diff -r 4b0563bc4b03 -r 7d8949da7d99 Quot/Nominal/LamEx2.thy
--- a/Quot/Nominal/LamEx2.thy	Wed Feb 24 17:32:43 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,567 +0,0 @@
-theory LamEx
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs"
-begin
-
-atom_decl name
-
-datatype rlam =
-  rVar "name"
-| rApp "rlam" "rlam"
-| rLam "name" "rlam"
-
-fun
-  rfv :: "rlam \<Rightarrow> atom set"
-where
-  rfv_var: "rfv (rVar a) = {atom a}"
-| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
-| rfv_lam: "rfv (rLam a t) = (rfv t) - {atom a}"
-
-instantiation rlam :: pt
-begin
-
-primrec
-  permute_rlam
-where
-  "permute_rlam pi (rVar a) = rVar (pi \<bullet> a)"
-| "permute_rlam pi (rApp t1 t2) = rApp (permute_rlam pi t1) (permute_rlam pi t2)"
-| "permute_rlam pi (rLam a t) = rLam (pi \<bullet> a) (permute_rlam pi t)"
-
-instance
-apply default
-apply(induct_tac [!] x)
-apply(simp_all)
-done
-
-end
-
-instantiation rlam :: fs
-begin
-
-lemma neg_conj:
-  "\<not>(P \<and> Q) \<longleftrightarrow> (\<not>P) \<or> (\<not>Q)"
-  by simp
-
-lemma infinite_Un:
-  "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
-  by simp
-
-instance
-apply default
-apply(induct_tac x)
-(* var case *)
-apply(simp add: supp_def)
-apply(fold supp_def)[1]
-apply(simp add: supp_at_base)
-(* app case *)
-apply(simp only: supp_def)
-apply(simp only: permute_rlam.simps) 
-apply(simp only: rlam.inject) 
-apply(simp only: neg_conj)
-apply(simp only: Collect_disj_eq)
-apply(simp only: infinite_Un)
-apply(simp only: Collect_disj_eq)
-apply(simp)
-(* lam case *)
-apply(simp only: supp_def)
-apply(simp only: permute_rlam.simps) 
-apply(simp only: rlam.inject) 
-apply(simp only: neg_conj)
-apply(simp only: Collect_disj_eq)
-apply(simp only: infinite_Un)
-apply(simp only: Collect_disj_eq)
-apply(simp)
-apply(fold supp_def)[1]
-apply(simp add: supp_at_base)
-done
-
-end
-
-
-(* for the eqvt proof of the alpha-equivalence *)
-declare permute_rlam.simps[eqvt]
-
-lemma rfv_eqvt[eqvt]:
-  shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
-apply(induct t)
-apply(simp_all)
-apply(simp add: permute_set_eq atom_eqvt)
-apply(simp add: union_eqvt)
-apply(simp add: Diff_eqvt)
-apply(simp add: permute_set_eq atom_eqvt)
-done
-
-inductive
-    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: "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s)) \<Longrightarrow> rLam a t \<approx> rLam b s"
-print_theorems
-thm alpha.induct
-
-lemma a3_inverse:
-  assumes "rLam a t \<approx> rLam b s"
-  shows "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s))"
-using assms
-apply(erule_tac alpha.cases)
-apply(auto)
-done
-
-text {* should be automatic with new version of eqvt-machinery *}
-lemma alpha_eqvt:
-  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(rule alpha_gen_atom_eqvt)
-apply(rule rfv_eqvt)
-apply assumption
-done
-
-lemma alpha_refl:
-  shows "t \<approx> t"
-apply(induct t rule: rlam.induct)
-apply(simp add: a1)
-apply(simp add: a2)
-apply(rule a3)
-apply(rule_tac x="0" in exI)
-apply(rule alpha_gen_refl)
-apply(assumption)
-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 alpha_gen_compose_sym)
-  apply(erule alpha_eqvt)
-  done
-
-lemma alpha_trans:
-  shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
-apply(induct arbitrary: t3 rule: alpha.induct)
-apply(simp add: a1)
-apply(rotate_tac 4)
-apply(erule alpha.cases)
-apply(simp_all add: a2)
-apply(erule alpha.cases)
-apply(simp_all)
-apply(rule a3)
-apply(erule alpha_gen_compose_trans)
-apply(assumption)
-apply(erule alpha_eqvt)
-done
-
-lemma alpha_equivp:
