--- a/Nominal/Ex/BetaCR.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,378 +0,0 @@
-theory BetaCR
-imports CR
-begin
-
-(* TODO1: Does not work:*)
-
-(* equivariance beta_star *)
-
-(* proved manually below. *)
-
-lemma eqvt_helper: "x1 \<longrightarrow>b* x2 \<Longrightarrow>  (p \<bullet> x1) \<longrightarrow>b* (p \<bullet> x2)"
-  by (erule beta_star.induct)
-     (metis beta.eqvt bs2 bs1)+
-
-lemma [eqvt]: "p \<bullet> (x1 \<longrightarrow>b* x2) = ((p \<bullet> x1) \<longrightarrow>b* (p \<bullet> x2))"
-  apply rule
-  apply (drule permute_boolE)
-  apply (erule eqvt_helper)
-  apply (intro permute_boolI)
-  apply (drule_tac p="-p" in eqvt_helper)
-  by simp
-
-definition
-  equ :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<approx> _")
-where
-  "t \<approx> s \<longleftrightarrow> (\<exists>r. t \<longrightarrow>b* r \<and> s \<longrightarrow>b* r)"
-
-lemma equ_refl:
-  shows "t \<approx> t"
-  unfolding equ_def by auto
-
-lemma equ_sym:
-  assumes "t \<approx> s"
-  shows "s \<approx> t"
-  using assms unfolding equ_def
-  by auto
-
-lemma equ_trans:
-  assumes "A \<approx> B" "B \<approx> C"
-  shows "A \<approx> C"
-  using assms unfolding equ_def
-proof (elim exE conjE)
-  fix D E
-  assume a: "A \<longrightarrow>b* D" "B \<longrightarrow>b* D" "B \<longrightarrow>b* E" "C \<longrightarrow>b* E"
-  then obtain F where "D \<longrightarrow>b* F" "E \<longrightarrow>b* F" using CR_for_Beta_star by blast
-  then have "A \<longrightarrow>b* F \<and> C \<longrightarrow>b* F" using a bs3 by blast
-  then show "\<exists>F. A \<longrightarrow>b* F \<and> C \<longrightarrow>b* F" by blast
-qed
-
-lemma App_beta: "A \<longrightarrow>b* B \<Longrightarrow> C \<longrightarrow>b* D \<Longrightarrow> App A C \<longrightarrow>b* App B D"
-  apply (erule beta_star.induct)
-  apply auto
-  apply (erule beta_star.induct)
-  apply auto
-  done
-
-lemma Lam_beta: "A \<longrightarrow>b* B \<Longrightarrow> Lam [x]. A \<longrightarrow>b* Lam [x]. B"
-  by (erule beta_star.induct) auto
-
-lemma Lam_rsp: "A \<approx> B \<Longrightarrow> Lam [x]. A \<approx> Lam [x]. B"
-  unfolding equ_def
-  apply auto
-  apply (rule_tac x="Lam [x]. r" in exI)
-  apply (auto simp add: Lam_beta)
-  done
-
-lemma [quot_respect]:
-  shows "(op = ===> equ) Var Var"
-  and   "(equ ===> equ ===> equ) App App"
-  and   "(op = ===> equ ===> equ) CR.Lam CR.Lam"
-  unfolding fun_rel_def equ_def
-  apply auto
-  apply (rule_tac x="App r ra" in exI)
-  apply (auto simp add: App_beta)
-  apply (rule_tac x="Lam [x]. r" in exI)
-  apply (auto simp add: Lam_beta)
-  done
-
-lemma beta_subst1_pre: "B \<longrightarrow>b C \<Longrightarrow> A [x ::= B] \<longrightarrow>b* A [x ::= C]"
-  by (nominal_induct A avoiding: x B C rule: lam.strong_induct)
-     (auto simp add: App_beta Lam_beta)
-
-lemma beta_subst1: "B \<longrightarrow>b* C \<Longrightarrow> A [x ::= B] \<longrightarrow>b* A [x ::= C]"
-  apply (erule beta_star.induct)
-  apply auto
-  apply (subgoal_tac "A [x ::= M2] \<longrightarrow>b* A [x ::= M3]")
-  apply (rotate_tac 1)
-  apply (erule bs3)
-  apply assumption
-  apply (simp add: beta_subst1_pre)
-  done
-
-lemma beta_subst2_pre:
-  assumes "A \<longrightarrow>b B" shows "A [x ::= C] \<longrightarrow>b* B [x ::= C]"
