--- a/Quot/Nominal/LFex.thy Wed Feb 24 10:08:54 2010 +0100
+++ b/Quot/Nominal/LFex.thy Wed Feb 24 10:25:59 2010 +0100
@@ -69,7 +69,7 @@
[ [[]], [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
notation
alpha_rkind ("_ \<approx>ki _" [100, 100] 100)
-and alpha_rty ("_ \<approx>rty _" [100, 100] 100)
+and alpha_rty ("_ \<approx>ty _" [100, 100] 100)
and alpha_rtrm ("_ \<approx>tr _" [100, 100] 100)
thm fv_rkind_fv_rty_fv_rtrm.simps alpha_rkind_alpha_rty_alpha_rtrm.intros
local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alphas_inj}, []), (build_alpha_inj @{thms alpha_rkind_alpha_rty_alpha_rtrm.intros} @{thms rkind.distinct rty.distinct rtrm.distinct rkind.inject rty.inject rtrm.inject} @{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases} ctxt)) ctxt)) *}
@@ -107,56 +107,56 @@
| "fv_rtrm (Lam A x M) = (fv_rty A) \<union> ((fv_rtrm M) - {atom x})"
inductive
- arkind :: "rkind \<Rightarrow> rkind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
-and arty :: "rty \<Rightarrow> rty \<Rightarrow> bool" ("_ \<approx>rty _" [100, 100] 100)
-and artrm :: "rtrm \<Rightarrow> rtrm \<Rightarrow> bool" ("_ \<approx>tr _" [100, 100] 100)
+ alpha_rkind :: "rkind \<Rightarrow> rkind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
+and alpha_rty :: "rty \<Rightarrow> rty \<Rightarrow> bool" ("_ \<approx>ty _" [100, 100] 100)
+and alpha_rtrm :: "rtrm \<Rightarrow> rtrm \<Rightarrow> bool" ("_ \<approx>tr _" [100, 100] 100)
where
a1: "(Type) \<approx>ki (Type)"
-| a2: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, K) \<approx>gen arkind fv_rkind pi ({atom b}, K'))\<rbrakk> \<Longrightarrow> (KPi A a K) \<approx>ki (KPi A' b K')"
-| a3: "i = j \<Longrightarrow> (TConst i) \<approx>rty (TConst j)"
-| a4: "\<lbrakk>A \<approx>rty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>rty (TApp A' M')"
-| a5: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, B) \<approx>gen arty fv_rty pi ({atom b}, B'))\<rbrakk> \<Longrightarrow> (TPi A a B) \<approx>rty (TPi A' b B')"
+| a2: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (({atom a}, K) \<approx>gen alpha_rkind fv_rkind pi ({atom b}, K'))\<rbrakk> \<Longrightarrow> (KPi A a K) \<approx>ki (KPi A' b K')"
+| a3: "i = j \<Longrightarrow> (TConst i) \<approx>ty (TConst j)"
+| a4: "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>ty (TApp A' M')"
+| a5: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (({atom a}, B) \<approx>gen alpha_rty fv_rty pi ({atom b}, B'))\<rbrakk> \<Longrightarrow> (TPi A a B) \<approx>ty (TPi A' b B')"
| a6: "i = j \<Longrightarrow> (Const i) \<approx>tr (Const j)"
| a7: "x = y \<Longrightarrow> (Var x) \<approx>tr (Var y)"
| a8: "\<lbrakk>M \<approx>tr M'; N \<approx>tr N'\<rbrakk> \<Longrightarrow> (App M N) \<approx>tr (App M' N')"
-| a9: "\<lbrakk>A \<approx>rty A'; \<exists>pi. (({atom a}, M) \<approx>gen artrm fv_rtrm pi ({atom b}, M'))\<rbrakk> \<Longrightarrow> (Lam A a M) \<approx>tr (Lam A' b M')"
+| a9: "\<lbrakk>A \<approx>ty A'; \<exists>pi. (({atom a}, M) \<approx>gen alpha_rtrm fv_rtrm pi ({atom b}, M'))\<rbrakk> \<Longrightarrow> (Lam A a M) \<approx>tr (Lam A' b M')"
-lemma arkind_arty_artrm_inj:
+lemma alpha_rkind_alpha_rty_alpha_rtrm_inj:
"(Type) \<approx>ki (Type) = True"
- "((KPi A a K) \<approx>ki (KPi A' b K')) = ((A \<approx>rty A') \<and> (\<exists>pi. ({atom a}, K) \<approx>gen arkind fv_rkind pi ({atom b}, K')))"
- "(TConst i) \<approx>rty (TConst j) = (i = j)"
- "(TApp A M) \<approx>rty (TApp A' M') = (A \<approx>rty A' \<and> M \<approx>tr M')"
- "((TPi A a B) \<approx>rty (TPi A' b B')) = ((A \<approx>rty A') \<and> (\<exists>pi. (({atom a}, B) \<approx>gen arty fv_rty pi ({atom b}, B'))))"
+ "((KPi A a K) \<approx>ki (KPi A' b K')) = ((A \<approx>ty A') \<and> (\<exists>pi. ({atom a}, K) \<approx>gen alpha_rkind fv_rkind pi ({atom b}, K')))"
