--- a/src/HOL/Nominal/Examples/SOS.thy Mon Mar 19 15:58:02 2007 +0100
+++ b/src/HOL/Nominal/Examples/SOS.thy Mon Mar 19 19:28:27 2007 +0100
@@ -1,10 +1,12 @@
(* "$Id$" *)
(* *)
-(* Formalisation of some typical SOS-proofs from a *)
-(* challenge suggested by Adam Chlipala. *)
+(* Formalisation of some typical SOS-proofs *)
(* *)
-(* We thank Nick Benton for hellping us with the *)
-(* termination-proof for evaluation . *)
+(* This work arose from challenge suggested by Adam *)
+(* Chlipala suggested on the POPLmark mailing list. *)
+(* *)
+(* We thank Nick Benton for helping us with the *)
+(* termination-proof for evaluation. *)
theory SOS
imports "Nominal"
@@ -87,6 +89,8 @@
by (induct rule: lookup.induct)
(auto simp add: fresh_list_cons fresh_prod fresh_atm)
+text {* Parallel Substitution *}
+
consts
psubst :: "(name\<times>trm) list \<Rightarrow> trm \<Rightarrow> trm" ("_<_>" [95,95] 105)
@@ -101,20 +105,19 @@
"\<theta><(InL e)> = InL (\<theta><e>)"
"\<theta><(InR e)> = InR (\<theta><e>)"
"\<lbrakk>y\<noteq>x; x\<sharp>(e,e\<^isub>2,\<theta>); y\<sharp>(e,e\<^isub>1,\<theta>)\<rbrakk>
- \<Longrightarrow> \<theta><(Case e of inl x \<rightarrow> e\<^isub>1 | inr y \<rightarrow> e\<^isub>2)> =
- (Case (\<theta><e>) of inl x \<rightarrow> (\<theta><e\<^isub>1>) | inr y \<rightarrow> (\<theta><e\<^isub>2>))"
- apply(finite_guess add: lookup_eqvt)+
- apply(rule TrueI)+
- apply(simp add: abs_fresh)+
- apply(fresh_guess add: fs_name1 lookup_eqvt)+
- done
+ \<Longrightarrow> \<theta><Case e of inl x \<rightarrow> e\<^isub>1 | inr y \<rightarrow> e\<^isub>2> = (Case (\<theta><e>) of inl x \<rightarrow> (\<theta><e\<^isub>1>) | inr y \<rightarrow> (\<theta><e\<^isub>2>))"
+apply(finite_guess add: lookup_eqvt)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)+
+apply(fresh_guess add: fs_name1 lookup_eqvt)+
+done
lemma psubst_eqvt[eqvt]:
fixes pi::"name prm"
and t::"trm"
shows "pi\<bullet>(\<theta><t>) = (pi\<bullet>\<theta>)<(pi\<bullet>t)>"
- by (nominal_induct t avoiding: \<theta> rule: trm.induct)
- (perm_simp add: fresh_bij lookup_eqvt)+
+by (nominal_induct t avoiding: \<theta> rule: trm.induct)
+ (perm_simp add: fresh_bij lookup_eqvt)+
lemma fresh_psubst:
fixes z::"name"
@@ -142,7 +145,7 @@
and "\<lbrakk>z\<noteq>x; x\<sharp>(y,e,e\<^isub>2,t'); z\<sharp>(y,e,e\<^isub>1,t')\<rbrakk>
\<Longrightarrow> (Case e of inl x \<rightarrow> e\<^isub>1 | inr z \<rightarrow> e\<^isub>2)[y::=t'] =
(Case (e[y::=t']) of inl x \<rightarrow> (e\<^isub>1[y::=t']) | inr z \<rightarrow> (e\<^isub>2[y::=t']))"
- by (simp_all add: fresh_list_cons fresh_list_nil)
+by (simp_all add: fresh_list_cons fresh_list_nil)
lemma subst_eqvt[eqvt]:
fixes pi::"name prm"
@@ -151,60 +154,6 @@
by (nominal_induct t avoiding: x t' rule: trm.induct)
(perm_simp add: fresh_bij)+
-
-lemma subst_rename:
- fixes c::"name"
- and t\<^isub>1::"trm"
- assumes "c\<sharp>t\<^isub>1"
- shows "t\<^isub>1[a::=t\<^isub>2] = ([(c,a)]\<bullet>t\<^isub>1)[c::=t\<^isub>2]"
- using assms
- apply(nominal_induct t\<^isub>1 avoiding: a c t\<^isub>2 rule: trm.induct)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)
- apply(simp (no_asm_use))
- apply(rule sym)
- apply(rule trans)
- apply(rule subst)
- apply(simp add: perm_bij)
- apply(simp add: fresh_prod)
- apply(simp add: fresh_bij)
- apply(simp add: calc_atm fresh_atm)
- apply(simp add: fresh_prod)
- apply(simp add: fresh_bij)