-  using assms
-  apply (nominal_induct avoiding: x C rule: beta.strong_induct)
-  apply (auto simp add: App_beta fresh_star_def fresh_Pair Lam_beta)[3]
-  apply (subst substitution_lemma)
-  apply (auto simp add: fresh_star_def fresh_Pair fresh_at_base)[2]
-  apply (auto simp add: fresh_star_def fresh_Pair)
-  apply (rule beta_star.intros)
-  defer
-  apply (subst beta.intros)
-  apply (auto simp add: fresh_fact)
-  done
-
-lemma beta_subst2: "A \<longrightarrow>b* B \<Longrightarrow> A [x ::= C] \<longrightarrow>b* B [x ::= C]"
-  apply (erule beta_star.induct)
-  apply auto
-  apply (subgoal_tac "M2[x ::= C] \<longrightarrow>b* M3[x ::= C]")
-  apply (rotate_tac 1)
-  apply (erule bs3)
-  apply assumption
-  apply (simp add: beta_subst2_pre)
-  done
-
-lemma beta_subst: "A \<longrightarrow>b* B \<Longrightarrow> C \<longrightarrow>b* D \<Longrightarrow> A [x ::= C] \<longrightarrow>b* B [x ::= D]"
-  using beta_subst1 beta_subst2 bs3 by metis
-
-lemma subst_rsp_pre:
-  "x \<approx> y \<Longrightarrow> xb \<approx> ya \<Longrightarrow> x [xa ::= xb] \<approx> y [xa ::= ya]"
-  unfolding equ_def
-  apply auto
-  apply (rule_tac x="r [xa ::= ra]" in exI)
-  apply (simp add: beta_subst)
-  done
-
-lemma [quot_respect]:
-  shows "(equ ===> op = ===> equ ===> equ) subst subst"
-unfolding fun_rel_def by (auto simp add: subst_rsp_pre)
-
-lemma [quot_respect]:
-  shows "(op = ===> equ ===> equ) permute permute"
-  unfolding fun_rel_def equ_def
-  apply (auto intro: eqvts)
-  apply (rule_tac x="x \<bullet> r" in exI)
-  using eqvts(1) permute_boolI by metis
-
-quotient_type qlam = lam / equ
-  by (auto intro!: equivpI reflpI sympI transpI simp add: equ_refl equ_sym)
-     (erule equ_trans, assumption)
-
-quotient_definition "QVar::name \<Rightarrow> qlam" is Var
-quotient_definition "QApp::qlam \<Rightarrow> qlam \<Rightarrow> qlam" is App
-quotient_definition QLam ("QLam [_]._")
-  where "QLam::name \<Rightarrow> qlam \<Rightarrow> qlam" is CR.Lam
-
-lemmas qlam_strong_induct = lam.strong_induct[quot_lifted]
-lemmas qlam_induct = lam.induct[quot_lifted]
-
-instantiation qlam :: pt
-begin
-
-quotient_definition "permute_qlam::perm \<Rightarrow> qlam \<Rightarrow> qlam" 
-  is "permute::perm \<Rightarrow> lam \<Rightarrow> lam"
-
-instance
-apply default
-apply(descending)
-apply(simp)
-apply(rule equ_refl)
-apply(descending)
-apply(simp)
-apply(rule equ_refl)
-done
-
-end
-
-lemma qlam_perm[simp, eqvt]:
-  shows "p \<bullet> (QVar x) = QVar (p \<bullet> x)"
-  and "p \<bullet> (QApp t s) = QApp (p \<bullet> t) (p \<bullet> s)"
-  and "p \<bullet> (QLam [x].t) = QLam [p \<bullet> x].(p \<bullet> t)"
-  by(descending, simp add: equ_refl)+
-
-lemma qlam_supports:
-  shows "{atom x} supports (QVar x)"
-  and   "supp (t, s) supports (QApp t s)"
-  and   "supp (x, t) supports (QLam [x].t)"
-unfolding supports_def fresh_def[symmetric]
-by (auto simp add: swap_fresh_fresh)
-
-lemma qlam_fs:
-  fixes t::"qlam"
-  shows "finite (supp t)"
-apply(induct t rule: qlam_induct)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp add: supp_Pair)
-apply(rule supports_finite)
-apply(rule qlam_supports)
-apply(simp add: supp_Pair finite_supp)
-done
-
-instantiation qlam :: fs
-begin
-
-instance
-apply(default)