+ "(TConst i) \<approx>ty (TConst j) = (i = j)"
+ "(TApp A M) \<approx>ty (TApp A' M') = (A \<approx>ty A' \<and> M \<approx>tr M')"
+ "((TPi A a B) \<approx>ty (TPi A' b B')) = ((A \<approx>ty A') \<and> (\<exists>pi. (({atom a}, B) \<approx>gen alpha_rty fv_rty pi ({atom b}, B'))))"
"(Const i) \<approx>tr (Const j) = (i = j)"
"(Var x) \<approx>tr (Var y) = (x = y)"
"(App M N) \<approx>tr (App M' N') = (M \<approx>tr M' \<and> N \<approx>tr N')"
- "((Lam A a M) \<approx>tr (Lam A' b M')) = ((A \<approx>rty A') \<and> (\<exists>pi. (({atom a}, M) \<approx>gen artrm fv_rtrm pi ({atom b}, M'))))"
+ "((Lam A a M) \<approx>tr (Lam A' b M')) = ((A \<approx>ty A') \<and> (\<exists>pi. (({atom a}, M) \<approx>gen alpha_rtrm fv_rtrm pi ({atom b}, M'))))"
apply -
-apply (simp add: arkind_arty_artrm.intros)
+apply (simp add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule arkind.cases) apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rkind.cases) apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule arty.cases) apply simp apply simp apply simp
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rty.cases) apply simp apply simp apply simp
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule arty.cases) apply simp apply simp apply simp
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rty.cases) apply simp apply simp apply simp
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule arty.cases) apply simp apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rty.cases) apply simp apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rtrm.cases) apply simp apply simp apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rtrm.cases) apply simp apply simp apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rtrm.cases) apply simp apply simp apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
-apply rule apply (erule artrm.cases) apply simp apply simp apply simp apply blast
-apply (simp only: arkind_arty_artrm.intros)
+apply rule apply (erule alpha_rtrm.cases) apply simp apply simp apply simp apply blast
+apply (simp only: alpha_rkind_alpha_rty_alpha_rtrm.intros)
done
lemma rfv_eqvt[eqvt]:
@@ -170,11 +170,11 @@
lemma alpha_eqvt:
"t1 \<approx>ki s1 \<Longrightarrow> (pi \<bullet> t1) \<approx>ki (pi \<bullet> s1)"
- "t2 \<approx>rty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>rty (pi \<bullet> s2)"
+ "t2 \<approx>ty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>ty (pi \<bullet> s2)"
"t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
-apply(induct rule: arkind_arty_artrm.inducts)
-apply (simp_all add: arkind_arty_artrm.intros)
-apply (simp_all add: arkind_arty_artrm_inj)
+apply(induct rule: alpha_rkind_alpha_rty_alpha_rtrm.inducts)
+apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply (rule alpha_gen_atom_eqvt)
apply (simp add: rfv_eqvt)
apply assumption
@@ -191,19 +191,19 @@
and A::"rty"
and M::"rtrm"
shows "K \<approx>ki K"
- and "A \<approx>rty A"
+ and "A \<approx>ty A"
and "M \<approx>tr M"
apply(induct K and A and M rule: rkind_rty_rtrm.inducts)
- apply(auto intro: arkind_arty_artrm.intros)
- apply (rule a2)
+ apply(auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(2))
apply auto
apply(rule_tac x="0" in exI)
apply(simp_all add: fresh_star_def fresh_zero_perm alpha_gen)
- apply (rule a5)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(5))
apply auto
apply(rule_tac x="0" in exI)
apply(simp_all add: fresh_star_def fresh_zero_perm alpha_gen)
- apply (rule a9)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(9))
apply auto
apply(rule_tac x="0" in exI)
apply(simp_all add: fresh_star_def fresh_zero_perm alpha_gen)
@@ -212,11 +212,11 @@
lemma al_sym:
fixes K K'::"rkind" and A A'::"rty" and M M'::"rtrm"
shows "K \<approx>ki K' \<Longrightarrow> K' \<approx>ki K"