- apply(simp add: calc_atm fresh_atm)
- apply(rule sym)
- apply(rule trans)
- apply(rule subst)
- apply(simp add: fresh_atm)
- apply(simp)
- apply(simp)
- apply(simp (no_asm_use) add: trm.inject)
- apply(rule conjI)
- apply(blast)
- apply(rule conjI)
- apply(rotate_tac 12)
- apply(drule_tac x="a" in meta_spec)
- apply(rotate_tac 14)
- apply(drule_tac x="c" in meta_spec)
- apply(rotate_tac 14)
- apply(drule_tac x="t\<^isub>2" in meta_spec)
- apply(simp add: calc_atm fresh_atm alpha abs_fresh)
- apply(rotate_tac 13)
- apply(drule_tac x="a" in meta_spec)
- apply(rotate_tac 14)
- apply(drule_tac x="c" in meta_spec)
- apply(rotate_tac 14)
- apply(drule_tac x="t\<^isub>2" in meta_spec)
- apply(simp add: calc_atm fresh_atm alpha abs_fresh)
- done
-
lemma fresh_subst:
fixes z::"name"
and t\<^isub>1::"trm"
@@ -234,24 +183,6 @@
by (nominal_induct L avoiding: x P rule: trm.induct)
(auto simp add: fresh_atm abs_fresh)
-lemma subst_fun_eq:
- fixes u::trm
- assumes "[x].t\<^isub>1 = [y].t\<^isub>2"
- shows "t\<^isub>1[x::=u] = t\<^isub>2[y::=u]"
-proof -
- {
- assume "x=y" and "t\<^isub>1=t\<^isub>2"
- then have ?thesis using assms by simp
- }
- moreover
- {
- assume h1:"x \<noteq> y" and h2:"t\<^isub>1=[(x,y)]\<bullet>t\<^isub>2" and h3:"x \<sharp> t\<^isub>2"
- then have "([(x,y)]\<bullet>t\<^isub>2)[x::=u] = t\<^isub>2[y::=u]" by (simp add: subst_rename)
- then have ?thesis using h2 by simp
- }
- ultimately show ?thesis using alpha assms by blast
-qed
-
lemma psubst_empty[simp]:
shows "[]<t> = t"
by (nominal_induct t rule: trm.induct, auto simp add:fresh_list_nil)
@@ -271,7 +202,7 @@
by (nominal_induct t avoiding: a e rule: trm.induct)
(auto simp add: fresh_atm abs_fresh fresh_nat)
-text {* Typing *}
+text {* Typing-Judgements *}
inductive2
valid :: "(name \<times> 'a::pt_name) list \<Rightarrow> bool"
@@ -335,7 +266,7 @@
(x\<^isub>1,Data(S\<^isub>1))#\<Gamma> \<turnstile> e\<^isub>1 : T; (x\<^isub>2,Data(S\<^isub>2))#\<Gamma> \<turnstile> e\<^isub>2 : T\<rbrakk>
\<Longrightarrow> \<Gamma> \<turnstile> (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2) : T"
-lemma typing_valid:
+lemma typing_implies_valid:
assumes "\<Gamma> \<turnstile> t : T"
shows "valid \<Gamma>"
using assms
@@ -535,7 +466,7 @@
have "[(x\<^isub>1',c)]\<bullet>((x\<^isub>1',Data \<sigma>\<^isub>1)# \<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1' : T" using h1 typing_eqvt by blast
then have x:"(c,Data \<sigma>\<^isub>1)#( [(x\<^isub>1',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1': T" using f'
by (auto simp add: fresh_atm calc_atm)
- have "x\<^isub>1' \<sharp> \<Gamma>" using h1 typing_valid by auto
+ have "x\<^isub>1' \<sharp> \<Gamma>" using h1 typing_implies_valid by auto
then have "(c,Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> [(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f' x e1' by (auto simp add: perm_fresh_fresh)
then have "[(x\<^isub>1,c)]\<bullet>((c,Data \<sigma>\<^isub>1)#\<Gamma>) \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using typing_eqvt by blast
then have "([(x\<^isub>1,c)]\<bullet>(c,Data \<sigma>\<^isub>1)) #\<Gamma> \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f f'
@@ -548,7 +479,7 @@
have "[(x\<^isub>2',c)]\<bullet>((x\<^isub>2',Data \<sigma>\<^isub>2)# \<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2' : T" using h2 typing_eqvt by blast