-apply(rule qlam_fs)
-done
-
-end
-
-quotient_definition subst_qlam ("_[_ ::q= _]")
-  where "subst_qlam::qlam \<Rightarrow> name \<Rightarrow> qlam \<Rightarrow> qlam" is subst  
-
-definition
-  "Supp t = \<Inter>{supp s | s. s \<approx> t}"
-
-lemma Supp_rsp:
-  "a \<approx> b \<Longrightarrow> Supp a = Supp b"
-  unfolding Supp_def
-  apply(rule_tac f="Inter" in arg_cong)
-  apply(auto)
-  apply (metis equ_trans)
-  by (metis equivp_def qlam_equivp)
-
-lemma [quot_respect]:
-  shows "(equ ===> op=) Supp Supp"
-  unfolding fun_rel_def by (auto simp add: Supp_rsp)
-
-quotient_definition "supp_qlam::qlam \<Rightarrow> atom set" 
-  is "Supp::lam \<Rightarrow> atom set"
-
-lemma Supp_supp:
-  "Supp t \<subseteq> supp t"
-unfolding Supp_def
-apply(auto)
-apply(drule_tac x="supp t" in spec)
-apply(auto simp add: equ_refl)
-done
-
-lemma supp_subst:
-  shows "supp (t[x ::= s]) \<subseteq> (supp t - {atom x}) \<union> supp s"
-  by (induct t x s rule: subst.induct) (auto simp add: lam.supp supp_at_base)
-
-lemma supp_beta: "x \<longrightarrow>b r \<Longrightarrow> supp r \<subseteq> supp x"
-  apply (induct rule: beta.induct)
-  apply (simp_all add: lam.supp)
-  apply blast+
-  using supp_subst by blast
-
-lemma supp_beta_star: "x \<longrightarrow>b* r \<Longrightarrow> supp r \<subseteq> supp x"
-  apply (erule beta_star.induct)
-  apply auto
-  using supp_beta by blast
-
-lemma supp_equ: "x \<approx> y \<Longrightarrow> \<exists>z. z \<approx> x \<and> supp z \<subseteq> supp x \<inter> supp y"
-  unfolding equ_def
-  apply (simp (no_asm) only: equ_def[symmetric])
-  apply (elim exE)
-  apply (rule_tac x=r in exI)
-  apply rule
-  apply (metis bs1 equ_def)
-  using supp_beta_star by blast
-
-lemma supp_psubset: "Supp x \<subset> supp x \<Longrightarrow> \<exists>t. t \<approx> x \<and> supp t \<subset> supp x"
-proof -
-  assume "Supp x \<subset> supp x"
-  then obtain a where "a \<in> supp x" "a \<notin> Supp x" by blast
-  then obtain y where nin: "y \<approx> x" "a \<notin> supp y" unfolding Supp_def by blast
-  then obtain t where eq: "t \<approx> x" "supp t \<subseteq> supp x \<inter> supp y"
-    using supp_equ equ_sym by blast
-  then have "supp t \<subset> supp x" using nin
-    by (metis `(a\<Colon>atom) \<in> supp (x\<Colon>lam)` le_infE psubset_eq set_rev_mp)
-  then show "\<exists>t. t \<approx> x \<and> supp t \<subset> supp x" using eq by blast
-qed
-
-lemma Supp_exists: "\<exists>t. supp t = Supp t \<and> t \<approx> x"
-proof -
-  have "\<And>fs x. supp x = fs \<Longrightarrow> \<exists>t. supp t = Supp t \<and> t \<approx> x"
-  proof -
-    fix fs
-    show "\<And>x. supp x = fs \<Longrightarrow> \<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x"
-    proof (cases "finite fs")
-      case True
-      assume fs: "finite fs"
-      then show "\<And>x. supp x = fs \<Longrightarrow> \<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x"
-      proof (induct fs rule: finite_psubset_induct, clarify)
-        fix A :: "atom set" fix x :: lam
-        assume IH: "\<And>B xa. \<lbrakk>B \<subset> supp x; supp xa = B\<rbrakk> \<Longrightarrow> \<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> xa"
-        show "\<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x"