-  shows "equivp alpha"
-  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_all add: alpha_gen.simps)
-  done
-
-quotient_type lam = rlam / alpha
-  by (rule alpha_equivp)
-
-quotient_definition
-  "Var :: name \<Rightarrow> lam"
-is
-  "rVar"
-
-quotient_definition
-   "App :: lam \<Rightarrow> lam \<Rightarrow> lam"
-is
-  "rApp"
-
-quotient_definition
-  "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
-is
-  "rLam"
-
-quotient_definition
-  "fv :: lam \<Rightarrow> atom set"
-is
-  "rfv"
-
-lemma perm_rsp[quot_respect]:
-  "(op = ===> alpha ===> alpha) permute permute"
-  apply(auto)
-  apply(rule alpha_eqvt)
-  apply(simp)
-  done
-
-lemma rVar_rsp[quot_respect]:
-  "(op = ===> alpha) rVar rVar"
-  by (auto intro: a1)
-
-lemma rApp_rsp[quot_respect]: 
-  "(alpha ===> alpha ===> alpha) rApp rApp"
-  by (auto intro: a2)
-
-lemma rLam_rsp[quot_respect]: 
-  "(op = ===> alpha ===> alpha) rLam rLam"
-  apply(auto)
-  apply(rule a3)
-  apply(rule_tac x="0" in exI)
-  unfolding fresh_star_def 
-  apply(simp add: fresh_star_def fresh_zero_perm alpha_gen.simps)
-  apply(simp add: alpha_rfv)
-  done
-
-lemma rfv_rsp[quot_respect]: 
-  "(alpha ===> op =) rfv rfv"
-apply(simp add: alpha_rfv)
-done
-
-
-section {* lifted theorems *}
-
-lemma lam_induct:
-  "\<lbrakk>\<And>name. P (Var name);
-    \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
-    \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> 
-    \<Longrightarrow> P lam"
-  apply (lifting rlam.induct)
-  done
-
-instantiation lam :: pt
-begin
-
-quotient_definition
-  "permute_lam :: perm \<Rightarrow> lam \<Rightarrow> lam"
-is
-  "permute :: perm \<Rightarrow> rlam \<Rightarrow> rlam"
-
-lemma permute_lam [simp]:
-  shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
-  and   "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
-  and   "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
-apply(lifting permute_rlam.simps)
-done
-
-instance
-apply default
-apply(induct_tac [!] x rule: lam_induct)
-apply(simp_all)
-done
-
-end
-
-lemma fv_lam [simp]: 
-  shows "fv (Var a) = {atom a}"
-  and   "fv (App t1 t2) = fv t1 \<union> fv t2"
-  and   "fv (Lam a t) = fv t - {atom a}"
-apply(lifting rfv_var rfv_app rfv_lam)
-done
-
-lemma fv_eqvt:
-  shows "(p \<bullet> fv t) = fv (p \<bullet> t)"
-apply(lifting rfv_eqvt)
-done
-
-lemma a1: 
-  "a = b \<Longrightarrow> Var a = Var b"
-  by  (lifting a1)
-
-lemma a2: 
-  "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
-  by  (lifting a2)
-
-lemma alpha_gen_rsp_pre:
-  assumes a5: "\<And>t s. R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s)"
-  and     a1: "R s1 t1"
-  and     a2: "R s2 t2"
-  and     a3: "\<And>a b c d. R a b \<Longrightarrow> R c d \<Longrightarrow> R1 a c = R2 b d"
-  and     a4: "\<And>x y. R x y \<Longrightarrow> fv1 x = fv2 y"
-  shows   "(a, s1) \<approx>gen R1 fv1 pi (b, s2) = (a, t1) \<approx>gen R2 fv2 pi (b, t2)"
-apply (simp add: alpha_gen.simps)
-apply (simp only: a4[symmetric, OF a1] a4[symmetric, OF a2])
-apply auto
-apply (subst a3[symmetric])
-apply (rule a5)
-apply (rule a1)
-apply (rule a2)
-apply (assumption)
-apply (subst a3)
-apply (rule a5)
-apply (rule a1)
-apply (rule a2)
-apply (assumption)
-done
-
-lemma [quot_respect]: "(prod_rel op = alpha ===>
-           (alpha ===> alpha ===> op =) ===> (alpha ===> op =) ===> op = ===> prod_rel op = alpha ===> op =)
-           alpha_gen alpha_gen"
-apply simp
-apply clarify
-apply (rule alpha_gen_rsp_pre[of "alpha",OF alpha_eqvt])
-apply auto
-done
-
-(* pi_abs would be also sufficient to prove the next lemma *)
