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theory PreEqu = Main + Swap + Terms + Disagreement + Fresh:
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consts
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equ :: "(fresh_envs \<times> trm \<times> trm) set"
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syntax
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"_equ_judge" :: "fresh_envs \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool" (" _ \<turnstile> _ \<approx> _" [80,80,80] 80)
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translations
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"nabla \<turnstile> t1 \<approx> t2" \<rightleftharpoons> "(nabla,t1,t2) \<in> equ"
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inductive equ
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intros
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equ_abst_ab[intro!]: "\<lbrakk>a\<noteq>b;(nabla \<turnstile> a \<sharp> t2);(nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<rbrakk>
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\<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst b t2)"
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equ_abst_aa[intro!]: "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst a t2)"
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equ_unit[intro!]: "(nabla \<turnstile> Unit \<approx> Unit)"
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equ_atom[intro!]: "a=b\<Longrightarrow>nabla \<turnstile> Atom a \<approx> Atom b"
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equ_susp[intro!]: "(\<forall> c \<in> ds pi1 pi2. (c,X) \<in> nabla) \<Longrightarrow> (nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X)"
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equ_paar[intro!]: "\<lbrakk>(nabla \<turnstile> t1\<approx>t2);(nabla \<turnstile> s1\<approx>s2)\<rbrakk> \<Longrightarrow> (nabla \<turnstile> Paar t1 s1 \<approx> Paar t2 s2)"
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equ_func[intro!]: "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Func F t1 \<approx> Func F t2)"
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lemma equ_atom_elim[elim!]: "nabla\<turnstile>Atom a \<approx> Atom b \<Longrightarrow> a=b"
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apply(ind_cases "nabla \<turnstile> Atom a \<approx> Atom b", auto)
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done
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lemma equ_susp_elim[elim!]: "(nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X)
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\<Longrightarrow> (\<forall> c \<in> ds pi1 pi2. (c,X)\<in> nabla)"
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apply(ind_cases "nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X", auto)
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done
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lemma equ_paar_elim[elim!]: "(nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2) \<Longrightarrow>
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(nabla \<turnstile> s1 \<approx> s2)\<and>(nabla \<turnstile> t1 \<approx> t2)"
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apply(ind_cases "nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2", auto)
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done
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lemma equ_func_elim[elim!]: "(nabla \<turnstile> Func F t1 \<approx> Func F t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"
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apply(ind_cases "nabla \<turnstile> Func F t1 \<approx> Func F t2", auto)
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done
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lemma equ_abst_aa_elim[elim!]: "(nabla \<turnstile> Abst a t1 \<approx> Abst a t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"
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apply(ind_cases "nabla \<turnstile> Abst a t1 \<approx> Abst a t2", auto)
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done
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lemma equ_abst_ab_elim[elim!]: "\<lbrakk>(nabla \<turnstile> Abst a t1 \<approx> Abst b t2);a\<noteq>b\<rbrakk> \<Longrightarrow>
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(nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<and>(nabla\<turnstile>a\<sharp>t2)"
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apply(ind_cases "(nabla \<turnstile> Abst a t1 \<approx> Abst b t2)", auto)
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done
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lemma equ_depth: "nabla \<turnstile> t1 \<approx> t2 \<Longrightarrow> depth t1 = depth t2"
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apply(erule equ.induct)
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apply(simp_all)
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done
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lemma rev_pi_pi_equ: "(nabla\<turnstile>swap (rev pi) (swap pi t)\<approx>t)"
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apply(induct_tac t)
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apply(auto)
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-- Susp
