[STATEMENT]
lemma interpret_floatarith_eventually_isDERIV:
  assumes iD: "\<And>i j. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)"
  assumes m: "max_Var_floatarith fa \<le> DIM('a::executable_euclidean_space) + length params"
  shows "\<forall>i < DIM('a). \<forall>\<^sub>F (x::'a) in at x. isDERIV i fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
using iD m
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa (list_of_eucl x @ params)
max_Var_floatarith fa \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
proof (induction fa)
[PROOF STATE]
proof (state)
goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
case (Inverse fa)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)
max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)
max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params
[PROOF STEP]
have "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)
max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
have iD: "i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)" "interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0" for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
[PROOF STEP]
using Inverse.prems(1)[OF ]
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)

goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
[PROOF STEP]
by force+
[PROOF STATE]
proof (state)
this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa (list_of_eucl x @ params)
interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
from Inverse
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)
max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params
[PROOF STEP]
have m: "max_Var_floatarith fa \<le> DIM('a) + length params"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Inverse fa) (list_of_eucl x @ params)
max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. max_Var_floatarith fa \<le> DIM('a) + length params
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
max_Var_floatarith fa \<le> DIM('a) + length params

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
from has_derivative_continuous[OF interpret_floatarith_FDERIV_floatarith_append, OF iD(1) m]
[PROOF STATE]
proof (chain)
picking this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x"
[PROOF STATE]
proof (prove)
using this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0"
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
[PROOF STEP]
using iD(2) tendsto_imp_eventually_ne
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
\<lbrakk>(?f \<longlongrightarrow> ?c) ?F; ?c \<noteq> ?c'\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F z in ?F. ?f z \<noteq> ?c'

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
[PROOF STEP]
by (auto simp: isCont_def)
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0

goal (18 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Inverse fa) (list_of_eucl x @ params); max_Var_floatarith (Inverse fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 7. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 9. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 10. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
A total of 18 subgoals...
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<noteq> 0

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)
[PROOF STEP]
by (auto elim: eventually_elim2)
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Inverse fa) (list_of_eucl x @ params)

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
case (Sqrt fa)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)
max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)
max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params
[PROOF STEP]
have "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)
max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
have iD: "i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)" "interpret_floatarith fa (list_of_eucl x @ params) > 0" for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
using Sqrt.prems(1)[OF ]
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)

goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
by force+
[PROOF STATE]
proof (state)
this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa (list_of_eucl x @ params)
0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
from Sqrt
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)
max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params
[PROOF STEP]
have m: "max_Var_floatarith fa \<le> DIM('a) + length params"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Sqrt fa) (list_of_eucl x @ params)
max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. max_Var_floatarith fa \<le> DIM('a) + length params
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
max_Var_floatarith fa \<le> DIM('a) + length params

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
from has_derivative_continuous[OF interpret_floatarith_FDERIV_floatarith_append, OF iD(1) m]
[PROOF STATE]
proof (chain)
picking this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x"
[PROOF STATE]
proof (prove)
using this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) > 0"
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
using iD(2) order_tendstoD
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
0 < interpret_floatarith fa (list_of_eucl x @ params)
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?a < ?y\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?a < ?f x
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?y < ?a\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?f x < ?a

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
by (auto simp: isCont_def)
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (17 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Sqrt fa) (list_of_eucl x @ params); max_Var_floatarith (Sqrt fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
A total of 17 subgoals...
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)
[PROOF STEP]
by (auto elim: eventually_elim2)
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Sqrt fa) (list_of_eucl x @ params)

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
case (Powr fa1 fa2)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)
max_Var_floatarith (Powr fa1 fa2) \<le> DIM('a) + length params

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)
max_Var_floatarith (Powr fa1 fa2) \<le> DIM('a) + length params
[PROOF STEP]
have "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)"
    "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)
max_Var_floatarith (Powr fa1 fa2) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params) &&& \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
have iD: "i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params)" "interpret_floatarith fa1 (list_of_eucl x @ params) > 0"
    for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
[PROOF STEP]
using Powr.prems(1)
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)

goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
[PROOF STEP]
by force+
[PROOF STATE]
proof (state)
this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa1 (list_of_eucl x @ params)
0 < interpret_floatarith fa1 (list_of_eucl x @ params)

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
from Powr
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)
max_Var_floatarith (Powr fa1 fa2) \<le> DIM('a) + length params
[PROOF STEP]
have m: "max_Var_floatarith fa1 \<le> DIM('a) + length params"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Powr fa1 fa2) (list_of_eucl x @ params)
max_Var_floatarith (Powr fa1 fa2) \<le> DIM('a) + length params

goal (1 subgoal):
 1. max_Var_floatarith fa1 \<le> DIM('a) + length params
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
max_Var_floatarith fa1 \<le> DIM('a) + length params