then have x:"(c,Data \<sigma>\<^isub>2)#([(x\<^isub>2',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2': T" using f'
by (auto simp add: fresh_atm calc_atm)
- have "x\<^isub>2' \<sharp> \<Gamma>" using h2 typing_valid by auto
+ have "x\<^isub>2' \<sharp> \<Gamma>" using h2 typing_implies_valid by auto
then have "(c,Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> [(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f' x e2' by (auto simp add: perm_fresh_fresh)
then have "[(x\<^isub>2,c)]\<bullet>((c,Data \<sigma>\<^isub>2)#\<Gamma>) \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using typing_eqvt by blast
then have "([(x\<^isub>2,c)]\<bullet>(c,Data \<sigma>\<^isub>2))#\<Gamma> \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f f'
@@ -589,7 +520,7 @@
assumes a: "(x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma> \<turnstile> e : T"
shows "(x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma> \<turnstile> e : T"
proof -
- from a have "valid ((x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma>)" by (simp add: typing_valid)
+ from a have "valid ((x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma>)" by (simp add: typing_implies_valid)
then have "x\<^isub>1\<noteq>x\<^isub>2" "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" "valid \<Gamma>"
by (auto simp: fresh_list_cons fresh_atm[symmetric])
then have "valid ((x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma>)"
@@ -607,7 +538,6 @@
thus "t\<^isub>1=t\<^isub>2" by (simp only: type_unicity_in_context)
qed
-
lemma typing_substitution:
fixes \<Gamma>::"(name \<times> ty) list"
assumes "(x,T')#\<Gamma> \<turnstile> e : T"
@@ -630,7 +560,7 @@
have "(y,T) \<in> set ((x,T')#\<Gamma>)" using h1 by auto
then have "(y,T) \<in> set \<Gamma>" using as by auto
moreover
- have "valid \<Gamma>" using h2 by (simp only: typing_valid)
+ have "valid \<Gamma>" using h2 by (simp only: typing_implies_valid)
ultimately show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T" using as by (simp add: t_Var)
qed
next
@@ -642,7 +572,7 @@
using vc by (auto simp add: fresh_list_cons)
then have pr2'':"(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" by (simp add: context_exchange)
have ih: "\<lbrakk>(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2; (y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> (y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" by fact
- have "valid \<Gamma>" using pr1 by (simp add: typing_valid)
+ have "valid \<Gamma>" using pr1 by (simp add: typing_implies_valid)
then have "valid ((y,T\<^isub>1)#\<Gamma>)" using vc by auto
then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using pr1 by (auto intro: weakening)
then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" using ih pr2'' by simp
@@ -667,11 +597,11 @@
from h3 have h3': "(x,T')#(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3 : T" by (rule context_exchange)
have "\<Gamma> \<turnstile> t\<^isub>1[x::=e'] : Data (DSum S\<^isub>1 S\<^isub>2)" using h1 ih1 as1 by simp
moreover
- have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using h2' by (auto dest: typing_valid)
+ have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using h2' by (auto dest: typing_implies_valid)
then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2[x::=e'] : T" using ih2 h2' by simp
moreover
- have "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using h3' by (auto dest: typing_valid)
