Fixed wrong rename.
--- a/Quot/Examples/LamEx2.thy Mon Feb 01 15:46:25 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,632 +0,0 @@
-theory LamEx
-imports Nominal "../QuotMain" "../QuotList"
-begin
-
-atom_decl name
-
-datatype rlam =
- rVar "name"
-| rApp "rlam" "rlam"
-| rLam "name" "rlam"
-
-fun
- rfv :: "rlam \<Rightarrow> name set"
-where
- rfv_var: "rfv (rVar a) = {a}"
-| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
-| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
-
-overloading
- perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked)
-begin
-
-fun
- perm_rlam
-where
- "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"
-| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"
-| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"
-
-end
-
-declare perm_rlam.simps[eqvt]
-
-instance rlam::pt_name
- apply(default)
- apply(induct_tac [!] x rule: rlam.induct)
- apply(simp_all add: pt_name2 pt_name3)
- done
-
-instance rlam::fs_name
- apply(default)
- apply(induct_tac [!] x rule: rlam.induct)
- apply(simp add: supp_def)
- apply(fold supp_def)
- apply(simp add: supp_atm)
- apply(simp add: supp_def Collect_imp_eq Collect_neg_eq)
- apply(simp add: supp_def)
- apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric])
- apply(fold supp_def)
- 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)
-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::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"
-
-
-
-
-(* 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:
- 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="[]" 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"
-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)
-
-
-quotient_definition
- "Var :: name \<Rightarrow> lam"
-as
- "rVar"
-
-quotient_definition
- "App :: lam \<Rightarrow> lam \<Rightarrow> lam"
-as
- "rApp"
-
-quotient_definition
- "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
-as
- "rLam"
-
-quotient_definition
- "fv :: lam \<Rightarrow> name set"
-as
- "rfv"
-
-(* definition of overloaded permutation function *)
-(* for the lifted type lam *)
-overloading
- perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked)
-begin
-
-quotient_definition
- "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
-as
- "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"
-
-end
-
-lemma perm_rsp[quot_respect]:
- "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
- apply(auto)
- (* this is propably true if some type conditions are imposed ;o) *)
- sorry
-
-lemma fresh_rsp:
- "(op = ===> alpha ===> op =) fresh fresh"
- apply(auto)
- (* this is probably only true if some type conditions are imposed *)
- sorry
-
-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="[]" 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"
-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"
- by (lifting rlam.induct)
-
-lemma perm_lam [simp]:
- fixes pi::"'a prm"
- 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 perm_rlam.simps)
-done
-
-instance lam::pt_name
-apply(default)
-apply(induct_tac [!] x rule: lam_induct)
-apply(simp_all add: pt_name2 pt_name3)
-done
-
-lemma fv_lam [simp]:
- shows "fv (Var a) = {a}"
- and "fv (App t1 t2) = fv t1 \<union> fv t2"
- and "fv (Lam a t) = fv t - {a}"
-apply(lifting rfv_var rfv_app rfv_lam)
-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 a3:
- "\<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>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>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 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 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
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-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)::name set)"
- by (simp add: supp_def)
-
-lemma var_supp:
- shows "(supp (Var a)) = {a::name}"
- using var_supp1 by (simp add: supp_atm)
-
-lemma app_supp:
- shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"
-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
-
-
-instance lam::fs_name
-apply(default)
-apply(induct_tac x rule: lam_induct)
-apply(simp add: var_supp)
-apply(simp add: app_supp)
-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)"
-apply(simp add: fresh_def)
-apply(simp add: lam_supp abs_supp)
-apply(auto)
-done
-
-lemma lam_induct_strong:
- fixes a::"'a::fs_name"
- 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; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
- shows "P a lam"
-proof -
- have "\<And>(pi::name prm) 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::name prm) a. P a (pi \<bullet> lam1)" by fact
- have b2: "\<And>(pi::name prm) 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::name prm) a. P a (pi \<bullet> lam)" by fact
- obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
- apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"])
- apply(simp_all add: fs_name1)
- done
- from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))"
- apply -
- apply(rule a3)
- apply(blast)
- apply(simp)
- done
- have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
- apply(rule perm_fresh_fresh)
- using fr
- apply(simp add: fresh_lam)
- apply(simp add: fresh_lam)
- done
- show "P a (pi \<bullet> Lam name lam)"
- apply (simp)
- apply(subst eq[symmetric])
- using p
- apply(simp only: perm_lam pt_name2 swap_simps)
- done
- qed
- then have "P a (([]::name prm) \<bullet> lam)" by blast
- then show "P a lam" by simp
-qed
-
-
-lemma var_fresh:
- fixes a::"name"
- shows "(a \<sharp> (Var b)) = (a \<sharp> b)"
- apply(simp add: fresh_def)
- apply(simp add: var_supp1)
- done
-
-(* lemma hom_reg: *)
-
-lemma rlam_rec_eqvt:
- fixes pi::"name prm"
- and f1::"name \<Rightarrow> ('a::pt_name)"
- shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
-apply(induct t)
-apply(simp_all)
-apply(simp add: perm_fun_def)
-apply(perm_simp)
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-back
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(simp)
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-back
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(simp)
-done
-
-
-lemma rlam_rec_respects:
- assumes f1: "f_var \<in> Respects (op= ===> op=)"
- and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
- and f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)"
- shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)"
-apply(simp add: mem_def)
-apply(simp add: Respects_def)
-apply(rule allI)
-apply(rule allI)
-apply(rule impI)
-apply(erule alpha.induct)
-apply(simp)
-apply(simp)
-using f2
-apply(simp add: mem_def)
-apply(simp add: Respects_def)
-using f3[simplified mem_def Respects_def]
-apply(simp)
-apply(case_tac "a=b")
-apply(clarify)
-apply(simp)
-(* probably true *)
-sorry
-
-function
- term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
- (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
- ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a"
-where
- "term1_hom var app abs' (rVar x) = (var x)"
-| "term1_hom var app abs' (rApp t u) =
- app t u (term1_hom var app abs' t) (term1_hom var app abs' u)"
-| "term1_hom var app abs' (rLam x u) =
- abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))"
-apply(pat_completeness)
-apply(auto)
-done
-
-lemma pi_size:
- fixes pi::"name prm"
- and t::"rlam"
- shows "size (pi \<bullet> t) = size t"
-apply(induct t)
-apply(auto)
-done
-
-termination term1_hom
- apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)")
-apply(auto simp add: pi_size)
-done
-
-lemma lam_exhaust:
- "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk>
- \<Longrightarrow> P"
-apply(lifting rlam.exhaust)
-done
-
-(* THIS IS NOT TRUE, but it lets prove the existence of the hom function *)
-lemma lam_inject':
- "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))"
-sorry
-
-function
- hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
- (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
- ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a"
-where
- "hom f_var f_app f_lam (Var x) = f_var x"
-| "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)"
-| "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))"
-defer
-apply(simp_all add: lam_inject') (* inject, distinct *)
-apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
-apply(rule refl)
-apply(rule ext)
-apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
-apply simp_all
-apply(erule conjE)+
-apply(rule_tac x="b" in cong)
-apply simp_all
-apply auto
-apply(rule_tac y="b" in lam_exhaust)
-apply simp_all
-apply auto
-apply meson
-apply(simp_all add: lam_inject')
-apply metis
-done
-
-termination hom
- apply -
-(*
-ML_prf {* Size.size_thms @{theory} "LamEx.lam" *}
-*)
-sorry
-
-thm hom.simps
-
-lemma term1_hom_rsp:
- "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk>
- \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)"
-apply(simp)
-apply(rule allI)+
-apply(rule impI)
-apply(erule alpha.induct)
-apply(auto)[1]
-apply(auto)[1]
-apply(simp)
-apply(erule conjE)+
-apply(erule exE)+
-apply(erule conjE)+
-apply(clarify)
-sorry
-
-lemma hom: "
-\<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =).
-\<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =).
-\<exists>hom\<in>Respects (alpha ===> op =).
- ((\<forall>x. hom (rVar x) = f_var x) \<and>
- (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
- (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-apply(rule allI)
-apply(rule ballI)+
-apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI)
-apply(simp_all)
-apply(simp only: in_respects)
-apply(rule term1_hom_rsp)
-apply(assumption)+
-done
-
-lemma hom':
-"\<exists>hom.