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apply(drule_tac ds_cancel_pi_left[of _ "rev pi @ pi" _ "[]", THEN mp, simplified])
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apply(simp only: ds_rev_pi_pi)
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apply(simp only: ds_def)
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apply(force)
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done
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lemma equ_pi_right: "\<forall>pi. (\<forall>a\<in>ds [] pi. nabla\<turnstile>a\<sharp>t) \<longrightarrow> (nabla\<turnstile>t\<approx>swap pi t)"
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apply(induct_tac t)
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apply(simp_all)
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-- Abst
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apply(rule allI)
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apply(case_tac "(swapas pi list)=list")
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apply(simp)
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apply(rule impI)
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apply(rule equ_abst_aa)
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apply(drule_tac x="pi" in spec)
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apply(subgoal_tac "(\<forall>a\<in>ds [] pi. nabla \<turnstile> a \<sharp> trm)")--A
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apply(force)
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--A
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apply(rule ballI)
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apply(drule_tac x=a in bspec)
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apply(assumption)
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apply(case_tac "list\<noteq>a")
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apply(force dest!: fresh_abst_ab_elim)
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apply(simp add: ds_def)
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apply(rule impI)
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apply(rule equ_abst_ab)
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apply(force)
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apply(drule_tac x="swapas (rev pi) list" in bspec)
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apply(simp add: ds_def)
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apply(rule conjI)
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apply(subgoal_tac "swapas (rev pi) list \<in> atms (rev pi)") --B
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apply(simp)
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--B
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apply(drule swapas_pi_ineq_a[THEN mp])
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apply(rule swapas_pi_in_atms)
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apply(simp)
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apply(clarify)
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apply(drule swapas_rev_pi_b)
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apply(simp)
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apply(force dest!: fresh_abst_ab_elim swapas_rev_pi_b intro!: fresh_swap_right[THEN mp])
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apply(drule_tac x="(list, swapas pi list)#pi" in spec)
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apply(subgoal_tac "(\<forall>a\<in>ds [] ((list, swapas pi list) # pi). nabla \<turnstile> a \<sharp> trm)")--C
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apply(force simp add: swap_append[THEN sym])
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--C
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apply(rule ballI)
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apply(drule_tac x="a" in bspec)
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apply(rule_tac b="list" in ds_7)
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apply(force)
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apply(assumption)
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apply(case_tac "list=a")
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apply(simp)
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apply(simp only: ds_def mem_Collect_eq)
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apply(erule conjE)
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apply(subgoal_tac "a\<noteq>swapas pi a")
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apply(simp)
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apply(force)
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apply(force dest!: fresh_abst_ab_elim)
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-- Susp
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apply(rule allI)
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apply(rule impI)
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apply(rule equ_susp)
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apply(rule ballI)
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apply(subgoal_tac "swapas list1 c\<in>ds [] pi")--A
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apply(force dest!: fresh_susp_elim)
