theory Lambda+
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imports "../NewParser"+
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begin+
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+
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atom_decl name+
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+
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nominal_datatype lam =+
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  Var "name"+
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| App "lam" "lam"+
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| Lam x::"name" l::"lam"  bind_set x in l+
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+
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thm lam.fv+
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thm lam.supp+
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lemmas supp_fn' = lam.fv[simplified lam.supp]+
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+
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declare lam.perm[eqvt]+
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+
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+
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section {* Strong Induction Principles*}+
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+
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(*+
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  Old way of establishing strong induction+
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  principles by chosing a fresh name.+
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*)+
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lemma+
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  fixes c::"'a::fs"+
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  assumes a1: "\<And>name c. P c (Var name)"+
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  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"+
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  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"+
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  shows "P c lam"+
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proof -+
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  have "\<And>p. P c (p \<bullet> lam)"+
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    apply(induct lam arbitrary: c rule: lam.induct)+
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    apply(perm_simp)+
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    apply(rule a1)+
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    apply(perm_simp)+
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    apply(rule a2)+
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    apply(assumption)+
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    apply(assumption)+
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    apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")+
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    defer+
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    apply(simp add: fresh_def)+
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    apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)+
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    apply(simp add: supp_Pair finite_supp)+
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    apply(blast)+
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    apply(erule exE)+
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    apply(rule_tac t="p \<bullet> Lam name lam" and +
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                   s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)+
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    apply(simp del: lam.perm)+
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    apply(subst lam.perm)+
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    apply(subst (2) lam.perm)+
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    apply(rule flip_fresh_fresh)+
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    apply(simp add: fresh_def)+
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    apply(simp only: supp_fn')+
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    apply(simp)+
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    apply(simp add: fresh_Pair)+
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    apply(simp)+
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    apply(rule a3)+
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    apply(simp add: fresh_Pair)+
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    apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)+
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    apply(simp)+
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    done+
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  then have "P c (0 \<bullet> lam)" by blast+
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  then show "P c lam" by simp+
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qed+
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+
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(* +
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  New way of establishing strong induction+
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  principles by using a appropriate permutation.+
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*)+
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lemma+
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  fixes c::"'a::fs"+
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  assumes a1: "\<And>name c. P c (Var name)"+
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  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"+
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  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"+
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  shows "P c lam"+
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proof -+
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  have "\<And>p. P c (p \<bullet> lam)"+
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    apply(induct lam arbitrary: c rule: lam.induct)+
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    apply(perm_simp)+
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    apply(rule a1)+
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    apply(perm_simp)+
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    apply(rule a2)+
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    apply(assumption)+
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    apply(assumption)+
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    apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")+
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    apply(erule exE)+
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    apply(rule_tac t="p \<bullet> Lam name lam" and +
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                   s="q \<bullet> p \<bullet> Lam name lam" in subst)+
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    defer+
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    apply(simp)+
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    apply(rule a3)+
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    apply(simp add: eqvts fresh_star_def)+
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    apply(drule_tac x="q + p" in meta_spec)+
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    apply(simp)+
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    apply(rule at_set_avoiding2)+
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    apply(simp add: finite_supp)+
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    apply(simp add: finite_supp)+
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    apply(simp add: finite_supp)+
