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theory Unification = Main + Terms + Fresh + Equ + Substs + Mgu:
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(* problems to which no reduction applies *)
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consts stuck :: "problem_type set"
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defs
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stuck_def: "stuck \<equiv> { P1. \<not>(\<exists>P2 nabla s. P1 \<Turnstile>(nabla,s)\<Rightarrow>P2)}"
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(* all problems which are stuck and have no unifier *)
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consts fail :: "problem_type set"
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inductive fail
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intros
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[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Abst a t#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Func F t#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Paar t1 t2#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Abst a t\<approx>?Susp pi X#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Func F t\<approx>?Susp pi X#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Paar t1 t2\<approx>?Susp pi X#xs,ys)\<in>fail"
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[intro!,dest!]: "([],a\<sharp>? Atom a#ys)\<in>fail"
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[intro!]: "a\<noteq>b\<Longrightarrow>(Atom a\<approx>? Atom b#xs,ys)\<in>fail"
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[intro!,dest!]: "(Abst a t\<approx>?Unit#xs,ys)\<in>fail"
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[intro!,dest!]: "(Unit\<approx>?Abst a t#xs,ys)\<in>fail"
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[intro!,dest!]: "(Abst a t\<approx>?Atom b#xs,ys)\<in>fail"
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[intro!,dest!]: "(Atom b\<approx>?Abst a t#xs,ys)\<in>fail"
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[intro!,dest!]: "(Abst a t\<approx>?Paar t1 t2#xs,ys)\<in>fail"
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[intro!,dest!]: "(Paar t1 t2\<approx>?Abst a t#xs,ys)\<in>fail"
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[intro!,dest!]: "(Abst a t\<approx>?Func F t1#xs,ys)\<in>fail"
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[intro!,dest!]: "(Func F t1\<approx>?Abst a t#xs,ys)\<in>fail"
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[intro!,dest!]: "(Unit\<approx>?Atom b#xs,ys)\<in>fail"
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[intro!,dest!]: "(Atom b\<approx>?Unit#xs,ys)\<in>fail"
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[intro!,dest!]: "(Unit\<approx>?Paar t1 t2#xs,ys)\<in>fail"
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[intro!,dest!]: "(Paar t1 t2\<approx>?Unit#xs,ys)\<in>fail"
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[intro!,dest!]: "(Unit\<approx>?Func F t1#xs,ys)\<in>fail"
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[intro!,dest!]: "(Func F t1\<approx>?Unit#xs,ys)\<in>fail"
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[intro!,dest!]: "(Atom a\<approx>?Paar t1 t2#xs,ys)\<in>fail"
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[intro!,dest!]: "(Paar t1 t2\<approx>?Atom a#xs,ys)\<in>fail"
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[intro!,dest!]: "(Atom a\<approx>?Func F t1#xs,ys)\<in>fail"
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[intro!,dest!]: "(Func F t1\<approx>?Atom a#xs,ys)\<in>fail"
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[intro!,dest!]: "(Func F t\<approx>?Paar t1 t2#xs,ys)\<in>fail"
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[intro!,dest!]: "(Paar t1 t2\<approx>?Func F t#xs,ys)\<in>fail"
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[intro!]: "\<lbrakk>F1\<noteq>F2\<rbrakk>\<Longrightarrow>(Func F1 t1\<approx>?Func F2 t2#xs,ys)\<in>fail"
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(* the results that are interesting are the stuck ones *)
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consts
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results :: "problem_type \<Rightarrow> problem_type set"
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defs
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results_def:
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"results P1 \<equiv> if P1\<in>stuck then {P1} else {P2. \<exists>nabla s. P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<and> P2\<in>stuck}"
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(* a "failed" problem has no unifier *)
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lemma fail_then_empty:
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"(P1\<in>fail) \<Longrightarrow> (U P1={})"
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apply(erule fail.cases)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t1\<in>psub_trms (Abst a (subst b t)).\<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t1)")
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apply(simp)
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apply(drule equ_sym)
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apply(clarify)
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apply(drule_tac "t1.0"="Abst a (subst b t)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)")
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apply(simp)
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apply(drule equ_sym)
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apply(clarify)
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apply(drule_tac "t1.0"="Func F (subst b t)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(erule disjE)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
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\<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
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apply(simp)
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apply(drule equ_sym)
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apply(clarify)
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apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2a" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
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\<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
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apply(simp)
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apply(drule equ_sym)
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apply(clarify)
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apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2a" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t3\<in>psub_trms (Abst a (subst b t)).