-lemma replam_eqvt: "pi \<bullet> (rep_lam x) = rep_lam (pi \<bullet> x)"
-apply (unfold rep_lam_def)
-sorry
-
-lemma [quot_preserve]: "(prod_fun id rep_lam --->
-           (abs_lam ---> abs_lam ---> id) ---> (abs_lam ---> id) ---> id ---> (prod_fun id rep_lam) ---> id)
-           alpha_gen = alpha_gen"
-apply (simp add: expand_fun_eq alpha_gen.simps Quotient_abs_rep[OF Quotient_lam])
-apply (simp add: replam_eqvt)
-apply (simp only: Quotient_abs_rep[OF Quotient_lam])
-apply auto
-done
-
-
-lemma alpha_prs [quot_preserve]: "(rep_lam ---> rep_lam ---> id) alpha = (op =)"
-apply (simp add: expand_fun_eq)
-apply (simp add: Quotient_rel_rep[OF Quotient_lam])
-done
-
-lemma a3:
-  "\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s) \<Longrightarrow> Lam a t = Lam b s"
-  apply (unfold alpha_gen)
-  apply (lifting a3[unfolded alpha_gen])
-  done
-
-
-lemma a3_inv:
-  "Lam a t = Lam b s \<Longrightarrow> \<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s)"
-  apply (unfold alpha_gen)
-  apply (lifting a3_inverse[unfolded alpha_gen])
-  done
-
-lemma alpha_cases: 
-  "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
-    \<And>t1 t2 s1 s2. \<lbrakk>a1 = App t1 s1; a2 = App t2 s2; t1 = t2; s1 = s2\<rbrakk> \<Longrightarrow> P;
-    \<And>a t b s. \<lbrakk>a1 = Lam a t; a2 = Lam b s; \<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s)\<rbrakk>
-   \<Longrightarrow> P\<rbrakk>
-    \<Longrightarrow> P"
-unfolding alpha_gen
-apply (lifting alpha.cases[unfolded alpha_gen])
-done
-
-(* not sure whether needed *)
-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>a t b s. \<exists>pi. ({atom a}, t) \<approx>gen (\<lambda>x1 x2. x1 = x2 \<and> qxb x1 x2) fv pi ({atom b}, s) \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
-    \<Longrightarrow> qxb qx qxa"
-unfolding alpha_gen by (lifting alpha.induct[unfolded alpha_gen])
-
-(* should they lift automatically *)
-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(regularize)
-prefer 2
-apply(regularize)
-prefer 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
-
-thm a3_inv
-lemma Lam_pseudo_inject:
-  shows "(Lam a t = Lam b s) = (\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s))"
-apply(rule iffI)
-apply(rule a3_inv)
-apply(assumption)
-apply(rule a3)
-apply(assumption)
-done
-
-lemma rlam_distinct:
-  shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"
-  and   "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"
-  and   "\<not>(rVar nam \<approx> rLam nam' rlam')"
-  and   "\<not>(rLam nam' rlam' \<approx> rVar nam)"
-  and   "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"
-  and   "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"
-apply auto
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-apply (erule alpha.cases)
-apply (simp_all only: rlam.distinct)
-done
-
-lemma lam_distinct[simp]:
-  shows "Var nam \<noteq> App lam1' lam2'"
-  and   "App lam1' lam2' \<noteq> Var nam"
-  and   "Var nam \<noteq> Lam nam' lam'"
-  and   "Lam nam' lam' \<noteq> Var nam"
-  and   "App lam1 lam2 \<noteq> Lam nam' lam'"
-  and   "Lam nam' lam' \<noteq> App lam1 lam2"
-apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6))
-done
-
-lemma var_supp1:
-  shows "(supp (Var a)) = (supp a)"
-  apply (simp add: supp_def)
-  done
-
-lemma var_supp:
-  shows "(supp (Var a)) = {a:::name}"
-  using var_supp1 by (simp add: supp_at_base)
-
-lemma app_supp:
-  shows "supp (App t1 t2) = (supp t1) \<union> (supp t2)"
-apply(simp only: supp_def lam_inject)
-apply(simp add: Collect_imp_eq Collect_neg_eq)
-done
-