-        proof (cases "supp x = Supp x")
-          assume "supp x = Supp x"
-          then show "\<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x"
-            by (rule_tac x="x" in exI) (simp add: equ_refl)
-        next
-          assume "supp x \<noteq> Supp x"
-          then have "Supp x \<subset> supp x" using Supp_supp by blast
-          then obtain y where a1: "supp y \<subset> supp x" "y \<approx> x"
-            using supp_psubset by blast
-          then obtain t where "supp t = Supp t \<and> t \<approx> y"
-            using IH[of "supp y" y] by blast
-          then show "\<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x" using a1 equ_trans
-            by blast
-        qed
-      qed
-    next
-      case False
-      fix x :: lam
-      assume "supp x = fs" "infinite fs"
-      then show "\<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x"
-        by (auto simp add: finite_supp)
-    qed simp
-  qed
-  then show "\<exists>t\<Colon>lam. supp t = Supp t \<and> t \<approx> x" by blast
-qed
-
-lemma subst3: "x \<noteq> y \<and> atom x \<notin> Supp s \<Longrightarrow> Lam [x]. t [y ::= s] \<approx> Lam [x]. (t [y ::= s])"
-proof -
-  assume as: "x \<noteq> y \<and> atom x \<notin> Supp s"
-  obtain s' where s: "supp s' = Supp s'" "s' \<approx> s" using Supp_exists[of s] by blast
-  then have lhs: "Lam [x]. t [y ::= s] \<approx> Lam [x]. t [y ::= s']" using subst_rsp_pre equ_refl equ_sym by blast
-  have supp: "Supp s' = Supp s" using Supp_rsp s(2) by blast
-  have lhs_rhs: "Lam [x]. t [y ::= s'] = Lam [x]. (t [y ::= s'])"
-    by (simp add: fresh_def supp_Pair s supp as supp_at_base)
-  have "t [y ::= s'] \<approx> t [y ::= s]"
-    using subst_rsp_pre[OF equ_refl s(2)] .
-  with Lam_rsp have rhs: "Lam [x]. (t [y ::= s']) \<approx> Lam [x]. (t [y ::= s])" .
-  show ?thesis 
-    using lhs[unfolded lhs_rhs] rhs equ_trans by blast
-qed
-
-thm subst3[quot_lifted]
-
-lemma Supp_Lam:
-  "atom a \<notin> Supp (Lam [a].t)"
-proof -
-  have "atom a \<notin> supp (Lam [a].t)" by (simp add: lam.supp)
-  then show ?thesis using Supp_supp
-    by blast
-qed
-
-lemma [eqvt]: "(p \<bullet> (a \<approx> b)) = ((p \<bullet> a) \<approx> (p \<bullet> b))"
-  unfolding equ_def
-  by perm_simp rule
-
-lemma [eqvt]: "p \<bullet> (Supp x) = Supp (p \<bullet> x)"
-  unfolding Supp_def
-  by perm_simp rule
-
-lemmas s = Supp_Lam[quot_lifted]
-
-lemma var_beta_pre: "s \<longrightarrow>b* r \<Longrightarrow> s = Var name \<Longrightarrow> r = Var name"
-  apply (induct rule: beta_star.induct)
-  apply simp
-  apply (subgoal_tac "M2 = Var name")
-  apply (thin_tac "M1 \<longrightarrow>b* M2")
-  apply (thin_tac "M1 = Var name")
-  apply (thin_tac "M1 = Var name \<Longrightarrow> M2 = Var name")
-  defer
-  apply simp
-  apply (erule_tac beta.cases)
-  apply simp_all
-  done
-
-lemma var_beta: "Var name \<longrightarrow>b* r \<longleftrightarrow> r = Var name"
-  by (auto simp add: var_beta_pre)
-
-lemma qlam_eq_iff:
-  "(QVar n = QVar m) = (n = m)"
-  apply descending unfolding equ_def var_beta by auto
-
-lemma "supp (QVar n) = {atom n}"
-  unfolding supp_def
-  apply simp
-  unfolding qlam_eq_iff
-  apply (fold supp_def)
-  apply (simp add: supp_at_base)
-  done
-
-end
-
-
-