- ((\<forall>x. hom (Var x) = f_var x) \<and>
- (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and>
- (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-apply (lifting hom)
-done
-
-(* test test
-lemma raw_hom_correct:
- assumes f1: "f_var \<in> Respects (op= ===> op=)"
- and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
- and f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)"
- shows "\<exists>!hom\<in>Respects (alpha ===> op =).
- ((\<forall>x. hom (rVar x) = f_var x) \<and>
- (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
- (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-unfolding Bex1_def
-apply(rule ex1I)
-sorry
-*)
-
-
-end
-
--- a/Quot/Nominal/LamEx.thy Mon Feb 01 15:46:25 2010 +0100
+++ b/Quot/Nominal/LamEx.thy Mon Feb 01 15:57:37 2010 +0100
@@ -1,5 +1,5 @@
theory LamEx
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "Abs" "../QuotProd"
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "Abs"
begin
@@ -26,6 +26,13 @@
apply(simp)
done
+lemma fresh_minus_perm:
+ fixes p::perm
+ shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
+ apply(simp add: fresh_def)
+ apply(simp only: supp_minus_perm)
+ done
+
lemma fresh_plus:
fixes p q::perm
shows "\<lbrakk>a \<sharp> p; a \<sharp> q\<rbrakk> \<Longrightarrow> a \<sharp> (p + q)"
@@ -146,13 +153,12 @@
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"
-
-thm alpha.induct
+| a3: "\<exists>pi. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)
+ \<Longrightarrow> rLam a t \<approx> rLam b s"
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))"
+ shows "\<exists>pi. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)"
using assms
apply(erule_tac alpha.cases)
apply(auto)
@@ -166,11 +172,11 @@
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(simp add: alpha_gen.simps)
-apply(erule conjE)+
-apply(rule conjI)+
+apply(rule conjI)
apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
apply(simp add: eqvts atom_eqvt)
apply(rule conjI)
@@ -187,43 +193,24 @@
apply(simp add: a2)
apply(rule a3)
apply(rule_tac x="0" in exI)
-apply(rule alpha_gen_refl)
-apply(assumption)
+apply(simp_all add: fresh_star_def fresh_zero_perm)
done
-lemma fresh_minus_perm:
- fixes p::perm
- shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
- apply(simp add: fresh_def)
- apply(simp only: supp_minus_perm)
- done
-
-lemma alpha_gen_atom_sym:
- assumes a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"
- shows "\<exists>pi. ({atom a}, t) \<approx>gen \<lambda>x1 x2. R x1 x2 \<and> R x2 x1 f pi ({atom b}, s) \<Longrightarrow>
- \<exists>pi. ({atom b}, s) \<approx>gen R f pi ({atom a}, t)"
- apply(erule exE)
- apply(rule_tac x="- pi" in exI)
- apply(simp add: alpha_gen.simps)
- apply(erule conjE)+
- apply(rule conjI)
- apply(simp add: fresh_star_def fresh_minus_perm)
- apply(subgoal_tac "R (- pi \<bullet> s) ((- pi) \<bullet> (pi \<bullet> t))")
- apply simp
- apply(rule a)
- 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(rule alpha_gen_atom_sym)
- apply(rule alpha_eqvt)
- apply(assumption)+
- done
+apply(induct rule: alpha.induct)
+apply(simp add: a1)
+apply(simp add: a2)
+apply(rule a3)
+apply(erule exE)
+apply(rule_tac x="- pi" in exI)
+apply(simp)
+apply(simp add: fresh_star_def fresh_minus_perm)
+apply(erule conjE)+
+apply(rotate_tac 3)
+apply(drule_tac pi="- pi" in alpha_eqvt)
+apply(simp)
+done
lemma alpha_trans:
shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
@@ -238,13 +225,11 @@
apply(rotate_tac 1)
apply(erule alpha.cases)
apply(simp_all)
-apply(simp add: alpha_gen.simps)
apply(erule conjE)+
apply(erule exE)+
apply(erule conjE)+
apply(rule a3)
apply(rule_tac x="pia + pi" in exI)
-apply(simp add: alpha_gen.simps)
apply(simp add: fresh_star_plus)
apply(drule_tac x="- pia \<bullet> sa" in spec)
apply(drule mp)
@@ -266,9 +251,89 @@
lemma alpha_rfv:
shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
apply(induct rule: alpha.induct)
- apply(simp_all add: alpha_gen.simps)
+ apply(simp_all)
done
+inductive
+ alpha2 :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100)
+where
+ a21: "a = b \<Longrightarrow> (rVar a) \<approx>2 (rVar b)"
+| a22: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx>2 rApp t2 s2"