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\<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t3)")
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apply(simp)
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apply(clarify)
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apply(drule_tac "t1.0"="Abst a (subst b t)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)")
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apply(simp)
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apply(clarify)
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apply(drule_tac "t1.0"="Func F (subst b t)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def)
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apply(rule allI)+
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apply(rule impI)
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apply(erule disjE)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
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\<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
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apply(simp)
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apply(clarify)
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apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2a" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp])
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apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)).
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\<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)")
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apply(simp)
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apply(clarify)
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apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and
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"t2.0"="subst b (Susp pi X)" and
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"t3.0"="swap pia t2a" in equ_trans)
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apply(simp)
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apply(best)
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apply(rule psub_trm_not_equ)
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apply(simp add: all_solutions_def, fast dest!: fresh.cases)
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apply(simp add: all_solutions_def, fast dest!: equ.cases)+
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done
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(* the only stuck problems are the "failed" problems and the empty problem *)
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lemma stuck_equiv: "stuck = {([],[])}\<union>fail"
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apply(subgoal_tac "([],[])\<in>stuck")
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apply(subgoal_tac "\<forall>P\<in>fail. P\<in>stuck")
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apply(subgoal_tac "\<forall>P\<in>stuck. P=([],[]) \<or> P\<in>fail")
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apply(force)
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apply(rule ballI)
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apply(thin_tac "([], []) \<in> stuck")
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apply(thin_tac "\<forall>P\<in>fail. P \<in> stuck")
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apply(simp add: stuck_def)
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apply(clarify)
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apply(case_tac a)
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apply(simp)
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apply(case_tac b)
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apply(simp)
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apply(simp)
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apply(case_tac aa)
