# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1263610560 -3600
# Node ID 4646198988909cf35bf26b5dfbe71d15a2a4da88
# Parent  4e0b89d886ac7744d4b8ecd4f143080fd368962c
used "new" alpha-equivalence relation (according to new scheme); proved equivalence theorems and so on

diff -r 4e0b89d886ac -r 464619898890 Quot/Examples/LamEx.thy
--- a/Quot/Examples/LamEx.thy	Sat Jan 16 02:09:38 2010 +0100
+++ b/Quot/Examples/LamEx.thy	Sat Jan 16 03:56:00 2010 +0100
@@ -29,6 +29,8 @@
 
 end
 
+declare perm_rlam.simps[eqvt]
+
 instance rlam::pt_name
 apply(default)
 apply(induct_tac [!] x rule: rlam.induct)
@@ -48,34 +50,126 @@
 apply(simp add: supp_atm)
 done
 
+declare set_diff_eqvt[eqvt]
+
+lemma rfv_eqvt[eqvt]:
+  fixes pi::"name prm"
+  shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
+apply(induct t)
+apply(simp_all)
+apply(simp add: perm_set_eq)
+apply(simp add: union_eqvt)
+apply(simp add: set_diff_eqvt)
+apply(simp add: perm_set_eq)
+done
+
 inductive
-  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
+    alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
 where
   a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
-| a3: "\<lbrakk>t \<approx> ([(a,b)] \<bullet> s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"
+| a3: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b) 
+       \<Longrightarrow> rLam a t \<approx> rLam b s"
 
-lemma helper:
-  fixes t::"rlam"
-  and   a::"name"
-  shows "[(a, a)] \<bullet> t = t"
-by (induct t)
-   (auto simp add: calc_atm)
+(* should be automatic with new version of eqvt-machinery *)
+lemma alpha_eqvt:
+  fixes pi::"name prm"
+  shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)"
+apply(induct rule: alpha.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(simp)
+apply(rule a3)
+apply(erule conjE)
+apply(erule exE)
+apply(erule conjE)
+apply(rule_tac x="pi \<bullet> pia" in exI)
+apply(rule conjI)
+apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1])
+apply(perm_simp add: eqvts)
+apply(rule conjI)
+apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1])
+apply(perm_simp add: eqvts)
+apply(rule conjI)
+apply(subst perm_compose[symmetric])
+apply(simp)
+apply(subst perm_compose[symmetric])
+apply(simp)
+done
 
 lemma alpha_refl:
-  fixes t::"rlam"
   shows "t \<approx> t"
-  apply(induct t rule: rlam.induct)
-  apply(simp add: a1)
-  apply(simp add: a2)
-  apply(rule a3)
-  apply(simp add: helper)
-  apply(simp)
-  done
+apply(induct t rule: rlam.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(rule a3)
+apply(rule_tac x="[]" in exI)
+apply(simp_all add: fresh_star_def fresh_list_nil)
+done
+
+lemma alpha_sym:
+  shows "t \<approx> s \<Longrightarrow> s \<approx> t"
+apply(induct rule: alpha.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(rule a3)
+apply(erule exE)
+apply(rule_tac x="rev pi" in exI)
+apply(simp)
+apply(simp add: fresh_star_def fresh_list_rev)
+apply(rule conjI)
+apply(erule conjE)+
+apply(rotate_tac 3)
+apply(drule_tac pi="rev pi" in alpha_eqvt)
+apply(perm_simp)
+apply(rule pt_bij2[OF pt_name_inst at_name_inst])
+apply(simp)
+done
+
+lemma alpha_trans:
+  shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
+apply(induct arbitrary: t3 rule: alpha.induct)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(simp add: a1)
+apply(rotate_tac 4)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(simp add: a2)
+apply(rotate_tac 1)
+apply(erule alpha.cases)
+apply(simp_all)
+apply(erule conjE)+
+apply(erule exE)+
+apply(erule conjE)+
+apply(rule a3)
+apply(rule_tac x="pia @ pi" in exI)
+apply(simp add: fresh_star_def fresh_list_append)
+apply(simp add: pt_name2)
+apply(drule_tac x="rev pia \<bullet> sa" in spec)
+apply(drule mp)
+apply(rotate_tac 8)
+apply(drule_tac pi="rev pia" in alpha_eqvt)
+apply(perm_simp)
+apply(rotate_tac 11)
+apply(drule_tac pi="pia" in alpha_eqvt)
+apply(perm_simp)
+done
 