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--A
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apply(rule ds_cancel_pi_left[THEN mp])
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apply(simp)
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-- Unit
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apply(force)
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-- Atom
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apply(rule allI)
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apply(rule impI)
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apply(case_tac "(swapas pi list) = list")
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apply(force)
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apply(drule ds_elem)
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apply(force dest!: fresh_atom_elim)
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-- Paar
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apply(force dest!: fresh_paar_elim)
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-- Func
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apply(force)
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done
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lemma pi_comm: "nabla\<turnstile>(swap (pi@[(a,b)]) t)\<approx>(swap ([(swapas pi a, swapas pi b)]@pi) t)"
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apply(induct_tac t)
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apply(simp_all)
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-- Abst
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apply(force simp add: swapas_comm)
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-- Susp
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apply(rule equ_susp)
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apply(rule ballI)
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apply(simp only: ds_def)
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apply(simp only: mem_Collect_eq)
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apply(erule conjE)
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apply(subgoal_tac "swapas (pi@[(a,b)]) (swapas list1 c) =
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swapas ([(swapas pi a,swapas pi b)]@pi) (swapas list1 c)")--A
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apply(simp add: swapas_append[THEN sym])
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--A
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apply(simp only: swapas_comm)
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-- Units
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apply(rule equ_unit)
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-- Atom
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apply(force dest!: swapas_rev_pi_b swapas_rev_pi_a simp add: swapas_append)
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--Paar
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apply(force)
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--Func
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apply(force)
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done
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lemma l3_jud: "(nabla \<turnstile> t1\<approx>t2) \<Longrightarrow> (nabla \<turnstile> a\<sharp>t1) \<longrightarrow> (nabla \<turnstile> a\<sharp>t2)"
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apply(erule equ.induct)
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apply(simp_all)
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--Abst.ab
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apply(rule impI)
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apply(case_tac "aa=a")
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apply(force)
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apply(case_tac "b=a")
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apply(force)
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apply(force dest!: fresh_abst_ab_elim fresh_swap_left[THEN mp])
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-- Abst.aa
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apply(case_tac "a=aa")
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apply(force)
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apply(force dest!: fresh_abst_ab_elim)
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-- Susp
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apply(rule impI)
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apply(drule fresh_susp_elim, rule fresh_susp)
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apply(case_tac "swapas (rev pi1) a = swapas (rev pi2) a")
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apply(simp)
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apply(drule_tac x="swapas (rev pi2) a" in bspec)
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apply(rule ds_cancel_pi_left[THEN mp])
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apply(subgoal_tac "swapas (pi1@(rev pi2)) a \<noteq> a")--A
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apply(drule ds_elem)
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apply(force simp add: ds_def swapas_append)
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--A