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    apply(perm_simp)+
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    apply(simp add: fresh_star_def fresh_def supp_fn')+
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    apply(rule supp_perm_eq)+
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    apply(simp)+
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    done+
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  then have "P c (0 \<bullet> lam)" by blast+
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  then show "P c lam" by simp+
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qed+
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+
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section {* Typing *}+
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+
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nominal_datatype ty =+
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  TVar string+
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| TFun ty ty+
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+
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notation+
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 TFun ("_ \<rightarrow> _") +
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+
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declare ty.perm[eqvt]+
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+
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inductive+
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  valid :: "(name \<times> ty) list \<Rightarrow> bool"+
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where+
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  "valid []"+
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| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"+
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+
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inductive+
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  typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60) +
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where+
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    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"+
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  | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"+
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  | t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"+
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+
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equivariance valid+
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equivariance typing+
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+
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thm valid.eqvt+
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thm typing.eqvt+
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thm eqvts+
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thm eqvts_raw+
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+
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thm typing.induct[of "\<Gamma>" "t" "T", no_vars]+
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+
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lemma+
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  fixes c::"'a::fs"+
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  assumes a: "\<Gamma> \<turnstile> t : T" +
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  and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"+
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  and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk> +
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           \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"+
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  and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>atom x \<sharp> c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk> +
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           \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"+
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  shows "P c \<Gamma> t T"+
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proof -+
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  from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"+
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  proof (induct)+
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    case (t_Var \<Gamma> x T p c)+
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    then show ?case+
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      apply -+
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      apply(perm_strict_simp)+
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      apply(rule a1)+
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      apply(drule_tac p="p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(rotate_tac 1)+
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      apply(drule_tac p="p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      done+
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  next+
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    case (t_App \<Gamma> t1 T1 T2 t2 p c)+
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    then show ?case+
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      apply -+
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      apply(perm_strict_simp)+
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      apply(rule a2)+
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      apply(drule_tac p="p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(assumption)+
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      apply(rotate_tac 2)+
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      apply(drule_tac p="p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(assumption)+
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      done+
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  next+
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    case (t_Lam x \<Gamma> T1 t T2 p c)+
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    then show ?case+
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      apply -+
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      apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and> +
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        supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")+
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      apply(erule exE)+
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      apply(rule_tac t="p \<bullet> \<Gamma>" and  s="(q + p) \<bullet> \<Gamma>" in subst)+
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      apply(simp only: permute_plus)+
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      apply(rule supp_perm_eq)+
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      apply(simp add: supp_Pair fresh_star_union)+
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      apply(rule_tac t="p \<bullet> Lam x t" and  s="(q + p) \<bullet> Lam x t" in subst)+
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      apply(simp only: permute_plus)+
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      apply(rule supp_perm_eq)+
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      apply(simp add: supp_Pair fresh_star_union)+
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      apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and  s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)+
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      apply(simp only: permute_plus)+
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      apply(rule supp_perm_eq)+