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
from has_derivative_continuous[OF interpret_floatarith_FDERIV_floatarith_append, OF iD(1) m]
[PROOF STATE]
proof (chain)
picking this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x
[PROOF STEP]
have "isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x"
[PROOF STATE]
proof (prove)
using this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x
[PROOF STEP]
have "\<forall>\<^sub>F x in at x. interpret_floatarith fa1 (list_of_eucl x @ params) > 0"
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
[PROOF STEP]
using iD(2) order_tendstoD
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa1 (list_of_eucl x @ params)) x
0 < interpret_floatarith fa1 (list_of_eucl x @ params)
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?a < ?y\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?a < ?f x
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?y < ?a\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?f x < ?a

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
[PROOF STEP]
by (auto simp: isCont_def)
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)

goal (16 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 16 subgoals...
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
apply safe
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>i. \<lbrakk>\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params); i < DIM('a)\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
subgoal for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params); i < DIM('a)\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
apply (safe dest!: spec[of _ i])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params); i < DIM('a); \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
subgoal premises prems
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
using prems(1,3,4)
[PROOF STATE]
proof (prove)
using this:
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa1 (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params)

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)
[PROOF STEP]
by eventually_elim auto
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Powr fa1 fa2) (list_of_eucl x @ params)

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
case (Ln fa)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)
max_Var_floatarith (Ln fa) \<le> DIM('a) + length params

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)
max_Var_floatarith (Ln fa) \<le> DIM('a) + length params
[PROOF STEP]
have "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)
max_Var_floatarith (Ln fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
have iD: "i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)" "interpret_floatarith fa (list_of_eucl x @ params) > 0" for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
using Ln.prems(1)[OF ]
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)

goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
by force+
[PROOF STATE]
proof (state)
this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa (list_of_eucl x @ params)
0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
from Ln
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)
max_Var_floatarith (Ln fa) \<le> DIM('a) + length params
[PROOF STEP]
have m: "max_Var_floatarith fa \<le> DIM('a) + length params"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Ln fa) (list_of_eucl x @ params)
max_Var_floatarith (Ln fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. max_Var_floatarith fa \<le> DIM('a) + length params
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
max_Var_floatarith fa \<le> DIM('a) + length params

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
from has_derivative_continuous[OF interpret_floatarith_FDERIV_floatarith_append, OF iD(1) m]
[PROOF STATE]
proof (chain)
picking this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x"
[PROOF STATE]
proof (prove)
using this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have "\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) > 0"
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
using iD(2) order_tendstoD
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
0 < interpret_floatarith fa (list_of_eucl x @ params)
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?a < ?y\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?a < ?f x
\<lbrakk>(?f \<longlongrightarrow> ?y) ?F; ?y < ?a\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in ?F. ?f x < ?a

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
by (auto simp: isCont_def)
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (15 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 15 subgoals...
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. 0 < interpret_floatarith fa (list_of_eucl x @ params)

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Ln fa) (list_of_eucl x @ params)
[PROOF STEP]
by (auto elim: eventually_elim2)
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Ln fa) (list_of_eucl x @ params)

goal (14 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 14 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (14 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 14 subgoals...
[PROOF STEP]
case (Power fa m)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (fa ^\<^sub>e m) (list_of_eucl x @ params)
max_Var_floatarith (fa ^\<^sub>e m) \<le> DIM('a) + length params

goal (14 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 14 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (fa ^\<^sub>e m) (list_of_eucl x @ params)
max_Var_floatarith (fa ^\<^sub>e m) \<le> DIM('a) + length params
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (fa ^\<^sub>e m) (list_of_eucl x @ params)
max_Var_floatarith (fa ^\<^sub>e m) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (fa ^\<^sub>e m) (list_of_eucl x @ params)
[PROOF STEP]
by (cases m) auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (fa ^\<^sub>e m) (list_of_eucl x @ params)

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
case (Floor fa)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)
max_Var_floatarith (Floor fa) \<le> DIM('a) + length params

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)
max_Var_floatarith (Floor fa) \<le> DIM('a) + length params
[PROOF STEP]
have "\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)
max_Var_floatarith (Floor fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
have iD: "i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)"
    "interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>" for i
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>
[PROOF STEP]
using Floor.prems(1)[OF ]
[PROOF STATE]
proof (prove)
using this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)

goal (1 subgoal):
 1. (i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)) &&& interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>
[PROOF STEP]
by force+
[PROOF STATE]
proof (state)
this:
?i < DIM('a) \<Longrightarrow> isDERIV ?i fa (list_of_eucl x @ params)
interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
from Floor
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)
max_Var_floatarith (Floor fa) \<le> DIM('a) + length params
[PROOF STEP]
have m: "max_Var_floatarith fa \<le> DIM('a) + length params"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
?i < DIM('a) \<Longrightarrow> isDERIV ?i (Floor fa) (list_of_eucl x @ params)
max_Var_floatarith (Floor fa) \<le> DIM('a) + length params

goal (1 subgoal):
 1. max_Var_floatarith fa \<le> DIM('a) + length params
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
max_Var_floatarith fa \<le> DIM('a) + length params