+ have "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using h3' by (auto dest: typing_implies_valid)
then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3[x::=e'] : T" using ih3 h3' by simp
ultimately have "\<Gamma> \<turnstile> Case (t\<^isub>1[x::=e']) of inl x\<^isub>1 \<rightarrow> (t\<^isub>2[x::=e']) | inr x\<^isub>2 \<rightarrow> (t3[x::=e']) : T"
@@ -699,6 +629,27 @@
nominal_inductive big
+lemma big_eqvt[eqvt]:
+ fixes pi::"name prm"
+ assumes a: "t \<Down> t'"
+ shows "(pi\<bullet>t) \<Down> (pi\<bullet>t')"
+ using a
+ apply(induct)
+ apply(auto simp add: eqvt)
+ apply(rule_tac x="pi\<bullet>x" in b_App)
+ apply(auto simp add: eqvt fresh_bij fresh_prod)
+ done
+
+lemma big_eqvt':
+ fixes pi::"name prm"
+ assumes a: "(pi\<bullet>t) \<Down> (pi\<bullet>t')"
+ shows "t \<Down> t'"
+using a
+apply -
+apply(drule_tac pi="rev pi" in big_eqvt)
+apply(perm_simp)
+done
+
lemma fresh_preserved:
fixes x::name
fixes t::trm
@@ -732,24 +683,20 @@
assumes a: "t \<Down> t'"
and a1: "\<And>x e c. P c (Lam [x].e) (Lam [x].e)"
and a2: "\<And>x e\<^isub>1 e\<^isub>2 e\<^isub>2' e' e\<^isub>1' c.
- \<lbrakk>x\<sharp>(e\<^isub>1,e\<^isub>2,e',c); e\<^isub>1\<Down>Lam [x].e\<^isub>1'; (\<And>c. P c e\<^isub>1 (Lam [x].e\<^isub>1'));
- e\<^isub>2\<Down>e\<^isub>2'; (\<And>c. P c e\<^isub>2 e\<^isub>2'); e\<^isub>1'[x::=e\<^isub>2']\<Down>e'; (\<And>c. P c (e\<^isub>1'[x::=e\<^isub>2']) e')\<rbrakk>
+ \<lbrakk>x\<sharp>(e\<^isub>1,e\<^isub>2,e',c); \<And>c. P c e\<^isub>1 (Lam [x].e\<^isub>1'); \<And>c. P c e\<^isub>2 e\<^isub>2'; \<And>c. P c (e\<^isub>1'[x::=e\<^isub>2']) e'\<rbrakk>
\<Longrightarrow> P c (App e\<^isub>1 e\<^isub>2) e'"
and a3: "\<And>n c. P c (Const n) (Const n)"
- and a4: "\<And>e\<^isub>1 e\<^isub>1' e\<^isub>2 e\<^isub>2' c.
- \<lbrakk>e\<^isub>1 \<Down> e\<^isub>1'; (\<And>c. P c e\<^isub>1 e\<^isub>1'); e\<^isub>2 \<Down> e\<^isub>2'; (\<And>c. P c e\<^isub>2 e\<^isub>2')\<rbrakk>
+ and a4: "\<And>e\<^isub>1 e\<^isub>1' e\<^isub>2 e\<^isub>2' c. \<lbrakk>\<And>c. P c e\<^isub>1 e\<^isub>1'; \<And>c. P c e\<^isub>2 e\<^isub>2'\<rbrakk>
\<Longrightarrow> P c (Pr e\<^isub>1 e\<^isub>2) (Pr e\<^isub>1' e\<^isub>2')"
- and a5: "\<And>e e\<^isub>1 e\<^isub>2 c. \<lbrakk>e \<Down> Pr e\<^isub>1 e\<^isub>2; (\<And>c. P c e (Pr e\<^isub>1 e\<^isub>2))\<rbrakk> \<Longrightarrow> P c (Fst e) e\<^isub>1"
- and a6: "\<And>e e\<^isub>1 e\<^isub>2 c. \<lbrakk>e \<Down> Pr e\<^isub>1 e\<^isub>2; (\<And>c. P c e (Pr e\<^isub>1 e\<^isub>2))\<rbrakk> \<Longrightarrow> P c (Snd e) e\<^isub>2"
- and a7: "\<And>e e' c. \<lbrakk>e \<Down> e'; (\<And>c. P c e e')\<rbrakk> \<Longrightarrow> P c (InL e) (InL e')"
- and a8: "\<And>e e' c. \<lbrakk>e \<Down> e'; (\<And>c. P c e e')\<rbrakk> \<Longrightarrow> P c (InR e) (InR e')"
+ and a5: "\<And>e e\<^isub>1 e\<^isub>2 c. \<lbrakk>\<And>c. P c e (Pr e\<^isub>1 e\<^isub>2)\<rbrakk> \<Longrightarrow> P c (Fst e) e\<^isub>1"
+ and a6: "\<And>e e\<^isub>1 e\<^isub>2 c. \<lbrakk>\<And>c. P c e (Pr e\<^isub>1 e\<^isub>2)\<rbrakk> \<Longrightarrow> P c (Snd e) e\<^isub>2"
+ and a7: "\<And>e e' c. \<lbrakk>\<And>c. P c e e'\<rbrakk> \<Longrightarrow> P c (InL e) (InL e')"
+ and a8: "\<And>e e' c. \<lbrakk>\<And>c. P c e e'\<rbrakk> \<Longrightarrow> P c (InR e) (InR e')"
and a9: "\<And>x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' c.