 lemma alpha_equivp:
   shows "equivp alpha"
-sorry
+apply(rule equivpI)
+unfolding reflp_def symp_def transp_def
+apply(auto intro: alpha_refl alpha_sym alpha_trans)
+done
+
+lemma alpha_rfv:
+  shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
+apply(induct rule: alpha.induct)
+apply(simp)
+apply(simp)
+apply(simp)
+done
 
 quotient_type lam = rlam / alpha
   by (rule alpha_equivp)
@@ -114,13 +208,6 @@
 
 end
 
-lemma real_alpha:
-  assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"
-  shows "Lam a t = Lam b s"
-using a
-unfolding fresh_def supp_def
-sorry
-
 lemma perm_rsp[quot_respect]:
   "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
   apply(auto)
@@ -143,20 +230,18 @@
 lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"
   apply(auto)
   apply(rule a3)
-  apply(simp add: helper)
-  apply(simp)
+  apply(rule_tac x="[]" in exI)
+  unfolding fresh_star_def
+  apply(simp add: fresh_list_nil)
+  apply(simp add: alpha_rfv)
   done
 
 lemma rfv_rsp[quot_respect]: 
   "(alpha ===> op =) rfv rfv"
-  sorry
+apply(simp add: alpha_rfv)
+done
 
-lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"
-  apply (auto)
-  apply (erule alpha.cases)
-  apply (simp_all add: rlam.inject alpha_refl)
-  done
-
+section {* lifted theorems *}
 
 lemma lam_induct:
   "\<lbrakk>\<And>name. P (Var name);
@@ -196,36 +281,46 @@
   by  (lifting a2)
 
 lemma a3: 
-  "\<lbrakk>x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
+  "\<lbrakk>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk> 
+   \<Longrightarrow> Lam a t = Lam b s"
   by  (lifting a3)
 
 lemma alpha_cases: 
   "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
     \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
-    \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
+    \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; 
+         \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> P\<rbrakk>
     \<Longrightarrow> P"
   by (lifting alpha.cases)
 
 lemma alpha_induct: 
   "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);
     \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
-     \<And>x a b xa.
-        \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
+     \<And>t a s b.
+        \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and>
+         (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
     \<Longrightarrow> qxb qx qxa"
   by (lifting alpha.induct)
 
-lemma var_inject: 
-  "(Var a = Var b) = (a = b)"
-  by (lifting rvar_inject)
-
-
-lemma app_inject: 
-  "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
-sorry
+lemma lam_inject [simp]: 
+  shows "(Var a = Var b) = (a = b)"
+  and   "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
+apply(lifting rlam.inject(1) rlam.inject(2))
+apply(auto)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(simp add: alpha.a1)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(drule alpha.cases)
+apply(simp_all)
+apply(rule alpha.a2)
+apply(simp_all)
+done
 
 lemma var_supp1:
   shows "(supp (Var a)) = ((supp a)::name set)"
-apply(simp add: supp_def var_inject)
+apply(simp add: supp_def)
 done
 
 lemma var_supp:
@@ -236,13 +331,14 @@
 
 lemma app_supp:
   shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"
-apply(simp only: perm_lam supp_def app_inject)
+apply(simp only: perm_lam supp_def lam_inject)
 apply(simp add: Collect_imp_eq Collect_neg_eq)
 done
 
 lemma lam_supp:
   shows "supp (Lam x t) = ((supp ([x].t))::name set)"
 apply(simp add: supp_def)
+apply(simp add: abs_perm)
 sorry
 
 
@@ -251,7 +347,8 @@
 apply(induct_tac x rule: lam_induct)
 apply(simp add: var_supp)
 apply(simp add: app_supp)
-sorry
+apply(simp add: lam_supp abs_supp)
+done
 
 lemma fresh_lam:
   "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)"