used "new" alpha-equivalence relation (according to new scheme); proved equivalence theorems and so on
--- 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)"