (x ja)" in aadd_le_mono)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac x = "int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)" and y = "(vv j)
     (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)" and z = "vv j (x ja)"
      in aadd_le_mono)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja); int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac i = "an N + vv j (x ja)" and
       j = "int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja) + vv j (x ja)" and
       k = "vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a ((\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja)^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) +
          vv j (x ja)" in ale_trans, assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja); int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an N + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (x ja) + vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (subst aadd_commute)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja); int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an N + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)\<rbrakk> \<Longrightarrow> an m \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac x = "an m" and y = "vv j (x ja) + an N" in aless_imp_le)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja); int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an N + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an m \<le> vv j (x ja) + an N\<rbrakk> \<Longrightarrow> an m \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule_tac j = "vv j (x ja) + an N" in ale_trans[of "an m"],
                  assumption)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>L N ja. \<lbrakk>\<forall>m\<in>{h. h \<le> Suc n} - {ja}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); L < N; ja \<le> Suc n; ja \<noteq> j; int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>); vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; an m < vv j (x ja) + an N; 0 < vv ja (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); x j \<in> carrier K; an N \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja); an N + vv j (x ja) \<le> int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja); int N *\<^sub>a vv j ((\<Omega>\<^bsub>K vv Suc n\<^esub>) ja) + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an N + vv j (x ja) \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja); an m \<le> vv j (x ja) + an N\<rbrakk> \<Longrightarrow> vv j (x ja) + an N \<le> vv j (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>) + vv j (x ja)
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 4. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:aadd_commute)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N)\<rbrakk> \<Longrightarrow> \<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule allI, rule impI, subst comp_def)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N); ja \<le> Suc n\<rbrakk> \<Longrightarrow> vv j (x ja) \<noteq> - \<infinity>
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac a = ja in forall_spec, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N); ja \<le> Suc n; x ja \<in> carrier K\<rbrakk> \<Longrightarrow> vv j (x ja) \<noteq> - \<infinity>
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (frule_tac x = "x ja" in value_in_aug_inf[of "vv j"], assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K; \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K; (\<forall>ja\<le>Suc n. (vv j \<circ> x) ja \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>Suc n. an m < (vv j \<circ> x) l + an N); ja \<le> Suc n; x ja \<in> carrier K; vv j (x ja) \<in> Z\<^sub>\<infinity>\<rbrakk> \<Longrightarrow> vv j (x ja) \<noteq> - \<infinity>
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (simp add:aug_inf_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> \<forall>N. 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule allI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<plusminus> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
 2. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 3. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K\<rbrakk> \<Longrightarrow> -\<^sub>a 1\<^sub>r \<in> carrier K
 2. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption, rule Ring.ring_one, assumption)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j)\<rbrakk> \<Longrightarrow> \<forall>N ja. ja \<le> Suc n \<longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply ((rule allI)+, rule impI)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N ja. \<lbrakk>vals_nonequiv K (Suc n) vv; \<forall>l\<le>Suc n. x l \<in> carrier K; j \<le> Suc n; Ring K; aGroup K; valuation K (vv j); ja \<le> Suc n\<rbrakk> \<Longrightarrow> 1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) ja^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K
[PROOF STEP]
apply (rule_tac aa = "(\<Omega>\<^bsub>K vv (Suc n)\<^esub>) ja" in ApproximationTr1,
       simp add:Ostrowski_base_mem)
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done