[STATEMENT]
lemma af\<^sub>G_implies_af_eval\<^sub>G':
  assumes "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w)"
  assumes "\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>"
  assumes "\<And>\<psi> i. G \<psi> \<in> \<G> \<Longrightarrow> i < length w \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))"
  shows   "S \<Turnstile>\<^sub>P af \<phi> w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> w
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> w
[PROOF STEP]
proof (induction \<phi> arbitrary: w)
[PROOF STATE]
proof (state)
goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
case (LTLGlobal \<phi>)
[PROOF STATE]
proof (state)
this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
hence "G \<phi> \<in> \<G>"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. G \<phi> \<in> \<G>
[PROOF STEP]
unfolding af\<^sub>G_simps eval\<^sub>G.simps
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P (if G \<phi> \<in> \<G> then true else false)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. G \<phi> \<in> \<G>
[PROOF STEP]
by (cases "G \<phi> \<in> \<G>") simp+
[PROOF STATE]
proof (state)
this:
G \<phi> \<in> \<G>

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
hence "S \<Turnstile>\<^sub>P G \<phi>"
[PROOF STATE]
proof (prove)
using this:
G \<phi> \<in> \<G>

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P G \<phi>
[PROOF STEP]
using LTLGlobal
[PROOF STATE]
proof (prove)
using this:
G \<phi> \<in> \<G>
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P G \<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P G \<phi>

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P G \<phi>

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
{
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P G \<phi>

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
fix x
[PROOF STATE]
proof (state)
goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
assume "x \<in> set (map (af \<phi>) (suffixes w))"
[PROOF STATE]
proof (state)
this:
x \<in> set (map (af \<phi>) (suffixes w))

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x \<in> set (map (af \<phi>) (suffixes w))
[PROOF STEP]
obtain w' where "x = af \<phi> w'" and "w' \<in> set (suffixes w)"
[PROOF STATE]
proof (prove)
using this:
x \<in> set (map (af \<phi>) (suffixes w))

goal (1 subgoal):
 1. (\<And>w'. \<lbrakk>x = af \<phi> w'; w' \<in> set (suffixes w)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x = af \<phi> w'
w' \<in> set (suffixes w)

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x = af \<phi> w'
w' \<in> set (suffixes w)
[PROOF STEP]
obtain i where "w' = drop i w" and "i < length w"
[PROOF STATE]
proof (prove)
using this:
x = af \<phi> w'
w' \<in> set (suffixes w)

goal (1 subgoal):
 1. (\<And>i. \<lbrakk>w' = drop i w; i < length w\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by (auto simp add: suffixes_alt_def subsequence_def)
[PROOF STATE]
proof (state)
this:
w' = drop i w
i < length w

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
hence "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')"
[PROOF STATE]
proof (prove)
using this:
w' = drop i w
i < length w

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
[PROOF STEP]
using LTLGlobal.prems \<open>G \<phi> \<in> \<G>\<close>
[PROOF STATE]
proof (prove)
using this:
w' = drop i w
i < length w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))
G \<phi> \<in> \<G>

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
hence "S \<Turnstile>\<^sub>P x"
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P x
[PROOF STEP]
using LTLGlobal(1)[OF \<open>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')\<close>] LTLGlobal(3-4) drop_drop
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
\<lbrakk>\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w'
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))
drop ?n (drop ?m ?xs) = drop (?n + ?m) ?xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P x
[PROOF STEP]
unfolding \<open>x = af \<phi> w'\<close> \<open>w' = drop i w\<close>
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (drop i w))
\<lbrakk>\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> ia. \<lbrakk>G \<psi> \<in> \<G>; ia < length (drop i w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop ia (drop i w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> (drop i w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))
drop ?n (drop ?m ?xs) = drop (?n + ?m) ?xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> (drop i w)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P x