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apply(simp)
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apply(case_tac ba)
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apply(simp_all)
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apply(case_tac "ab=lista")
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(case_tac aa)
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apply(simp)
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apply(case_tac ab)
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apply(simp_all)
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apply(case_tac ba)
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apply(simp_all)
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apply(case_tac "lista=listb")
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apply(force)
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apply(force)
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apply(case_tac "occurs list2 (Abst lista trm)")
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apply(force)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(force)
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apply(case_tac ba)
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apply(simp_all)
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apply(case_tac "occurs list2 (Abst lista trm)")
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apply(force)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac "list2=list2a")
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apply(force)
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apply(case_tac "occurs list2 (Susp list1a list2a)")
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apply(simp)
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apply(drule_tac
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x="fst (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec)
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apply(drule_tac
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x="snd (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Susp list1a list2a))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac "occurs list2 Unit")
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apply(simp)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac "occurs list2 (Atom lista)")
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apply(simp)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac "occurs list2 (Paar trm1 trm2)")
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apply(force)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac "occurs list2 (Func lista trm)")
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apply(force)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec)
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apply(simp only: surjective_pairing[THEN sym])
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apply(force)
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apply(case_tac ba)
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apply(simp_all)
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apply(force)
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apply(case_tac "occurs list2 Unit")
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apply(simp)
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apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
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apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec)
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apply(drule_tac x="{}" in spec)
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284 |
apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec)
|
|
285 |
apply(simp only: surjective_pairing[THEN sym])
|
|
286 |
apply(force)
|
|
287 |
apply(force)
|
|
288 |
apply(force)
|
|
289 |
apply(force)
|
|
290 |
apply(force)
|
|
291 |
apply(case_tac ba)
|
|
292 |