+| a23: "(a = b \<and> t \<approx>2 s) \<or> (a \<noteq> b \<and> ((a \<leftrightarrow> b) \<bullet> t) \<approx>2 s \<and> atom b \<notin> rfv t)\<Longrightarrow> rLam a t \<approx>2 rLam b s"
+
+lemma fv_vars:
+ fixes a::name
+ assumes a1: "\<forall>x \<in> rfv t - {atom a}. pi \<bullet> x = x"
+ shows "(pi \<bullet> t) \<approx>2 ((a \<leftrightarrow> (pi \<bullet> a)) \<bullet> t)"
+using a1
+apply(induct t)
+apply(auto)
+apply(rule a21)
+apply(case_tac "name = a")
+apply(simp)
+apply(simp)
+defer
+apply(rule a22)
+apply(simp)
+apply(simp)
+apply(rule a23)
+apply(case_tac "a = name")
+apply(simp)
+oops
+
+
+lemma
+ assumes a1: "t \<approx>2 s"
+ shows "t \<approx> s"
+using a1
+apply(induct)
+apply(rule alpha.intros)
+apply(simp)
+apply(rule alpha.intros)
+apply(simp)
+apply(simp)
+apply(rule alpha.intros)
+apply(erule disjE)
+apply(rule_tac x="0" in exI)
+apply(simp add: fresh_star_def fresh_zero_perm)
+apply(erule conjE)+
+apply(drule alpha_rfv)
+apply(simp)
+apply(rule_tac x="(a \<leftrightarrow> b)" in exI)
+apply(simp)
+apply(erule conjE)+
+apply(rule conjI)
+apply(drule alpha_rfv)
+apply(drule sym)
+apply(simp)
+apply(simp add: rfv_eqvt[symmetric])
+defer
+apply(subgoal_tac "atom a \<sharp> (rfv t - {atom a})")
+apply(subgoal_tac "atom b \<sharp> (rfv t - {atom a})")
+
+defer
+sorry
+
+lemma
+ assumes a1: "t \<approx> s"
+ shows "t \<approx>2 s"
+using a1
+apply(induct)
+apply(rule alpha2.intros)
+apply(simp)
+apply(rule alpha2.intros)
+apply(simp)
+apply(simp)
+apply(clarify)
+apply(rule alpha2.intros)
+apply(frule alpha_rfv)
+apply(rotate_tac 4)
+apply(drule sym)
+apply(simp)
+apply(drule sym)
+apply(simp)
+oops
+
quotient_type lam = rlam / alpha
by (rule alpha_equivp)
@@ -313,7 +378,7 @@
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: fresh_star_def fresh_zero_perm)
apply(simp add: alpha_rfv)
done
@@ -376,60 +441,10 @@
"\<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
-
-lemma pi_rep: "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)
-apply (simp add: alpha_gen.simps)
-apply (simp add: pi_rep)
-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)
-sledgehammer
-sorry
-
-
-lemma a3:
- "\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s) \<Longrightarrow> Lam a t = Lam b s"
- apply (lifting a3)
+lemma a3:
+ "\<lbrakk>\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)\<rbrakk>
+ \<Longrightarrow> Lam a t = Lam b s"
+ apply (lifting a3)
done
lemma a3_inv:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Nominal/LamEx2.thy Mon Feb 01 15:57:37 2010 +0100
@@ -0,0 +1,632 @@
+theory LamEx
+imports Nominal "../QuotMain" "../QuotList"
+begin
+
+atom_decl name
+
+datatype rlam =
+ rVar "name"
+| rApp "rlam" "rlam"
+| rLam "name" "rlam"
+
+fun
+ rfv :: "rlam \<Rightarrow> name set"
+where
+ rfv_var: "rfv (rVar a) = {a}"
+| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
+| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
+
+overloading
+ perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked)
+begin
+
+fun
+ perm_rlam
+where
+ "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"
+| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"
+| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"
+
+end
+
+declare perm_rlam.simps[eqvt]
+
+instance rlam::pt_name
+ apply(default)
+ apply(induct_tac [!] x rule: rlam.induct)
+ apply(simp_all add: pt_name2 pt_name3)
+ done
+
+instance rlam::fs_name
+ apply(default)
+ apply(induct_tac [!] x rule: rlam.induct)
+ apply(simp add: supp_def)
+ apply(fold supp_def)
+ apply(simp add: supp_atm)
+ apply(simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ apply(simp add: supp_def)
+ apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric])
+ apply(fold supp_def)
+ 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)
+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::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"
+
+
+
+
+(* 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:
+ 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="[]" 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"
+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)
+
+
+quotient_definition
+ "Var :: name \<Rightarrow> lam"
+as
+ "rVar"
+
+quotient_definition