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apply(clarify)
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apply(simp only: swapas_append)
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apply(drule swapas_rev_pi_a)
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apply(force)
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apply(assumption)
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-- Paar
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apply(force dest!: fresh_paar_elim)
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-- Func
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apply(force dest!: fresh_func_elim)
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done
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lemma big: "\<forall>t1 t2 t3. (n=depth t1) \<longrightarrow>
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(((nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>t2\<approx>t1))\<and>
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(\<forall>pi. (nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>swap pi t1\<approx>swap pi t2))\<and>
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((nabla\<turnstile>t1\<approx>t2)\<and>(nabla\<turnstile>t2\<approx>t3)\<longrightarrow>(nabla\<turnstile>t1\<approx>t3)))"
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apply(induct_tac n rule: nat_less_induct)
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apply(rule allI)+apply(rule impI)
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apply(rule conjI)
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-- SYMMETRY
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apply(rule impI)
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apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
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apply(simp_all)
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-- Abst.ab
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apply(rule equ_abst_ab)
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apply(force) --abst.ab.first.premise
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apply(rule_tac "t1.1"="swap [(a,b)] t2a" in l3_jud[THEN mp])
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apply(drule_tac x="depth t1a" in spec)
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apply(simp)
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apply(rule fresh_swap_right[THEN mp])
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apply(simp) --abst.ab.second.premise
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apply(subgoal_tac "nabla \<turnstile> swap [(b, a)] t1a \<approx> t2a")
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apply(drule_tac x="depth t1a" in spec)
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apply(simp)
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apply(subgoal_tac "nabla \<turnstile> swap [(b,a)] t1a \<approx> swap ([(b,a)]@[(a,b)]) t2a") --A
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apply(subgoal_tac "nabla \<turnstile> swap ([(b,a)]@[(a,b)]) t2a \<approx> t2a") --B
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apply(drule_tac x="depth t1a" in spec)
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apply(simp (no_asm_use))
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apply(drule_tac x="swap [(b,a)] t1a" in spec)
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apply(simp (no_asm_use))
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apply(drule_tac x="swap [(b,a),(a,b)] t2a" in spec)
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apply(force)
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--B
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apply(subgoal_tac "nabla\<turnstile>t2a \<approx> swap ([(b, a)] @ [(a, b)]) t2a")--C
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apply(drule_tac x="depth t1a" in spec)
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apply(simp)
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apply(drule_tac x="t2a" in spec)
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apply(drule mp)
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apply(drule equ_depth)
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apply(force)
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apply(best)
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--C
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apply(rule equ_pi_right[THEN spec,THEN mp])
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apply(subgoal_tac "ds [] ([(b, a)] @ [(a, b)])={}")
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apply(simp)
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apply(simp add: ds_baab)
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--A
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apply(force simp only: swap_append)
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-- Abst.aa
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apply(force)
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-- Unit
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apply(rule equ_unit)
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-- Atom
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apply(force)