-(* supp for lam *)
-lemma lam_supp1:
-  shows "(supp (atom x, t)) supports (Lam x t) "
-apply(simp add: supports_def)
-apply(fold fresh_def)
-apply(simp add: fresh_Pair swap_fresh_fresh)
-apply(clarify)
-apply(subst swap_at_base_simps(3))
-apply(simp_all add: fresh_atom)
-done
-
-lemma lam_fsupp1:
-  assumes a: "finite (supp t)"
-  shows "finite (supp (Lam x t))"
-apply(rule supports_finite)
-apply(rule lam_supp1)
-apply(simp add: a supp_Pair supp_atom)
-done
-
-instance lam :: fs
-apply(default)
-apply(induct_tac x rule: lam_induct)
-apply(simp add: var_supp)
-apply(simp add: app_supp)
-apply(simp add: lam_fsupp1)
-done
-
-lemma supp_fv:
-  shows "supp t = fv t"
-apply(induct t rule: lam_induct)
-apply(simp add: var_supp)
-apply(simp add: app_supp)
-apply(subgoal_tac "supp (Lam name lam) = supp (Abs {atom name} lam)")
-apply(simp add: supp_Abs)
-apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
-apply(simp add: Lam_pseudo_inject)
-apply(simp add: Abs_eq_iff)
-apply(simp add: alpha_gen.simps)
-apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric])
-done
-
-lemma lam_supp2:
-  shows "supp (Lam x t) = supp (Abs {atom x} t)"
-apply(simp add: supp_def permute_set_eq atom_eqvt)
-apply(simp add: Lam_pseudo_inject)
-apply(simp add: Abs_eq_iff)
-apply(simp add: alpha_gen  supp_fv)
-done
-
-lemma lam_supp:
-  shows "supp (Lam x t) = ((supp t) - {atom x})"
-apply(simp add: lam_supp2)
-apply(simp add: supp_Abs)
-done
-
-lemma fresh_lam:
-  "(atom a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> atom a \<sharp> t)"
-apply(simp add: fresh_def)
-apply(simp add: lam_supp)
-apply(auto)
-done
-
-lemma lam_induct_strong:
-  fixes a::"'a::fs"
-  assumes a1: "\<And>name b. P b (Var name)"
-  and     a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"
-  and     a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
-  shows "P a lam"
-proof -
-  have "\<And>pi a. P a (pi \<bullet> lam)" 
-  proof (induct lam rule: lam_induct)
-    case (1 name pi)
-    show "P a (pi \<bullet> Var name)"
-      apply (simp)
-      apply (rule a1)
-      done
-  next
-    case (2 lam1 lam2 pi)
-    have b1: "\<And>pi a. P a (pi \<bullet> lam1)" by fact
-    have b2: "\<And>pi a. P a (pi \<bullet> lam2)" by fact
-    show "P a (pi \<bullet> App lam1 lam2)"
-      apply (simp)
-      apply (rule a2)
-      apply (rule b1)
-      apply (rule b2)
-      done
-  next
-    case (3 name lam pi a)
-    have b: "\<And>pi a. P a (pi \<bullet> lam)" by fact
-    obtain c::name where fr: "atom c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
-      apply(rule obtain_atom)
-      apply(auto)
-      sorry
-    from b fr have p: "P a (Lam c (((c \<leftrightarrow> (pi \<bullet> name)) + pi)\<bullet>lam))" 
-      apply -
-      apply(rule a3)
-      apply(blast)
-      apply(simp add: fresh_Pair)
-      done
-    have eq: "(atom c \<rightleftharpoons> atom (pi\<bullet>name)) \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
-      apply(rule swap_fresh_fresh)
-      using fr
-      apply(simp add: fresh_lam fresh_Pair)
-      apply(simp add: fresh_lam fresh_Pair)
-      done
-    show "P a (pi \<bullet> Lam name lam)" 
-      apply (simp)
-      apply(subst eq[symmetric])
-      using p
-      apply(simp only: permute_lam)
-      apply(simp add: flip_def)
-      done
-  qed
-  then have "P a (0 \<bullet> lam)" by blast
-  then show "P a lam" by simp 
-qed
-
-
-lemma var_fresh:
-  fixes a::"name"
-  shows "(atom a \<sharp> (Var b)) = (atom a \<sharp> b)"
-  apply(simp add: fresh_def)
-  apply(simp add: var_supp1)
-  done
-
-
-
-end
-