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
from has_derivative_continuous[OF interpret_floatarith_FDERIV_floatarith_append, OF iD(1) m]
[PROOF STATE]
proof (chain)
picking this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
have cont: "isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x"
[PROOF STATE]
proof (prove)
using this:
(\<And>i j. i < DIM('a) \<Longrightarrow> i < DIM('a)) \<Longrightarrow> isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
let ?i = "\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)"
[PROOF STATE]
proof (state)
goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
have "\<forall>\<^sub>F y in at x. ?i y > floor (?i x)" "\<forall>\<^sub>F y in at x. ?i y < ceiling (?i x)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<forall>\<^sub>F y in at x. real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl y @ params) &&& \<forall>\<^sub>F y in at x. interpret_floatarith fa (list_of_eucl y @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>
[PROOF STEP]
using cont
[PROOF STATE]
proof (prove)
using this:
isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x

goal (1 subgoal):
 1. \<forall>\<^sub>F y in at x. real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl y @ params) &&& \<forall>\<^sub>F y in at x. interpret_floatarith fa (list_of_eucl y @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>
[PROOF STEP]
by (auto simp: isCont_def eventually_floor_less eventually_less_ceiling iD(2))
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F y in at x. real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl y @ params)
\<forall>\<^sub>F y in at x. interpret_floatarith fa (list_of_eucl y @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<forall>\<^sub>F y in at x. real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl y @ params)
\<forall>\<^sub>F y in at x. interpret_floatarith fa (list_of_eucl y @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>
[PROOF STEP]
have "\<forall>\<^sub>F x in at x. ?i x \<notin> \<int>"
[PROOF STATE]
proof (prove)
using this:
\<forall>\<^sub>F y in at x. real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl y @ params)
\<forall>\<^sub>F y in at x. interpret_floatarith fa (list_of_eucl y @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>

goal (1 subgoal):
 1. \<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>
[PROOF STEP]
apply eventually_elim
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>xa. \<lbrakk>real_of_int \<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < interpret_floatarith fa (list_of_eucl xa @ params); interpret_floatarith fa (list_of_eucl xa @ params) < real_of_int \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>\<rbrakk> \<Longrightarrow> interpret_floatarith fa (list_of_eucl xa @ params) \<notin> \<int>
[PROOF STEP]
apply (auto simp: Ints_def)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>xa xaa. \<lbrakk>\<lfloor>interpret_floatarith fa (list_of_eucl x @ params)\<rfloor> < xaa; xaa < \<lceil>interpret_floatarith fa (list_of_eucl x @ params)\<rceil>; interpret_floatarith fa (list_of_eucl xa @ params) = real_of_int xaa\<rbrakk> \<Longrightarrow> False
[PROOF STEP]
by linarith
[PROOF STATE]
proof (state)
this:
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>

goal (13 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 13 subgoals...
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)
\<forall>\<^sub>F x in at x. interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>

goal (1 subgoal):
 1. \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Floor fa) (list_of_eucl x @ params)
[PROOF STEP]
by (auto elim: eventually_elim2)
[PROOF STATE]
proof (state)
this:
\<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Floor fa) (list_of_eucl x @ params)

goal (12 subgoals):
 1. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Add fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Add fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Add fa1 fa2) (list_of_eucl x @ params)
 2. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Minus fa) (list_of_eucl x @ params); max_Var_floatarith (Minus fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Minus fa) (list_of_eucl x @ params)
 3. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Mult fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Mult fa1 fa2) (list_of_eucl x @ params)
 4. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Cos fa) (list_of_eucl x @ params); max_Var_floatarith (Cos fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Cos fa) (list_of_eucl x @ params)
 5. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Arctan fa) (list_of_eucl x @ params); max_Var_floatarith (Arctan fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Arctan fa) (list_of_eucl x @ params)
 6. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Abs fa) (list_of_eucl x @ params); max_Var_floatarith (Abs fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Abs fa) (list_of_eucl x @ params)
 7. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Max fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Max fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Max fa1 fa2) (list_of_eucl x @ params)
 8. \<And>fa1 fa2. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa1 (list_of_eucl x @ params); max_Var_floatarith fa1 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa1 (list_of_eucl x @ params); \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa2 (list_of_eucl x @ params); max_Var_floatarith fa2 \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa2 (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Min fa1 fa2) (list_of_eucl x @ params); max_Var_floatarith (Min fa1 fa2) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Min fa1 fa2) (list_of_eucl x @ params)
 9. \<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i Pi (list_of_eucl x @ params); max_Var_floatarith Pi \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i Pi (list_of_eucl x @ params)
 10. \<And>fa. \<lbrakk>\<lbrakk>\<And>i. i < DIM('a) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params); max_Var_floatarith fa \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params); \<And>i. i < DIM('a) \<Longrightarrow> isDERIV i (Exp fa) (list_of_eucl x @ params); max_Var_floatarith (Exp fa) \<le> DIM('a) + length params\<rbrakk> \<Longrightarrow> \<forall>i<DIM('a). \<forall>\<^sub>F x in at x. isDERIV i (Exp fa) (list_of_eucl x @ params)
A total of 12 subgoals...
[PROOF STEP]
qed (fastforce intro: DIM_positive elim: eventually_elim2)+