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-- Susp
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apply(force simp only: ds_sym)
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-- Paar
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apply(rule equ_paar)
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apply(drule_tac x="depth t1a" in spec)
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apply(simp add: Suc_max_left)
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apply(drule_tac x="depth s1" in spec)
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apply(simp add: Suc_max_right)
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-- Func
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apply(best)
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-- ADD.PI
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apply(rule conjI)
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apply(rule impI)
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apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
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apply(simp_all)
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-- Abst.ab
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apply(rule allI)
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apply(rule equ_abst_ab)
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-- abst.ab.first.premise
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apply(clarify)
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apply(drule swapas_rev_pi_a)
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apply(simp)
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-- abst.ab.second.premise
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apply(rule fresh_swap_right[THEN mp])
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apply(simp)
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-- abst.ab.third.premise
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apply(subgoal_tac "nabla \<turnstile> swap pi t1a \<approx> swap (pi@[(a,b)]) t2a") --A
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apply(subgoal_tac "nabla \<turnstile> swap (pi@[(a,b)]) t2a \<approx> swap ([(swapas pi a,swapas pi b)]@pi) t2a") --B
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apply(drule_tac x="depth t1a" in spec)
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apply(simp (no_asm_use))
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apply(drule_tac x="swap pi t1a" in spec)
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apply(simp (no_asm_use))
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apply(drule_tac x="swap (pi@[(a,b)]) t2a" in spec)
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apply(drule conjunct2)+
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apply(drule_tac x="swap ((swapas pi a, swapas pi b) # pi) t2a" in spec)
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apply(simp add: swap_append[THEN sym])
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--B
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apply(rule pi_comm)
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apply(force simp only: swap_append)
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-- A
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apply(force simp only: swap_append)
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-- Unit
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apply(rule equ_unit)
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-- Atom
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apply(force)
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-- Susp
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apply(force simp only: ds_cancel_pi_front)
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-- Paar
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apply(rule allI)
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apply(rule equ_paar)
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apply(drule_tac x="depth t1a" in spec)
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apply(simp only: Suc_max_left)
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apply(drule_tac x="depth s1" in spec)
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apply(simp only: Suc_max_right)
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-- Func
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apply(best)
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-- TRANSITIVITY
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apply(rule impI)
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apply(erule conjE)
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apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")
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apply(simp_all)
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-- Abst.ab
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apply(ind_cases "nabla \<turnstile> Abst b t2a \<approx> t3")
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apply(simp)
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apply(case_tac "ba=a")
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apply(simp)
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apply(rule equ_abst_aa)