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
?x2 \<in> set (map (af \<phi>) (suffixes w)) \<Longrightarrow> S \<Turnstile>\<^sub>P ?x2

goal (10 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (G \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (G \<phi>) w
 9. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 10. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
S \<Turnstile>\<^sub>P G \<phi>
?x2 \<in> set (map (af \<phi>) (suffixes w)) \<Longrightarrow> S \<Turnstile>\<^sub>P ?x2
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P G \<phi>
?x2 \<in> set (map (af \<phi>) (suffixes w)) \<Longrightarrow> S \<Turnstile>\<^sub>P ?x2

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (G \<phi>) w
[PROOF STEP]
unfolding af_G eval\<^sub>G_And_map And_prop_entailment
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P G \<phi>
?x2 \<in> set (map (af \<phi>) (suffixes w)) \<Longrightarrow> S \<Turnstile>\<^sub>P ?x2

goal (1 subgoal):
 1. Ball (set (G \<phi> # map (af \<phi>) (suffixes w))) ((\<Turnstile>\<^sub>P) S)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (G \<phi>) w

goal (9 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 9. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (9 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 9. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
case (LTLFinal \<phi>)
[PROOF STATE]
proof (state)
this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (9 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 9. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))
[PROOF STEP]
obtain x where x_def: "x \<in> set (F \<phi> # map (eval\<^sub>G \<G> o af\<^sub>G \<phi>) (suffixes w))"
      and "S \<Turnstile>\<^sub>P x"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. (\<And>x. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
unfolding Or_prop_entailment af\<^sub>G_F eval\<^sub>G_Or_map
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
Bex (set (map (eval\<^sub>G \<G>) (F \<phi> # map (af\<^sub>G \<phi>) (suffixes w)))) ((\<Turnstile>\<^sub>P) S)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. (\<And>x. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by force
[PROOF STATE]
proof (state)
this:
x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w))
S \<Turnstile>\<^sub>P x

goal (9 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 9. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
hence "\<exists>y \<in> set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y"
[PROOF STATE]
proof (prove)
using this:
x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w))
S \<Turnstile>\<^sub>P x

goal (1 subgoal):
 1. \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
proof (cases "x \<noteq> F \<phi>")
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
case True
[PROOF STATE]
proof (state)
this:
x \<noteq> F \<phi>

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x \<noteq> F \<phi>
[PROOF STEP]
obtain w' where "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')" and "w' \<in> set (suffixes w)"
[PROOF STATE]
proof (prove)
using this:
x \<noteq> F \<phi>

goal (1 subgoal):
 1. (\<And>w'. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w'); w' \<in> set (suffixes w)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using x_def \<open>S \<Turnstile>\<^sub>P x\<close>
[PROOF STATE]
proof (prove)
using this:
x \<noteq> F \<phi>
x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w))
S \<Turnstile>\<^sub>P x

goal (1 subgoal):
 1. (\<And>w'. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w'); w' \<in> set (suffixes w)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
w' \<in> set (suffixes w)

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
hence "\<And>\<psi> i. G \<psi> \<in> \<G> \<Longrightarrow> i < length w' \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))"
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
w' \<in> set (suffixes w)

goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))
[PROOF STEP]
using LTLFinal.prems
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')
w' \<in> set (suffixes w)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))
[PROOF STEP]
by (auto simp add: suffixes_alt_def subsequence_def)
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
have "\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G \<psi>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G \<psi>)
[PROOF STEP]
using LTLFinal
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G \<psi>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G ?\<psi>)

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G ?\<psi>)
[PROOF STEP]
have "S \<Turnstile>\<^sub>P af \<phi> w'"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G ?\<psi>)

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> w'
[PROOF STEP]
using LTLFinal.IH[OF \<open>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w')\<close>]
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G ?\<psi>)
\<lbrakk>\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w'