- \<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2,c); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1,c); e \<Down> InL e'; (\<And>c. P c e (InL e'));
- e\<^isub>1[x\<^isub>1::=e'] \<Down> e''; (\<And>c. P c (e\<^isub>1[x\<^isub>1::=e']) e'')\<rbrakk>
+ \<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2,c); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1,c); \<And>c. P c e (InL e'); \<And>c. P c (e\<^isub>1[x\<^isub>1::=e']) e''\<rbrakk>
\<Longrightarrow> P c (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2) e''"
and a10:"\<And>x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' c.
- \<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2,c); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1,c); e \<Down> InR e'; (\<And>c. P c e (InR e'));
- e\<^isub>2[x\<^isub>2::=e'] \<Down> e''; (\<And>c. P c (e\<^isub>2[x\<^isub>2::=e']) e'')\<rbrakk>
+ \<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2,c); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1,c); \<And>c. P c e (InR e'); \<And>c. P c (e\<^isub>2[x\<^isub>2::=e']) e''\<rbrakk>
\<Longrightarrow> P c (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2) e''"
shows "P c t t'"
proof -
@@ -767,16 +714,6 @@
have f1: "(pi\<bullet>x)\<sharp>(pi\<bullet>e\<^isub>1,pi\<bullet>e\<^isub>2,pi\<bullet>e')" using f0 by (simp add: fresh_bij)
have f2: "y\<sharp>(?pi'\<bullet>e\<^isub>1,?pi'\<bullet>e\<^isub>2,?pi'\<bullet>e')" using f0
by (auto simp add: pt_name2 fresh_left calc_atm perm_pi_simp fresh_prod)
- have p1: "e\<^isub>1 \<Down> Lam [x].e\<^isub>1'" by fact
- then have "(?pi'\<bullet>e\<^isub>1)\<Down>(?pi'\<bullet>Lam [x].e\<^isub>1')" by (simp only: big_eqvt)
- moreover
- have p2: "e\<^isub>2 \<Down> e\<^isub>2'" by fact
- then have "(?pi'\<bullet>e\<^isub>2) \<Down> (?pi'\<bullet>e\<^isub>2')" by (simp only: big_eqvt)
- moreover
- have p3: "e\<^isub>1'[x::=e\<^isub>2'] \<Down> e'" by fact
- then have "(?pi'\<bullet>(e\<^isub>1'[x::=e\<^isub>2'])) \<Down> (?pi'\<bullet>e')" by (simp only: big_eqvt)
- then have "(?pi'\<bullet>e\<^isub>1')[y::=(?pi'\<bullet>e\<^isub>2')] \<Down> (?pi'\<bullet>e')" by (simp add: subst_eqvt calc_atm)
- moreover
have ih1: "\<And>c. P c (?pi'\<bullet>e\<^isub>1) (?pi'\<bullet>(Lam [x].e\<^isub>1'))" by fact
then have "\<And>c. P c (?pi'\<bullet>e\<^isub>1) (Lam [y].(?pi'\<bullet>e\<^isub>1'))" by (simp add: calc_atm)
moreover
@@ -799,36 +736,25 @@
show "P c (pi\<bullet>(Const n)) (pi\<bullet>(Const n))" using a3 by simp
next
case (b_Pr e\<^isub>1 e\<^isub>1' e\<^isub>2 e\<^isub>2' pi c)
- then show "P c (pi\<bullet>(Pr e\<^isub>1 e\<^isub>2)) (pi\<bullet>(Pr e\<^isub>1' e\<^isub>2'))" using a4
- by (simp, blast intro: big_eqvt)