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      apply(simp add: supp_Pair fresh_star_union)+
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      apply(simp (no_asm) only: eqvts)+
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      apply(rule a3)+
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      apply(simp only: eqvts permute_plus)+
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      apply(simp add: fresh_star_def)+
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      apply(drule_tac p="q + p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(rotate_tac 1)+
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      apply(drule_tac p="q + p" in permute_boolI)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(drule_tac x="d" in meta_spec)+
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      apply(drule_tac x="q + p" in meta_spec)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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      apply(assumption)+
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      apply(rule at_set_avoiding2)+
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      apply(simp add: finite_supp)+
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      apply(simp add: finite_supp)+
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      apply(simp add: finite_supp)+
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      apply(rule_tac p="-p" in permute_boolE)+
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      apply(perm_strict_simp add: permute_minus_cancel)+
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	(* supplied by the user *)+
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      apply(simp add: fresh_star_prod)+
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      apply(simp add: fresh_star_def)+
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      sorry+
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  qed+
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  then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .+
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  then show "P c \<Gamma> t T" by simp+
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qed+
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+
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+
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+
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+
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+
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+
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+
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inductive+
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  tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"  +
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  for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"+
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where+
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    aa: "tt r a a"+
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  | bb: "tt r a b ==> tt r a c"+
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+
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(* PROBLEM: derived eqvt-theorem does not conform with [eqvt]+
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equivariance tt+
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*)+
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+
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+
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inductive+
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 alpha_lam_raw'+
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where+
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  a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"+
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| a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>+
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   alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"+
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| a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>+
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   alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"+
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+
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equivariance alpha_lam_raw'+
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+
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thm eqvts_raw+
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+
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section {* size function *}+
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+
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lemma size_eqvt_raw:+
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  fixes t::"lam_raw"+
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  shows "size (pi \<bullet> t)  = size t"+
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  apply (induct rule: lam_raw.inducts)+
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  apply simp_all+
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  done+
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+
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instantiation lam :: size +
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begin+
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+
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quotient_definition+
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  "size_lam :: lam \<Rightarrow> nat"+
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is+
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  "size :: lam_raw \<Rightarrow> nat"+
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+
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lemma size_rsp:+
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  "alpha_lam_raw x y \<Longrightarrow> size x = size y"+
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  apply (induct rule: alpha_lam_raw.inducts)+
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  apply (simp_all only: lam_raw.size)+
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  apply (simp_all only: alphas)+
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  apply clarify+
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  apply (simp_all only: size_eqvt_raw)+
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  done+
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+
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lemma [quot_respect]:+
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  "(alpha_lam_raw ===> op =) size size"+
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  by (simp_all add: size_rsp)+
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+
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lemma [quot_preserve]:+
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  "(rep_lam ---> id) size = size"+
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  by (simp_all add: size_lam_def)+
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+
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instance+
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  by default+
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+
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end+
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+
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lemmas size_lam[simp] = +
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  lam_raw.size(4)[quot_lifted]+
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  lam_raw.size(5)[quot_lifted]+
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  lam_raw.size(6)[quot_lifted]+
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+
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(* is this needed? *)+