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> w'
[PROOF STEP]
using assms(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w'))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (G ?\<psi>)
\<lbrakk>\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w'\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w'))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w'
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> w'
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af \<phi> w'

goal (2 subgoals):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
 2. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<phi> w'

goal (1 subgoal):
 1. \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
using \<open>w' \<in> set (suffixes w)\<close>
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<phi> w'
w' \<in> set (suffixes w)

goal (1 subgoal):
 1. \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y

goal (1 subgoal):
 1. \<lbrakk>x \<in> set (F \<phi> # map (eval\<^sub>G \<G> \<circ> af\<^sub>G \<phi>) (suffixes w)); S \<Turnstile>\<^sub>P x; \<not> x \<noteq> F \<phi>\<rbrakk> \<Longrightarrow> \<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y
[PROOF STEP]
qed simp
[PROOF STATE]
proof (state)
this:
\<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y

goal (9 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (F \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (F \<phi>) w
 9. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
\<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (F \<phi>) w
[PROOF STEP]
unfolding af_F Or_prop_entailment eval\<^sub>G_Or_map
[PROOF STATE]
proof (prove)
using this:
\<exists>y\<in>set (F \<phi> # map (af \<phi>) (suffixes w)). S \<Turnstile>\<^sub>P y

goal (1 subgoal):
 1. Bex (set (F \<phi> # map (af \<phi>) (suffixes w))) ((\<Turnstile>\<^sub>P) S)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (F \<phi>) w

goal (8 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (8 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
case (LTLNext \<phi>)
[PROOF STATE]
proof (state)
this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (8 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi> w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 8. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
proof (cases w)
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
case (Cons x xs)
[PROOF STATE]
proof (state)
this:
w = x # xs

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
{
[PROOF STATE]
proof (state)
this:
w = x # xs

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
fix \<psi> i
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
assume "G \<psi> \<in> \<G>" and "Suc i < length (x#xs)"
[PROOF STATE]
proof (state)
this:
G \<psi> \<in> \<G>
Suc i < length (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
hence "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x#xs)))"
[PROOF STATE]
proof (prove)
using this:
G \<psi> \<in> \<G>
Suc i < length (x # xs)

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x # xs)))
[PROOF STEP]
using LTLNext.prems
[PROOF STATE]
proof (prove)
using this:
G \<psi> \<in> \<G>
Suc i < length (x # xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x # xs)))
[PROOF STEP]
unfolding Cons
[PROOF STATE]
proof (prove)
using this:
G \<psi> \<in> \<G>
Suc i < length (x # xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x # xs)))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x # xs)))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
hence "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))"
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop (Suc i) (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi>2 \<in> \<G>; Suc ?i2 < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi>2 (drop ?i2 xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
hence "\<And>\<psi> i. G \<psi> \<in> \<G> \<Longrightarrow> i < length xs \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))"
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi>2 \<in> \<G>; Suc ?i2 < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi>2 (drop ?i2 xs))

goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
 2. \<And>a list. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = a # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i xs))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
using LTLNext Cons
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i xs))
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))
w = x # xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (X \<phi>) w

goal (1 subgoal):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (X \<phi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (X \<phi>) w
[PROOF STEP]
qed simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (X \<phi>) w

goal (7 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (7 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
case (LTLUntil \<phi> \<psi>)
[PROOF STATE]
proof (state)
this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (7 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
 7. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 U \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 U \<phi>2) w
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w
[PROOF STEP]
proof (induction w)
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
case (Cons x xs)
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
{
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
assume "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))"
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
have "\<And>\<psi> i. G \<psi> \<in> \<G> \<Longrightarrow> i < length (x#xs) \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x#xs)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x # xs)))
[PROOF STEP]
using Cons
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x # xs)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))
[PROOF STEP]
have "S \<Turnstile>\<^sub>P af \<psi> (x # xs)"
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<psi> (x # xs)
[PROOF STEP]
using Cons.prems
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<psi> (x # xs)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af \<psi> (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
hence ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<psi> (x # xs)

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
unfolding af_U
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<psi> (x # xs)