+ then show "P c (pi\<bullet>(Pr e\<^isub>1 e\<^isub>2)) (pi\<bullet>(Pr e\<^isub>1' e\<^isub>2'))" using a4 by simp
next
case (b_Fst e e\<^isub>1 e\<^isub>2 pi c)
- have p1: "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
- then have "(pi\<bullet>e)\<Down>(pi\<bullet>(Pr e\<^isub>1 e\<^isub>2))" by (simp only: big_eqvt)
- moreover
- have ih1: "\<And>c. P c (pi\<bullet>e) (pi\<bullet>(Pr e\<^isub>1 e\<^isub>2))" by fact
- ultimately show "P c (pi\<bullet>(Fst e)) (pi\<bullet>e\<^isub>1)" using a5 by simp
+ then show "P c (pi\<bullet>(Fst e)) (pi\<bullet>e\<^isub>1)" using a5 by (simp, blast)
next
case (b_Snd e e\<^isub>1 e\<^isub>2 pi c)
- have p1: "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
- then have "(pi\<bullet>e)\<Down>(pi\<bullet>(Pr e\<^isub>1 e\<^isub>2))" by (simp only: big_eqvt)
- moreover
- have ih1: "\<And>c. P c (pi\<bullet>e) (pi\<bullet>(Pr e\<^isub>1 e\<^isub>2))" by fact
- ultimately show "P c (pi\<bullet>(Snd e)) (pi\<bullet>e\<^isub>2)" using a6 by simp
+ then show "P c (pi\<bullet>(Snd e)) (pi\<bullet>e\<^isub>2)" using a6 by (simp, blast)
next
case (b_InL e e' pi c)
- then show "P c (pi\<bullet>(InL e)) (pi\<bullet>(InL e'))" using a7
- by (simp, blast intro: big_eqvt)
+ then show "P c (pi\<bullet>(InL e)) (pi\<bullet>(InL e'))" using a7 by (simp)
next
case (b_InR e e' pi c)
- then show "P c (pi\<bullet>(InR e)) (pi\<bullet>(InR e'))" using a8
- by (simp, blast intro: big_eqvt)
+ then show "P c (pi\<bullet>(InR e)) (pi\<bullet>(InR e'))" using a8 by (simp)
next
case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' pi c)
obtain y\<^isub>1::"name" where fs1: "y\<^isub>1\<sharp>(pi\<bullet>x\<^isub>1,pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e\<^isub>2,pi\<bullet>e'',pi\<bullet>x\<^isub>2,c)"
- by (rule exists_fresh[OF fs_name1])
+ by (rule exists_fresh[OF fin_supp])
obtain y\<^isub>2::"name" where fs2: "y\<^isub>2\<sharp>(pi\<bullet>x\<^isub>2,pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e\<^isub>2,pi\<bullet>e'',pi\<bullet>x\<^isub>1,c,y\<^isub>1)"
- by (rule exists_fresh[OF fs_name1])
+ by (rule exists_fresh[OF fin_supp])
let ?sw1 = "[(pi\<bullet>x\<^isub>1,y\<^isub>1)]"
let ?sw2 = "[(pi\<bullet>x\<^isub>2,y\<^isub>2)]"
let ?pi' = "?sw2@?sw1@pi"
@@ -840,14 +766,6 @@
have f12: "(pi\<bullet>x\<^isub>2)\<sharp>(pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e'',pi\<bullet>x\<^isub>1)" using f02 by (simp add: fresh_bij)
have f22: "y\<^isub>2\<sharp>(?pi'\<bullet>e,?pi'\<bullet>e\<^isub>1,?pi'\<bullet>e'')" using f02 fs1 fs2
by (auto simp add: fresh_atm fresh_prod fresh_left calc_atm pt_name2 perm_pi_simp)
- have p1: "e \<Down> InL e'" by fact
- then have "(?pi'\<bullet>e) \<Down> (?pi'\<bullet>(InL e'))" by (simp only: big_eqvt)
- moreover