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lemma [measure_function]: +
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  "is_measure (size::lam\<Rightarrow>nat)" +
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  by (rule is_measure_trivial)+
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+
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section {* Matching *}+
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+
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definition+
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  MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"+
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where+
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  "MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"+
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+
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(*+
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lemma MATCH_eqvt:+
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  shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"+
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unfolding MATCH_def+
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apply(perm_simp the_eqvt)+
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apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})+
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apply(simp)+
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thm eqvts_raw +
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apply(subst if_eqvt)+
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apply(subst ex1_eqvt)+
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apply(subst permute_fun_def)+
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apply(subst ex_eqvt)+
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apply(subst permute_fun_def)+
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apply(subst eq_eqvt)+
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apply(subst permute_fun_app_eq[where f="M"])+
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apply(simp only: permute_minus_cancel)+
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apply(subst permute_prod.simps)+
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apply(subst permute_prod.simps)+
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apply(simp only: permute_minus_cancel)+
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apply(simp only: permute_bool_def)+
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apply(simp)+
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apply(subst ex1_eqvt)+
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apply(subst permute_fun_def)+
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apply(subst ex_eqvt)+
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apply(subst permute_fun_def)+
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apply(subst eq_eqvt)+
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+
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apply(simp only: eqvts)+
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apply(simp)+
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apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")+
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apply(drule sym)+
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apply(simp)+
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apply(rule impI)+
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apply(simp add: perm_bool)+
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apply(rule trans)+
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apply(rule pt_the_eqvt[OF pta at])+
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apply(assumption)+
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apply(simp add: pt_ex_eqvt[OF pt at])+
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apply(simp add: pt_eq_eqvt[OF ptb at])+
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apply(rule cheat)+
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apply(rule trans)+
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apply(rule pt_ex1_eqvt)+
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apply(rule pta)+
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apply(rule at)+
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apply(simp add: pt_ex_eqvt[OF pt at])+
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apply(simp add: pt_eq_eqvt[OF ptb at])+
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apply(subst pt_pi_rev[OF pta at])+
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apply(subst pt_fun_app_eq[OF pt at])+
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apply(subst pt_pi_rev[OF pt at])+
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apply(simp)+
−
done+
−
+
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lemma MATCH_cng:+
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  assumes a: "M1 = M2" "d1 = d2"+
−
  shows "MATCH M1 d1 x = MATCH M2 d2 x"+
−
using a by simp+
−
+
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lemma MATCH_eq:+
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  assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"+
−
  shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"+
−
using a+
−
unfolding MATCH_def+
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apply(subst if_P)+
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apply(rule_tac a="r x" in ex1I)+
−
apply(rule_tac x="x" in exI)+
−
apply(blast)+
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apply(erule exE)+
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apply(drule_tac x="q" in meta_spec)+
−
apply(auto)[1]+
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apply(rule the_equality)+
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apply(blast)+
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apply(erule exE)+
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apply(drule_tac x="q" in meta_spec)+
−
apply(auto)[1]+
−
done+
−
+
−
lemma MATCH_eq2:+
−
  assumes a: "t = l x1 x2" "G x1 x2" "\<And>x1' x2'. t = l x1' x2' \<Longrightarrow> G x1' x2' \<Longrightarrow> r x1' x2' = r x1 x2"+
−
  shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = r x1 x2"+
−
sorry+
−
+
−
lemma MATCH_neq:+
−
  assumes a: "\<And>x. t = l x \<Longrightarrow> G x \<Longrightarrow> False"+
−
  shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = d"+
−
using a+
−
unfolding MATCH_def+
−
apply(subst if_not_P)+
−
apply(blast)+
−
apply(rule refl)+
−
done+
−
+
−
lemma MATCH_neq2:+
−
  assumes a: "\<And>x1 x2. t = l x1 x2 \<Longrightarrow> G x1 x2 \<Longrightarrow> False"+
−
  shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = d"+
−
using a+
−
unfolding MATCH_def+
−
apply(subst if_not_P)+
−
apply(auto)+
−
done+
−
*)+
−
+
−
ML {*+
−
fun mk_avoids ctxt params name set =+
−
  let+
−
    val (_, ctxt') = ProofContext.add_fixes+
−
      (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;+
−
    fun mk s =+
−
      let+
−
        val t = Syntax.read_term ctxt' s;+
−
        val t' = list_abs_free (params, t) |>+
−
          funpow (length params) (fn Abs (_, _, t) => t)+
−
      in (t', HOLogic.dest_setT (fastype_of t)) end+
−
      handle TERM _ =>+
−
        error ("Expression " ^ quote s ^ " to be avoided in case " ^+
−
          quote name ^ " is not a set type");+
−
    fun add_set p [] = [p]+
−
      | add_set (t, T) ((u, U) :: ps) =+
−
          if T = U then+
−
            let val S = HOLogic.mk_setT T+
−
            in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps+
−
            end+
−
          else (u, U) :: add_set (t, T) ps+
−
  in+
−
    (mk #> add_set) set +
−
  end;+
−
*}+
−
+
−
+
−
ML {* +
−
  writeln (commas (map (Syntax.string_of_term @{context} o fst) +
−