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs and af \<phi> (x # xs) or af \<psi> (x # xs)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
{
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
assume "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)" and "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))"
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
have "\<And>\<psi> i. G \<psi> \<in> \<G> \<Longrightarrow> i < length (x#xs) \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x#xs)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x # xs)))
[PROOF STEP]
using Cons
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (x # xs)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))
[PROOF STEP]
have "S \<Turnstile>\<^sub>P af \<phi> (x # xs)" and "S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs"
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> (x # xs) &&& S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
[PROOF STEP]
using Cons
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))
\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length xs\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> ?w
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> ?w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length ?w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i ?w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> ?w
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length (x # xs)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i (x # xs)))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af \<phi> (x # xs) &&& S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs
[PROOF STEP]
by (blast, force)
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af \<phi> (x # xs)
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
hence ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<phi> (x # xs)
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
unfolding af_U
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P af \<phi> (x # xs)
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) xs and af \<phi> (x # xs) or af \<psi> (x # xs)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (2 subgoals):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
 2. \<And>a w. \<lbrakk>\<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (a # w)); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length (a # w)\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i (a # w)))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (a # w)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
using Cons(4)
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) (x # xs))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
unfolding af\<^sub>G_U
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (x # xs)) \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
\<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs); S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (x # xs))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) xs and af\<^sub>G \<phi> (x # xs) or af\<^sub>G \<psi> (x # xs))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) (x # xs)

goal (1 subgoal):
 1. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi> w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<psi> w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi> U \<psi>) []); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i []))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) []
[PROOF STEP]
qed simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (\<phi> U \<psi>) w

goal (6 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (6 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
case (LTLProp a)
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (6 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(x) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(x) w
 4. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 6. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
proof (cases w)
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(a) w
 2. \<And>aa list. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = aa # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
case (Cons x xs)
[PROOF STATE]
proof (state)
this:
w = x # xs

goal (2 subgoals):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(a) w
 2. \<And>aa list. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = aa # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
w = x # xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
using LTLProp
[PROOF STATE]
proof (prove)
using this:
w = x # xs
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
by (cases "a \<in> x") simp+
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af p(a) w

goal (1 subgoal):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G p(a) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af p(a) w
[PROOF STEP]
qed simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af p(a) w

goal (5 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 4. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (5 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 4. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
case (LTLPropNeg a)
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (5 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>x w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(x)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(x)) w
 4. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 5. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
proof (cases w)
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(a)) w
 2. \<And>aa list. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = aa # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
case (Cons x xs)
[PROOF STATE]
proof (state)
this:
w = x # xs

goal (2 subgoals):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(a)) w
 2. \<And>aa list. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = aa # list\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
w = x # xs

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
using LTLPropNeg
[PROOF STATE]
proof (prove)
using this:
w = x # xs
S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w)
G ?\<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G ?\<psi>
\<lbrakk>G ?\<psi> \<in> \<G>; ?i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G ?\<psi> (drop ?i w))

goal (1 subgoal):
 1. S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
by (cases "a \<in> x") simp+
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (np(a)) w

goal (1 subgoal):
 1. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w)); w = []\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (np(a)) w
[PROOF STEP]
qed simp
[PROOF STATE]
proof (state)
this:
S \<Turnstile>\<^sub>P af (np(a)) w

goal (4 subgoals):
 1. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G true w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af true w
 2. \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G false w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af false w
 3. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 and \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 and \<phi>2) w
 4. \<And>\<phi>1 \<phi>2 w. \<lbrakk>\<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>1 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>1 w; \<And>w. \<lbrakk>S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi>2 w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af \<phi>2 w; S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (\<phi>1 or \<phi>2) w); \<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>; \<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P af (\<phi>1 or \<phi>2) w
[PROOF STEP]
qed (unfold af_equals_af\<^sub>G_base_cases af\<^sub>G_decompose af_decompose, auto)