- and "A \<approx>rty A' \<Longrightarrow> A' \<approx>rty A"
+ and "A \<approx>ty A' \<Longrightarrow> A' \<approx>ty A"
and "M \<approx>tr M' \<Longrightarrow> M' \<approx>tr M"
- apply(induct K K' and A A' and M M' rule: arkind_arty_artrm.inducts)
- apply(simp_all add: arkind_arty_artrm.intros)
- apply (simp_all add: arkind_arty_artrm_inj)
+ apply(induct K K' and A A' and M M' rule: alpha_rkind_alpha_rty_alpha_rtrm.inducts)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply(erule alpha_gen_compose_sym)
apply(erule alpha_eqvt)
apply(erule alpha_gen_compose_sym)
@@ -228,42 +228,42 @@
lemma al_trans:
fixes K K' K''::"rkind" and A A' A''::"rty" and M M' M''::"rtrm"
shows "K \<approx>ki K' \<Longrightarrow> K' \<approx>ki K'' \<Longrightarrow> K \<approx>ki K''"
- and "A \<approx>rty A' \<Longrightarrow> A' \<approx>rty A'' \<Longrightarrow> A \<approx>rty A''"
+ and "A \<approx>ty A' \<Longrightarrow> A' \<approx>ty A'' \<Longrightarrow> A \<approx>ty A''"
and "M \<approx>tr M' \<Longrightarrow> M' \<approx>tr M'' \<Longrightarrow> M \<approx>tr M''"
- apply(induct K K' and A A' and M M' arbitrary: K'' and A'' and M'' rule: arkind_arty_artrm.inducts)
- apply(simp_all add: arkind_arty_artrm.intros)
- apply(erule arkind.cases)
- apply(simp_all add: arkind_arty_artrm.intros)
- apply(simp add: arkind_arty_artrm_inj)
+ apply(induct K K' and A A' and M M' arbitrary: K'' and A'' and M'' rule: alpha_rkind_alpha_rty_alpha_rtrm.inducts)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply(erule alpha_rkind.cases)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply(simp add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply(erule alpha_gen_compose_trans)
apply(assumption)
apply(erule alpha_eqvt)
apply(rotate_tac 4)
- apply(erule arty.cases)
- apply(simp_all add: arkind_arty_artrm.intros)
+ apply(erule alpha_rty.cases)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
apply(rotate_tac 3)
- apply(erule arty.cases)
- apply(simp_all add: arkind_arty_artrm.intros)
- apply(simp add: arkind_arty_artrm_inj)
+ apply(erule alpha_rty.cases)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply(simp add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply(erule alpha_gen_compose_trans)
apply(assumption)
apply(erule alpha_eqvt)
apply(rotate_tac 4)
- apply(erule artrm.cases)
- apply(simp_all add: arkind_arty_artrm.intros)
+ apply(erule alpha_rtrm.cases)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
apply(rotate_tac 3)
- apply(erule artrm.cases)
- apply(simp_all add: arkind_arty_artrm.intros)
- apply(simp add: arkind_arty_artrm_inj)
+ apply(erule alpha_rtrm.cases)
+ apply(simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply(simp add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply(erule alpha_gen_compose_trans)
apply(assumption)
apply(erule alpha_eqvt)
done
lemma alpha_equivps:
- shows "equivp arkind"
- and "equivp arty"
- and "equivp artrm"
+ shows "equivp alpha_rkind"
+ and "equivp alpha_rty"
+ and "equivp alpha_rtrm"
apply(rule equivpI)
unfolding reflp_def symp_def transp_def
apply(auto intro: al_refl al_sym al_trans)
@@ -275,12 +275,12 @@
apply(auto intro: al_refl al_sym al_trans)
done
-quotient_type RKIND = rkind / arkind
+quotient_type RKIND = rkind / alpha_rkind
by (rule alpha_equivps)
quotient_type
- RTY = rty / arty and
- RTRM = rtrm / artrm
+ RTY = rty / alpha_rty and
+ RTRM = rtrm / alpha_rtrm
by (auto intro: alpha_equivps)
quotient_definition
@@ -347,65 +347,65 @@
lemma alpha_rfv:
shows "(t \<approx>ki s \<longrightarrow> fv_rkind t = fv_rkind s) \<and>
- (t1 \<approx>rty s1 \<longrightarrow> fv_rty t1 = fv_rty s1) \<and>
+ (t1 \<approx>ty s1 \<longrightarrow> fv_rty t1 = fv_rty s1) \<and>
(t2 \<approx>tr s2 \<longrightarrow> fv_rtrm t2 = fv_rtrm s2)"
- apply(rule arkind_arty_artrm.induct)