apply(simp_all)
|
|
293 |
apply(force)
|
|
294 |
apply(case_tac "occurs list2 (Atom lista)")
|
|
295 |
apply(simp)
|
|
296 |
apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
|
|
297 |
apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec)
|
|
298 |
apply(drule_tac x="{}" in spec)
|
|
299 |
apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec)
|
|
300 |
apply(simp only: surjective_pairing[THEN sym])
|
|
301 |
apply(force)
|
|
302 |
apply(force)
|
|
303 |
apply(force)
|
|
304 |
apply(force)
|
|
305 |
apply(force)
|
|
306 |
apply(case_tac ba)
|
|
307 |
apply(simp_all)
|
|
308 |
apply(force)
|
|
309 |
apply(case_tac "occurs list2 (Paar trm1 trm2)")
|
|
310 |
apply(force)
|
|
311 |
apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
|
|
312 |
apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec)
|
|
313 |
apply(drule_tac x="{}" in spec)
|
|
314 |
apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec)
|
|
315 |
apply(simp only: surjective_pairing[THEN sym])
|
|
316 |
apply(force)
|
|
317 |
apply(force)
|
|
318 |
apply(force)
|
|
319 |
apply(force)
|
|
320 |
apply(force)
|
|
321 |
apply(case_tac ba)
|
|
322 |
apply(simp_all)
|
|
323 |
apply(force)
|
|
324 |
apply(case_tac "occurs list2 (Func lista trm)")
|
|
325 |
apply(force)
|
|
326 |
apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
|
|
327 |
apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec)
|
|
328 |
apply(drule_tac x="{}" in spec)
|
|
329 |
apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec)
|
|
330 |
apply(simp only: surjective_pairing[THEN sym])
|
|
331 |
apply(force)
|
|
332 |
apply(force)
|
|
333 |
apply(force)
|
|
334 |
apply(force)
|
|
335 |
apply(force)
|
|
336 |
apply(rule ballI)
|
|
337 |
apply(thin_tac "([], []) \<in> stuck")
|
|
338 |
apply(simp add: stuck_def)
|
|
339 |
apply(clarify)
|
|
340 |
apply(ind_cases "((a, b), (nabla, s), aa, ba) \<in> red_plus")
|
|
341 |
apply(ind_cases "((a, b), s, aa, ba) \<in> s_red")
|
|
342 |
apply(simp_all)
|
|
343 |
apply(ind_cases "((Unit, Unit) # aa, b) \<in> fail")
|
|
344 |
apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail")
|
|
345 |
apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail")
|
|
346 |
apply(simp)
|
|
347 |
apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail")
|
|
348 |
apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail")
|
|
349 |
apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail")
|
|
350 |
apply(simp)
|
|
351 |
apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail")
|
|
352 |
apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail")
|
|
353 |
apply(simp add: apply_subst_def)+
|
|
354 |
apply(case_tac "occurs X t1")
|
|
355 |
apply(simp)
|
|
356 |
apply(simp)
|
|
357 |
apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail")
|
|
358 |
apply(simp add: apply_subst_def)
|
|
359 |
apply(case_tac "occurs X t1")
|
|
360 |
apply(simp)
|
|
361 |
apply(simp)
|
|
362 |
apply(case_tac "occurs X t1")
|
|
363 |
apply(simp)
|
|
364 |
apply(simp)
|
|
365 |
apply(ind_cases "((a, b), nabla, aa, ba) \<in> c_red")
|
|
366 |
apply(simp_all)
|
|
367 |
apply(ind_cases "([], (ab, Unit) # ba) \<in> fail")
|
|
368 |
apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail")
|
|
369 |
apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail")
|
|
370 |
apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail")
|
|
371 |
apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail")
|
|
372 |
apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail")
|
|
373 |
apply(simp)
|
|
374 |
apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail")
|
|
375 |
apply(ind_cases "(a, b) \<turnstile> s1 \<leadsto> P2")
|
|
376 |
apply(simp_all)
|
|
377 |
apply(ind_cases "((Unit, Unit) # xs, b) \<in> fail")
|
|
378 |
apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail")
|
|
379 |
apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail")
|
|
380 |
apply(simp)
|
|
381 |
apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail")