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apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a \<approx> t2b") --A
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apply(best)
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--A
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apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx> swap ([(a,b)]@[(b,a)]) t2b") --B
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apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,a)]) t2b \<approx> t2b") --C
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apply(drule_tac x="depth t1a" in spec)
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apply(simp)
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328 |
apply(drule_tac x="swap [(a,b)] t2a" in spec)
|
|
329 |
apply(drule equ_depth)
|
|
330 |
apply(simp (no_asm_use))
|
|
331 |
apply(best)
|
|
332 |
--C
|
|
333 |
apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(a,b)]@[(b,a)]) t2b")--D
|
|
334 |
apply(drule_tac x="depth t1a" in spec)
|
|
335 |
apply(simp)
|
|
336 |
apply(drule_tac x="t2b" in spec)
|
|
337 |
apply(drule mp)
|
|
338 |
apply(force dest!: equ_depth)
|
|
339 |
apply(best)
|
|
340 |
--D
|
|
341 |
apply(rule equ_pi_right[THEN spec, THEN mp])
|
|
342 |
apply(simp add: ds_baab)
|
|
343 |
--B
|
|
344 |
apply(drule_tac x="depth t1a" in spec)
|
|
345 |
apply(simp)
|
|
346 |
apply(drule_tac x="t2a" in spec)
|
|
347 |
apply(drule equ_depth)
|
|
348 |
apply(simp)
|
|
349 |
apply(drule_tac x="swap [(b, a)] t2b" in spec)
|
|
350 |
apply(drule conjunct2)
|
|
351 |
apply(drule conjunct1)
|
|
352 |
apply(simp)
|
|
353 |
apply(drule_tac x="[(a,b)]" in spec)
|
|
354 |
apply(simp add: swap_append[THEN sym])
|
|
355 |
-- Abst.ab
|
|
356 |
apply(rule equ_abst_ab)
|
|
357 |
-- abst.ab.first.premise
|
|
358 |
apply(force)
|
|
359 |
-- abst.ab.second.premise
|
|
360 |
apply(rule_tac "t1.1"="swap [(b,ba)] t2a" in l3_jud[THEN mp])
|
|
361 |
apply(subgoal_tac "nabla \<turnstile> swap [(b,ba)] t2a \<approx> swap ([(b,ba)]@[(b, ba)]) t2b") --A
|
|
362 |
apply(subgoal_tac "nabla\<turnstile>swap ([(b,ba)]@[(b,ba)]) t2b \<approx> t2b") --B
|
|
363 |
apply(drule_tac x="depth t1a" in spec)
|
|
364 |
apply(simp)
|
|
365 |
apply(drule_tac x="swap [(b, ba)] t2a" in spec)
|
|
366 |
apply(drule mp)
|
|
367 |
apply(force dest!: equ_depth)
|
|
368 |
apply(best)
|
|
369 |
--B
|
|
370 |
apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(b,ba)] @ [(b,ba)]) t2b")--C
|
|
371 |
apply(drule_tac x="depth t1a" in spec)
|
|
372 |
apply(simp)
|
|
373 |
apply(drule_tac x="t2b" in spec)
|
|
374 |
apply(drule mp)
|
|
375 |
apply(force dest!: equ_depth)
|
|
376 |
apply(best)
|
|
377 |
-- C
|
|
378 |
apply(rule equ_pi_right[THEN spec, THEN mp])
|
|
379 |
apply(simp add: ds_abab)
|
|
380 |
--A
|
|
381 |
apply(drule_tac x="depth t1a" in spec)
|
|
382 |
apply(simp)
|
|
383 |
apply(drule_tac x="t2a" in spec)
|
|
384 |
apply(drule mp)
|
|
385 |
apply(force dest!: equ_depth)
|
|
386 |
apply(drule_tac x="swap [(b,ba)] t2b" in spec)
|
|
387 |
apply(drule conjunct2)
|
|
388 |
apply(drule conjunct1)
|
|
389 |
apply(simp)
|
|
390 |
apply(drule_tac x="[(b,ba)]" in spec)
|
|
391 |
apply(simp add: swap_append[THEN sym])
|
|
392 |
-- abst.ab.third.premise
|
|
393 |
apply(force intro!: fresh_swap_right[THEN mp])
|
|
394 |
-- very.complex
|
|
395 |
apply(subgoal_tac "nabla\<turnstile>t1a \<approx> swap ([(a,b)]@[(b,ba)]) t2b") --A
|
|
396 |
apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,ba)]) t2b \<approx> swap [(a,ba)] t2b") --B
|
|
397 |
apply(drule_tac x="depth t1a" in spec)
|
|
398 |
apply(simp (no_asm_use))
|
|
399 |
apply(best)
|
|
400 |
--B
|
|
401 |
apply(subgoal_tac "nabla\<turnstile>swap [(a, ba)] t2b \<approx> swap [(a,b),(b,ba)] t2b")--C
|
|
402 |
apply(drule_tac x="depth t1a" in spec)
|
|
403 |
apply(simp)
|
|
404 |
apply(drule_tac x="swap [(a, ba)] t2b" in spec)
|
|
405 |
apply(drule mp)
|
|
406 |
apply(force dest!: equ_depth)
|
|
407 |
apply(force)
|
|
408 |
apply(subgoal_tac "nabla\<turnstile>swap [(a,ba)] t2b\<approx> swap [(a,ba)] (swap [(a,ba),(a,b),(b,ba)] t2b)") --D
|
|
409 |
apply(subgoal_tac "nabla\<turnstile>swap (rev [(a,ba)]) (swap [(a,ba)] (swap [(a,b),(b,ba)] t2b))
|
|
410 |
\<approx>swap [(a,b),(b,ba)] t2b") --E
|
|
411 |
apply(simp add: swap_append[THEN sym])
|
|
412 |
apply(drule_tac x="depth t1a" in spec)
|
|
413 |
apply(simp)
|
|
414 |
apply(drule_tac x="swap [(a,ba)] t2b" in spec)
|
|
415 |
apply(drule mp)
|
|
416 |
apply(force dest!: equ_depth)
|
|
417 |
apply(drule_tac x="swap [(a, ba), (a, ba), (a, b), (b, ba)] t2b" in spec)
|
|
418 |
apply(drule conjunct2)+
|
|
419 |
apply(best)
|
|
420 |
-- D
|
|
421 |
apply(rule rev_pi_pi_equ)
|
|
422 |
-- E
|
|
423 |
apply(subgoal_tac "nabla\<turnstile>t2b\<approx>swap [(a, ba), (a, b), (b, ba)] t2b") --F
|
|
424 |
apply(drule_tac x="depth t1a" in spec)
|
|
425 |
apply(simp)
|
|
426 |
apply(drule_tac x="t2b" in spec)
|
|
427 |
apply(drule mp)
|
|
428 |
apply(force dest!: equ_depth)
|
|
429 |
apply(best)
|
|
430 |
--F
|
|
431 |
apply(rule equ_pi_right[THEN spec, THEN mp])
|
|
432 |
apply(subgoal_tac "ds [] [(a,ba),(a,b),(b,ba)]={a,b}") -- G
|
|
433 |
apply(simp)
|
|
434 |
apply(drule_tac "t1.1"="t2a" and "t2.1"="swap [(b, ba)] t2b" and a1="a" in l3_jud[THEN mp])
|
|
435 |