+ apply(rule alpha_rkind_alpha_rty_alpha_rtrm.induct)
apply(simp_all add: alpha_gen)
done
lemma perm_rsp[quot_respect]:
- "(op = ===> arkind ===> arkind) permute permute"
- "(op = ===> arty ===> arty) permute permute"
- "(op = ===> artrm ===> artrm) permute permute"
+ "(op = ===> alpha_rkind ===> alpha_rkind) permute permute"
+ "(op = ===> alpha_rty ===> alpha_rty) permute permute"
+ "(op = ===> alpha_rtrm ===> alpha_rtrm) permute permute"
by (simp_all add:alpha_eqvt)
lemma tconst_rsp[quot_respect]:
- "(op = ===> arty) TConst TConst"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+ "(op = ===> alpha_rty) TConst TConst"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma tapp_rsp[quot_respect]:
- "(arty ===> artrm ===> arty) TApp TApp"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+ "(alpha_rty ===> alpha_rtrm ===> alpha_rty) TApp TApp"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma var_rsp[quot_respect]:
- "(op = ===> artrm) Var Var"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+ "(op = ===> alpha_rtrm) Var Var"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma app_rsp[quot_respect]:
- "(artrm ===> artrm ===> artrm) App App"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+ "(alpha_rtrm ===> alpha_rtrm ===> alpha_rtrm) App App"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma const_rsp[quot_respect]:
- "(op = ===> artrm) Const Const"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9) done
+ "(op = ===> alpha_rtrm) Const Const"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma kpi_rsp[quot_respect]:
- "(arty ===> op = ===> arkind ===> arkind) KPi KPi"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
- apply (rule a2) apply assumption
+ "(alpha_rty ===> op = ===> alpha_rkind ===> alpha_rkind) KPi KPi"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(2)) apply simp_all
apply (rule_tac x="0" in exI)
apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
done
lemma tpi_rsp[quot_respect]:
- "(arty ===> op = ===> arty ===> arty) TPi TPi"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
- apply (rule a5) apply assumption
+ "(alpha_rty ===> op = ===> alpha_rty ===> alpha_rty) TPi TPi"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(5)) apply simp_all
apply (rule_tac x="0" in exI)
apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
done
lemma lam_rsp[quot_respect]:
- "(arty ===> op = ===> artrm ===> artrm) Lam Lam"
- apply (auto intro: a1 a2 a3 a4 a5 a6 a7 a8 a9)
- apply (rule a9) apply assumption
+ "(alpha_rty ===> op = ===> alpha_rtrm ===> alpha_rtrm) Lam Lam"
+ apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
+ apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(9)) apply simp_all
apply (rule_tac x="0" in exI)
apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
done
lemma fv_rty_rsp[quot_respect]:
- "(arty ===> op =) fv_rty fv_rty"
+ "(alpha_rty ===> op =) fv_rty fv_rty"
by (simp add: alpha_rfv)
lemma fv_rkind_rsp[quot_respect]:
- "(arkind ===> op =) fv_rkind fv_rkind"
+ "(alpha_rkind ===> op =) fv_rkind fv_rkind"
by (simp add: alpha_rfv)
lemma fv_rtrm_rsp[quot_respect]:
- "(artrm ===> op =) fv_rtrm fv_rtrm"
+ "(alpha_rtrm ===> op =) fv_rtrm fv_rtrm"
by (simp add: alpha_rfv)
thm rkind_rty_rtrm.induct
@@ -454,9 +454,9 @@
end
-lemmas ARKIND_ARTY_ARTRM_inducts = arkind_arty_artrm.inducts[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+lemmas ALPHA_RKIND_ALPHA_RTY_ALPHA_RTRM_inducts = alpha_rkind_alpha_rty_alpha_rtrm.inducts[unfolded alpha_gen, quot_lifted, folded alpha_gen]
-lemmas RKIND_RTY_RTRM_INJECT = arkind_arty_artrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
+lemmas RKIND_RTY_RTRM_INJECT = alpha_rkind_alpha_rty_alpha_rtrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
lemmas fv_kind_ty_trm = fv_rkind_fv_rty_fv_rtrm.simps[quot_lifted]