|
|
382 |
apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail")
|
|
383 |
apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail")
|
|
384 |
apply(simp)
|
|
385 |
apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail")
|
|
386 |
apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail")
|
|
387 |
apply(simp add: apply_subst_def)+
|
|
388 |
apply(case_tac "occurs X t1")
|
|
389 |
apply(simp)
|
|
390 |
apply(simp)
|
|
391 |
apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail")
|
|
392 |
apply(simp add: apply_subst_def)
|
|
393 |
apply(case_tac "occurs X t1")
|
|
394 |
apply(simp)
|
|
395 |
apply(simp)
|
|
396 |
apply(case_tac "occurs X t1")
|
|
397 |
apply(simp)
|
|
398 |
apply(simp)
|
|
399 |
apply(ind_cases "(a, b) \<turnstile> nabla1 \<rightarrow> P2")
|
|
400 |
apply(simp_all)
|
|
401 |
apply(ind_cases "([], (ab, Unit) # ba) \<in> fail")
|
|
402 |
apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail")
|
|
403 |
apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail")
|
|
404 |
apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail")
|
|
405 |
apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail")
|
|
406 |
apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail")
|
|
407 |
apply(simp)
|
|
408 |
apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail")
|
|
409 |
apply(simp add: stuck_def)
|
|
410 |
apply(rule allI)+
|
|
411 |
apply(clarify)
|
|
412 |
apply(ind_cases "(([], []), (nabla, s), a, b) \<in> red_plus")
|
|
413 |
apply(ind_cases "(([], []), s, a, b) \<in> s_red")
|
|
414 |
apply(ind_cases "(([], []), nabla, a, b) \<in> c_red")
|
|
415 |
apply(ind_cases "([], []) \<turnstile> s1 \<leadsto> P2")
|
|
416 |
apply(ind_cases "([], []) \<turnstile> nabla1 \<rightarrow> P2")
|
|
417 |
done
|
|
418 |
|
|
419 |
lemma u_empty_sred:
|
|
420 |
"P1\<turnstile>s\<leadsto>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
|
|
421 |
apply(rule impI)
|
|
422 |
apply(ind_cases "P1 \<turnstile> s \<leadsto> P2")
|
|
423 |
apply(rule impI, simp add: all_solutions_def)
|
|
424 |
apply(rule impI, simp add: all_solutions_def)
|
|
425 |
apply(fast dest!: equ_paar_elim)
|
|
426 |
apply(rule impI, simp add: all_solutions_def)
|
|
427 |
apply(fast dest!: equ_func_elim)
|
|
428 |
apply(rule impI, simp add: all_solutions_def)
|
|
429 |
apply(fast dest!: equ_abst_aa_elim)
|
|
430 |
apply(rule impI, simp add: all_solutions_def)
|
|
431 |
apply(force dest!: equ_abst_ab_elim simp add: subst_swap_comm[THEN sym])
|
|
432 |
apply(rule impI, simp add: all_solutions_def)
|
|
433 |
apply(rule impI, simp add: all_solutions_def)
|
|
434 |
apply(simp add: ds_list_equ_ds)
|
|
435 |
apply(rule allI)+
|
|
436 |
apply(rule impI)
|
|
437 |
apply(drule_tac x="a" in spec)
|
|
438 |
apply(drule_tac x="b" in spec)
|
|
439 |
apply(erule disjE)
|
|
440 |
apply(force)
|
|
441 |
apply(simp add: subst_susp)
|
|
442 |
apply(drule equ_pi1_pi2_dec[THEN mp])
|
|
443 |
apply(force simp add: subst_susp)
|
|
444 |
apply(auto)
|
|
445 |
apply(simp add: all_solutions_def)
|
|
446 |
apply(simp_all add: apply_subst_def)
|
|
447 |
apply(auto)
|
|
448 |
apply(drule_tac x="a" in spec)
|
|
449 |
apply(drule_tac x="b" in spec)
|
|
450 |
apply(drule unif_1)
|
|
451 |
apply(auto)
|
|
452 |
apply(drule_tac x="(aa,ba)" in bspec)
|
|
453 |
apply(assumption)
|
|
454 |
apply(simp)
|
|
455 |
apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a)
|
|
456 |
apply(simp add: subst_comp_expand)
|
|
457 |
apply(drule_tac x="(aa,ba)" in bspec)
|
|
458 |
apply(assumption)
|
|
459 |
apply(simp)
|
|
460 |
apply(drule_tac a="aa" and "t"="ba" in unif_2b)
|
|
461 |
apply(simp add: subst_comp_expand)
|
|
462 |
apply(simp add: all_solutions_def)
|
|
463 |
apply(auto)
|
|
464 |
apply(drule_tac x="a" in spec)
|
|
465 |
apply(drule_tac x="b" in spec)
|
|
466 |
apply(drule equ_sym)
|
|
467 |
apply(drule unif_1)
|
|
468 |
apply(auto)
|
|
469 |
apply(drule_tac x="(aa,ba)" in bspec)
|
|
470 |