+ "App :: lam \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "rApp"
+
+quotient_definition
+ "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "rLam"
+
+quotient_definition
+ "fv :: lam \<Rightarrow> name set"
+as
+ "rfv"
+
+(* definition of overloaded permutation function *)
+(* for the lifted type lam *)
+overloading
+ perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked)
+begin
+
+quotient_definition
+ "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
+as
+ "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"
+
+end
+
+lemma perm_rsp[quot_respect]:
+ "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
+ apply(auto)
+ (* this is propably true if some type conditions are imposed ;o) *)
+ sorry
+
+lemma fresh_rsp:
+ "(op = ===> alpha ===> op =) fresh fresh"
+ apply(auto)
+ (* this is probably only true if some type conditions are imposed *)
+ sorry
+
+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="[]" 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"
+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"
+ by (lifting rlam.induct)
+
+lemma perm_lam [simp]:
+ fixes pi::"'a prm"
+ 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 perm_rlam.simps)
+done
+
+instance lam::pt_name
+apply(default)
+apply(induct_tac [!] x rule: lam_induct)
+apply(simp_all add: pt_name2 pt_name3)
+done
+
+lemma fv_lam [simp]:
+ shows "fv (Var a) = {a}"
+ and "fv (App t1 t2) = fv t1 \<union> fv t2"
+ and "fv (Lam a t) = fv t - {a}"
+apply(lifting rfv_var rfv_app rfv_lam)
+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 a3:
+ "\<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>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>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 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 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
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+apply(erule alpha.cases)
+apply simp_all
+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)::name set)"
+ by (simp add: supp_def)
+
+lemma var_supp:
+ shows "(supp (Var a)) = {a::name}"
+ using var_supp1 by (simp add: supp_atm)
+
+lemma app_supp:
+ shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"
+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
+
+
+instance lam::fs_name
+apply(default)
+apply(induct_tac x rule: lam_induct)
+apply(simp add: var_supp)
+apply(simp add: app_supp)
+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)"
+apply(simp add: fresh_def)
+apply(simp add: lam_supp abs_supp)
+apply(auto)
+done
+
+lemma lam_induct_strong:
+ fixes a::"'a::fs_name"
+ 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; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
+ shows "P a lam"
+proof -
+ have "\<And>(pi::name prm) 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::name prm) a. P a (pi \<bullet> lam1)" by fact
+ have b2: "\<And>(pi::name prm) 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::name prm) a. P a (pi \<bullet> lam)" by fact
+ obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
+ apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"])
+ apply(simp_all add: fs_name1)
+ done
+ from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))"
+ apply -
+ apply(rule a3)
+ apply(blast)
+ apply(simp)
+ done
+ have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
+ apply(rule perm_fresh_fresh)
+ using fr
+ apply(simp add: fresh_lam)
+ apply(simp add: fresh_lam)
+ done
+ show "P a (pi \<bullet> Lam name lam)"
+ apply (simp)
+ apply(subst eq[symmetric])
+ using p
+ apply(simp only: perm_lam pt_name2 swap_simps)
+ done
+ qed
+ then have "P a (([]::name prm) \<bullet> lam)" by blast
+ then show "P a lam" by simp
+qed
+
+
+lemma var_fresh:
+ fixes a::"name"
+ shows "(a \<sharp> (Var b)) = (a \<sharp> b)"
+ apply(simp add: fresh_def)
+ apply(simp add: var_supp1)
+ done
+
+(* lemma hom_reg: *)
+
+lemma rlam_rec_eqvt:
+ fixes pi::"name prm"
+ and f1::"name \<Rightarrow> ('a::pt_name)"
+ shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
+apply(induct t)
+apply(simp_all)
+apply(simp add: perm_fun_def)
+apply(perm_simp)
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+back
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(simp)