apply(assumption)
|
|
436 |
apply(subgoal_tac "nabla \<turnstile> swapas (rev [(b,ba)]) a \<sharp> t2b") --H
|
|
437 |
apply(simp)
|
|
438 |
apply(case_tac "b=a")
|
|
439 |
apply(force)
|
|
440 |
apply(force)
|
|
441 |
--H
|
|
442 |
apply(rule fresh_swap_left[THEN mp])
|
|
443 |
apply(assumption)
|
|
444 |
--G
|
|
445 |
apply(rule ds_acabbc)
|
|
446 |
apply(assumption)+
|
|
447 |
--A
|
|
448 |
apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx>swap [(a,b)] (swap [(b,ba)] t2b)")--I
|
|
449 |
apply(drule_tac x="depth t1a" in spec)
|
|
450 |
apply(simp (no_asm_use))
|
|
451 |
apply(drule_tac x="t1a" in spec)
|
|
452 |
apply(simp (no_asm_use))
|
|
453 |
apply(drule_tac x="swap [(a,b)] t2a" in spec)
|
|
454 |
apply(drule conjunct2)+
|
|
455 |
apply(drule_tac x="swap [(a, b)] (swap [(b, ba)] t2b)" in spec)
|
|
456 |
apply(force simp add: swap_append[THEN sym])
|
|
457 |
--I
|
|
458 |
apply(drule_tac x="depth t1a" in spec)
|
|
459 |
apply(simp (no_asm_use))
|
|
460 |
apply(drule_tac x="t2a" in spec)
|
|
461 |
apply(drule mp)
|
|
462 |
apply(force dest!: equ_depth)
|
|
463 |
apply(drule_tac x="swap [(b,ba)] t2b" in spec)
|
|
464 |
apply(best)
|
|
465 |
-- Abst.ab
|
|
466 |
apply(simp)
|
|
467 |
apply(rule equ_abst_ab)
|
|
468 |
apply(assumption)
|
|
469 |
apply(drule_tac "t1.1"="t2a" and "t2.1"="t2b" and a1="a" in l3_jud[THEN mp])
|
|
470 |
apply(assumption)+
|
|
471 |
apply(subgoal_tac "nabla\<turnstile>swap [(a, b)] t2a\<approx>swap [(a, b)] t2b") --A
|
|
472 |
apply(best)
|
|
473 |
--A
|
|
474 |
apply(drule_tac x="depth t1a" in spec)
|
|
475 |
apply(simp (no_asm_use))
|
|
476 |
apply(drule_tac x="t2a" in spec)
|
|
477 |
apply(drule mp)
|
|
478 |
apply(force dest!: equ_depth)
|
|
479 |
apply(drule_tac x="t2b" in spec)
|
|
480 |
apply(best)
|
|
481 |
-- Abst
|
|
482 |
apply(ind_cases "nabla \<turnstile> Abst a t2a \<approx> t3")
|
|
483 |
apply(best)
|
|
484 |
apply(best)
|
|
485 |
-- Susp
|
|
486 |
apply(ind_cases "nabla \<turnstile> Susp pi2 X \<approx> t3")
|
|
487 |
apply(simp)
|
|
488 |
apply(rule equ_susp)
|
|
489 |
apply(rule ballI)
|
|
490 |
apply(drule_tac "pi2.1"="pi2" in ds_trans[THEN mp])
|
|
491 |
apply(force)
|
|
492 |
-- Paar
|
|
493 |
apply(ind_cases "nabla \<turnstile> Paar t2a s2 \<approx> t3")
|
|
494 |
apply(simp)
|
|
495 |
apply(rule equ_paar)
|
|
496 |
apply(drule_tac x="depth t1a" in spec)
|
|
497 |
apply(simp (no_asm_use) add: Suc_max_left)
|
|
498 |
apply(best)
|
|
499 |
apply(drule_tac x="depth s1" in spec)
|
|
500 |
apply(simp (no_asm_use) add: Suc_max_right)
|
|
501 |
apply(best)
|
|
502 |
-- Func
|
|
503 |
apply(ind_cases "nabla \<turnstile> Func F t2a \<approx> t3")
|
|
504 |
apply(best)
|
|
505 |
done
|
|
506 |
|
|
507 |
lemma pi_right_equ_help:
|
|
508 |
"\<forall>t. (n=depth t) \<longrightarrow> (\<forall>pi. nabla\<turnstile>t\<approx>swap pi t\<longrightarrow>(\<forall>a\<in> ds [] pi. nabla\<turnstile>a\<sharp>t))"
|
|
509 |
apply(induct_tac n rule: nat_less_induct)
|
|
510 |
apply(rule allI)+
|
|
511 |
apply(rule impI)
|
|
512 |
apply(rule allI)+
|
|
513 |
apply(rule impI)
|
|
514 |
apply(ind_cases "nabla \<turnstile> t \<approx> swap pi t")
|
|
515 |
apply(simp_all)
|
|
516 |
--Abst.ab
|
|
517 |
apply(rule ballI)
|
|
518 |
apply(case_tac "aa=a")
|
|
519 |
apply(force)
|
|
520 |
apply(rule fresh_abst_ab)
|
|
521 |
apply(case_tac "aa=swapas (rev pi) a")
|
|
522 |
apply(simp)
|
|
523 |
apply(drule fresh_swap_left[THEN mp])
|
|
524 |
apply(assumption)
|
|
525 |
apply(drule_tac x="depth t1" in spec)
|
|
526 |
apply(simp)
|
|
527 |
apply(drule_tac x="t1" in spec)
|
|
528 |
apply(simp add: swap_append[THEN sym])
|
|
529 |
apply(drule_tac x="[(a, swapas pi a)] @ pi" in spec)
|
|
530 |
apply(simp)
|
|
531 |
apply(case_tac "aa=swapas pi a")
|
|
532 |
apply(simp)
|
|
533 |
apply(drule_tac x="swapas pi a" in bspec)
|
|
534 |
apply(simp (no_asm) only: ds_def)
|
|
535 |
apply(simp only: mem_Collect_eq)
|
|
536 |
apply(rule conjI)
|
|
537 |
apply(simp)
|
|
538 |
apply(simp)
|
|
539 |
apply(rule impI)
|
|
540 |
apply(clarify)
|
|
541 |
apply(drule sym)
|
|
542 |
apply(drule swapas_rev_pi_a)
|
|
543 |
apply(force)
|
|
544 |
apply(assumption)
|
|
545 |
apply(drule_tac x="aa" in bspec)
|
|
546 |
apply(subgoal_tac "\<forall>A. aa\<in>A-{swapas pi a} \<and> aa\<noteq>swapas pi a \<longrightarrow>aa\<in>A")--A
|
|
547 |
apply(drule_tac x="ds [] ((a,swapas pi a) # pi)" in spec)
|
|
548 |
apply(simp add: ds_equality[THEN sym])
|
|
549 |
--A
|
|
550 |
apply(force)
|
|
551 |
apply(assumption)+
|
|
552 |
--Abst.aa
|
|
553 |
apply(rule ballI)
|
|
554 |
apply(case_tac "aa=a")
|
|
555 |
apply(force)
|
|
556 |
apply(best)
|
|
557 |
--Unit
|
|
558 |
apply(force)
|
|
559 |
--Atom
|
|
560 |
apply(force simp add: ds_def)
|
|
561 |
--Susp
|
|
562 |
apply(rule ballI)
|
|
563 |
apply(rule fresh_susp)
|
|
564 |
apply(drule_tac x="swapas (rev pi1) a" in bspec)
|
|
565 |
apply(rule ds_cancel_pi_right[of _ _ "[]" _, simplified, THEN mp])
|
|
566 |
apply(simp)
|
|
567 |
apply(assumption)
|
|
568 |
--Paar
|
|
569 |
apply(rule ballI)
|
|
570 |
apply(rule fresh_paar)
|
|
571 |
apply(drule_tac x="depth t1" in spec)
|
|
572 |
apply(force simp add: Suc_max_left)
|
|
573 |
apply(drule_tac x="depth s1" in spec)
|
|
574 |
apply(force simp add: Suc_max_right)
|
|
575 |
--Func
|
|
576 |
apply(best)
|
|
577 |
done
|
|
578 |
|
|
579 |
end
|