apply(assumption)
|
|
471 |
apply(simp)
|
|
472 |
apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a)
|
|
473 |
apply(simp add: subst_comp_expand)
|
|
474 |
apply(drule_tac x="(aa,ba)" in bspec)
|
|
475 |
apply(assumption)
|
|
476 |
apply(simp)
|
|
477 |
apply(drule_tac a="aa" and "t"="ba" in unif_2b)
|
|
478 |
apply(simp add: subst_comp_expand)
|
|
479 |
done
|
|
480 |
|
|
481 |
lemma u_empty_cred:
|
|
482 |
"P1\<turnstile>nabla\<rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
|
|
483 |
apply(rule impI)
|
|
484 |
apply(ind_cases "P1 \<turnstile>nabla\<rightarrow>P2")
|
|
485 |
apply(rule impI, simp add: all_solutions_def)
|
|
486 |
apply(rule impI, simp add: all_solutions_def)
|
|
487 |
apply(fast dest!: fresh_paar_elim)
|
|
488 |
apply(rule impI, simp add: all_solutions_def)
|
|
489 |
apply(fast dest!: fresh_func_elim)
|
|
490 |
apply(rule impI, simp add: all_solutions_def)
|
|
491 |
apply(rule impI, simp add: all_solutions_def)
|
|
492 |
apply(force dest!: fresh_abst_ab_elim)
|
|
493 |
apply(rule impI, simp add: all_solutions_def)
|
|
494 |
apply(rule impI, simp add: all_solutions_def)
|
|
495 |
done
|
|
496 |
|
|
497 |
lemma u_empty_red_plus:
|
|
498 |
"P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}"
|
|
499 |
apply(rule impI)
|
|
500 |
apply(erule red_plus.induct)
|
|
501 |
apply(drule u_empty_sred[THEN mp], assumption)
|
|
502 |
apply(drule u_empty_cred[THEN mp], assumption)
|
|
503 |
apply(drule u_empty_sred[THEN mp], force)
|
|
504 |
apply(drule u_empty_cred[THEN mp], force)
|
|
505 |
done
|
|
506 |
|
|
507 |
(* all problems that cannot be solved produce "failed" problems only *)
|
|
508 |
|
|
509 |
lemma empty_then_fail: "U P1={} \<longrightarrow> (\<forall>P\<in>results P1. P\<in>fail)"
|
|
510 |
apply(simp add: results_def)
|
|
511 |
apply(rule conjI)
|
|
512 |
apply(rule impI)
|
|
513 |
apply(rule impI)
|
|
514 |
apply(simp add: stuck_equiv)
|
|
515 |
apply(erule disjE)
|
|
516 |
apply(subgoal_tac "({},[])\<in>U ([],[])")
|
|
517 |
apply(simp)
|
|
518 |
apply(simp add: all_solutions_def)
|
|
519 |
apply(assumption)
|
|
520 |
apply(rule impI)+
|
|
521 |
apply(rule allI)+
|
|
522 |
apply(rule impI)
|
|
523 |
apply(erule conjE)
|
|
524 |
apply(simp add: stuck_equiv)
|
|
525 |
apply(auto)
|
|
526 |
apply(subgoal_tac "({},[])\<in>U ([],[])")
|
|
527 |
apply(drule_tac "nabla3.0"="nabla" and "nabla1.0"="{}" and "s1.0"="[]" in P1_from_P2_red_plus)
|
|
528 |
apply(simp add: ext_subst_def)
|
|
529 |
apply(auto)
|
|
530 |
apply(simp add: all_solutions_def)
|
|
531 |
done
|
|
532 |
|
|
533 |
(* if a problem can be solved then no "failed" problem is produced *)
|
|
534 |
|
|
535 |
lemma not_empty_then_not_fail: "U P1\<noteq>{} \<longrightarrow> \<not>(\<exists>P\<in>results P1. P\<in>fail)"
|
|
536 |
apply(rule impI)
|
|
537 |
apply(simp)
|
|
538 |
apply(rule ballI)
|
|
539 |
apply(clarify)
|
|
540 |
apply(simp add: results_def)
|
|
541 |
apply(case_tac "P1\<in>stuck")
|
|
542 |
apply(simp_all)
|
|
543 |
apply(drule fail_then_empty)
|
|
544 |
apply(simp)
|
|
545 |
apply(drule fail_then_empty)
|
|
546 |
apply(erule conjE)
|
|
547 |
apply(clarify)
|
|
548 |
apply(drule u_empty_red_plus[THEN mp])
|
|
549 |
apply(simp)
|
|
550 |
done
|
|
551 |
|
|
552 |
end
|
|
553 |
|
|
554 |
|
|
555 |
|
|
556 |
|
|
557 |
|
|
558 |
|
|
559 |
|
|
560 |
|
|
561 |
|
|
562 |
|
|
563 |
|
|
564 |
|
|
565 |
|
|
566 |
|
|
567 |
|
|
568 |
|
|
569 |
|
|
570 |
|
|
571 |
|
|
572 |
|
|
573 |
|
|
574 |
|
|
575 |
|
|
576 |
|
|
577 |
|
|
578 |
|
|
579 |
|
|
580 |
|
|
581 |
|
|
582 |
|
|
583 |
|
|
584 |
|
|
585 |
|
|
586 |
|
|
587 |
|
|
588 |
|
|
589 |
|
|
590 |
|
|
591 |
|
|
592 |
|
|
593 |
|
|
594 |
|
|
595 |
|
|
596 |
|
|
597 |
|
|
598 |
|
|
599 |
|
|
600 |
|
|
601 |
|
|
602 |
|
|
603 |
|
|
604 |
|
|
605 |
|
|
606 |
|
|
607 |
|
|
608 |
|
|
609 |
|
|
610 |
|
|
611 |
|
|
612 |
|
|
613 |
|
|
614 |
|
|
615 |
|
|
616 |
|
|
617 |
|
|
618 |
|
|
619 |
|
|
620 |
|
|
621 |
|
|
622 |
|
|
623 |
|
|
624 |
|
|
625 |
|