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+back
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
+apply(simp)
+done
+
+
+lemma rlam_rec_respects:
+ assumes f1: "f_var \<in> Respects (op= ===> op=)"
+ and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
+ and f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)"
+ shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)"
+apply(simp add: mem_def)
+apply(simp add: Respects_def)
+apply(rule allI)
+apply(rule allI)
+apply(rule impI)
+apply(erule alpha.induct)
+apply(simp)
+apply(simp)
+using f2
+apply(simp add: mem_def)
+apply(simp add: Respects_def)
+using f3[simplified mem_def Respects_def]
+apply(simp)
+apply(case_tac "a=b")
+apply(clarify)
+apply(simp)
+(* probably true *)
+sorry
+
+function
+ term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
+ (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
+ ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a"
+where
+ "term1_hom var app abs' (rVar x) = (var x)"
+| "term1_hom var app abs' (rApp t u) =
+ app t u (term1_hom var app abs' t) (term1_hom var app abs' u)"
+| "term1_hom var app abs' (rLam x u) =
+ abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))"
+apply(pat_completeness)
+apply(auto)
+done
+
+lemma pi_size:
+ fixes pi::"name prm"
+ and t::"rlam"
+ shows "size (pi \<bullet> t) = size t"
+apply(induct t)
+apply(auto)
+done
+
+termination term1_hom
+ apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)")
+apply(auto simp add: pi_size)
+done
+
+lemma lam_exhaust:
+ "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk>
+ \<Longrightarrow> P"
+apply(lifting rlam.exhaust)
+done
+
+(* THIS IS NOT TRUE, but it lets prove the existence of the hom function *)
+lemma lam_inject':
+ "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))"
+sorry
+
+function
+ hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
+ (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
+ ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a"
+where
+ "hom f_var f_app f_lam (Var x) = f_var x"
+| "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)"
+| "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))"
+defer
+apply(simp_all add: lam_inject') (* inject, distinct *)
+apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+apply(rule refl)
+apply(rule ext)
+apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
+apply simp_all
+apply(erule conjE)+
+apply(rule_tac x="b" in cong)
+apply simp_all
+apply auto
+apply(rule_tac y="b" in lam_exhaust)
+apply simp_all
+apply auto
+apply meson
+apply(simp_all add: lam_inject')
+apply metis
+done
+
+termination hom
+ apply -
+(*
+ML_prf {* Size.size_thms @{theory} "LamEx.lam" *}
+*)
+sorry
+
+thm hom.simps
+
+lemma term1_hom_rsp:
+ "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk>
+ \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)"
+apply(simp)
+apply(rule allI)+
+apply(rule impI)
+apply(erule alpha.induct)
+apply(auto)[1]
+apply(auto)[1]
+apply(simp)
+apply(erule conjE)+
+apply(erule exE)+
+apply(erule conjE)+
+apply(clarify)
+sorry
+
+lemma hom: "
+\<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =).
+\<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =).
+\<exists>hom\<in>Respects (alpha ===> op =).
+ ((\<forall>x. hom (rVar x) = f_var x) \<and>
+ (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+apply(rule allI)
+apply(rule ballI)+
+apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI)
+apply(simp_all)
+apply(simp only: in_respects)
+apply(rule term1_hom_rsp)
+apply(assumption)+
+done
+
+lemma hom':
+"\<exists>hom.
+ ((\<forall>x. hom (Var x) = f_var x) \<and>
+ (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+apply (lifting hom)
+done
+
+(* test test
+lemma raw_hom_correct:
+ assumes f1: "f_var \<in> Respects (op= ===> op=)"
+ and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
+ and f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)"
+ shows "\<exists>!hom\<in>Respects (alpha ===> op =).
+ ((\<forall>x. hom (rVar x) = f_var x) \<and>
+ (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
+ (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
+unfolding Bex1_def
+apply(rule ex1I)
+sorry
+*)
+
+
+end
+