    (mk_avoids @{context} [] "t_Var" "{x}" []))) +
−
*}+
−
+
−
+
−
ML {*+
−
+
−
fun prove_strong_ind (pred_name, avoids) ctxt = +
−
  Proof.theorem NONE (K I) [] ctxt+
−
+
−
local structure P = OuterParse and K = OuterKeyword in+
−
+
−
val _ =+
−
  OuterSyntax.local_theory_to_proof "nominal_inductive"+
−
    "proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal+
−
      (P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --+
−
        (P.$$$ ":" |-- P.and_list1 P.term))) []) >>  prove_strong_ind)+
−
+
−
end;+
−
+
−
*}+
−
+
−
(*+
−
nominal_inductive typing+
−
*)+
−
+
−
(* Substitution *)+
−
+
−
fun+
−
  subst_var_raw :: "lam_raw \<Rightarrow> name \<Rightarrow> name \<Rightarrow> lam_raw"+
−
where+
−
  "subst_var_raw (Var_raw x) y s = (if x=y then (Var_raw s) else (Var_raw x))"+
−
| "subst_var_raw (App_raw l r) y s = App_raw (subst_var_raw l y s) (subst_var_raw r y s)"+
−
| "subst_var_raw (Lam_raw x t) y s =+
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     (if x = y then Lam_raw x t else Lam_raw x (subst_var_raw t y s))"+
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+
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(* Should be true? *)+
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lemma "(alpha_lam_raw ===> op = ===> op = ===> alpha_lam_raw) subst_var_raw subst_var_raw"+
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  proof (intro fun_relI, (simp, clarify))+
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    fix a b ba bb+
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    assume a: "alpha_lam_raw a b"+
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    show "alpha_lam_raw (subst_var_raw a ba bb) (subst_var_raw b ba bb)" using a+
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      apply (induct a b rule: alpha_lam_raw.induct)+
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      apply (simp add: equivp_reflp[OF lam_equivp])+
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      apply (simp add: alpha_lam_raw.intros)+
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      apply auto+
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      apply (rule_tac[!] alpha_lam_raw.intros)+
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      apply (rule_tac[!] x="p" in exI) (* Need to do better *)+
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      apply (simp_all add: alphas)+
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      apply clarify+
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      apply simp+
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      sorry+
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    qed+
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+
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fun+
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  subst_raw :: "lam_raw \<Rightarrow> name \<Rightarrow> lam_raw \<Rightarrow> lam_raw"+
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where+
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  "subst_raw (Var_raw x) y s = (if x=y then s else (Var_raw x))"+
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| "subst_raw (App_raw l r) y s = App_raw (subst_raw l y s) (subst_raw r y s)"+
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| "subst_raw (Lam_raw x t) y s =+
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     (if x = y then t else+
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       (if atom x \<notin> (fv_lam_raw s) then (Lam_raw x (subst_raw t y s)) else undefined))"+
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+
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quotient_definition+
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  subst ("_ [ _ ::= _ ]" [100,100,100] 100)+
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where+
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  "subst :: lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" is "subst_raw"+
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+
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lemmas fv_rsp = quot_respect(10)[simplified,rulify]+
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+
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lemma subst_rsp_pre1:+
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  assumes a: "alpha_lam_raw a b"+
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  shows "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)"+
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  using a+
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  apply (induct a b arbitrary: c y rule: alpha_lam_raw.induct)+
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  apply (simp add: equivp_reflp[OF lam_equivp])+
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  apply (simp add: alpha_lam_raw.intros)+
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  apply (simp only: alphas)+
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  apply clarify+
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  apply (simp only: subst_raw.simps)+
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  sorry+
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+
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lemma subst_rsp_pre2:+
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  assumes a: "alpha_lam_raw a b"+
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  shows "alpha_lam_raw (subst_raw c y a) (subst_raw c y b)"+
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  sorry+
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+
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(* The below is definitely not true... *)+
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lemma [quot_respect]:+
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  "(alpha_lam_raw ===> op = ===> alpha_lam_raw ===> alpha_lam_raw) subst_raw subst_raw"+
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  proof (intro fun_relI, simp)+
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    fix a b c d :: lam_raw+
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    fix y :: name+
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    assume a: "alpha_lam_raw a b"+
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    assume b: "alpha_lam_raw c d"+
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    have c: "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)" using subst_rsp_pre1 a by simp+
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    then have d: "alpha_lam_raw (subst_raw b y c) (subst_raw b y d)" using subst_rsp_pre2 b by simp+
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    show "alpha_lam_raw (subst_raw a y c) (subst_raw b y d)"+
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      using c d equivp_transp[OF lam_equivp] by blast+
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  qed+
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+
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lemma simp3:+
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  "x \<noteq> y \<Longrightarrow> atom x \<notin> fv_lam_raw s \<Longrightarrow> subst_raw (Lam_raw x t) y s = Lam_raw x (subst_raw t y s)"+
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  by simp+
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+
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lemmas subst_simps = subst_raw.simps(1-2)[quot_lifted,no_vars]+
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  simp3[quot_lifted,simplified lam.supp,simplified fresh_def[symmetric], no_vars]+
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+
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end+
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+
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+
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+
−