- have p2: "e\<^isub>1[x\<^isub>1::=e'] \<Down> e''" by fact
- then have "(?pi'\<bullet>(e\<^isub>1[x\<^isub>1::=e'])) \<Down> (?pi'\<bullet>e'')" by (simp only: big_eqvt)
- then have "(?pi'\<bullet>e\<^isub>1)[y\<^isub>1::=(?pi'\<bullet>e')] \<Down> (?pi'\<bullet>e'')" using fs1 fs2
- by (auto simp add: calc_atm subst_eqvt fresh_prod fresh_atm del: append_Cons)
- moreover
have ih1: "\<And>c. P c (?pi'\<bullet>e) (?pi'\<bullet>(InL e'))" by fact
moreover
have ih2: "\<And>c. P c (?pi'\<bullet>(e\<^isub>1[x\<^isub>1::=e'])) (?pi'\<bullet>e'')" by fact
@@ -874,9 +792,9 @@
next
case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' pi c)
obtain y\<^isub>1::"name" where fs1: "y\<^isub>1\<sharp>(pi\<bullet>x\<^isub>1,pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e\<^isub>2,pi\<bullet>e'',pi\<bullet>x\<^isub>2,c)"
- by (rule exists_fresh[OF fs_name1])
+ by (rule exists_fresh[OF fin_supp])
obtain y\<^isub>2::"name" where fs2: "y\<^isub>2\<sharp>(pi\<bullet>x\<^isub>2,pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e\<^isub>2,pi\<bullet>e'',pi\<bullet>x\<^isub>1,c,y\<^isub>1)"
- by (rule exists_fresh[OF fs_name1])
+ by (rule exists_fresh[OF fin_supp])
let ?sw1 = "[(pi\<bullet>x\<^isub>1,y\<^isub>1)]"
let ?sw2 = "[(pi\<bullet>x\<^isub>2,y\<^isub>2)]"
let ?pi' = "?sw2@?sw1@pi"
@@ -888,14 +806,6 @@
have f12: "(pi\<bullet>x\<^isub>2)\<sharp>(pi\<bullet>e,pi\<bullet>e\<^isub>1,pi\<bullet>e'',pi\<bullet>x\<^isub>1)" using f02 by (simp add: fresh_bij)
have f22: "y\<^isub>2\<sharp>(?pi'\<bullet>e,?pi'\<bullet>e\<^isub>1,?pi'\<bullet>e'')" using f02 fs1 fs2
by (auto simp add: fresh_atm fresh_prod fresh_left calc_atm pt_name2 perm_pi_simp)
- have p1: "e \<Down> InR e'" by fact
- then have "(?pi'\<bullet>e) \<Down> (?pi'\<bullet>(InR e'))" by (simp only: big_eqvt)
- moreover
- have p2: "e\<^isub>2[x\<^isub>2::=e'] \<Down> e''" by fact
- then have "(?pi'\<bullet>(e\<^isub>2[x\<^isub>2::=e'])) \<Down> (?pi'\<bullet>e'')" by (simp only: big_eqvt)
- then have "(?pi'\<bullet>e\<^isub>2)[y\<^isub>2::=(?pi'\<bullet>e')] \<Down> (?pi'\<bullet>e'')" using fs1 fs2 f11 f12
- by (auto simp add: calc_atm subst_eqvt fresh_prod fresh_atm del: append_Cons)
- moreover
have ih1: "\<And>c. P c (?pi'\<bullet>e) (?pi'\<bullet>(InR e'))" by fact
moreover
have ih2: "\<And>c. P c (?pi'\<bullet>(e\<^isub>2[x\<^isub>2::=e'])) (?pi'\<bullet>e'')" by fact
@@ -937,24 +847,62 @@
done
lemma b_CaseL_elim[elim]:
- assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''" and "(\<And> t. \<not> e \<Down> InR t)"
+ assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
+ and "\<And> t. \<not> e \<Down> InR t"
+ and "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>1\<sharp>e"
obtains e' where "e \<Down> InL e'" and "e\<^isub>1[x\<^isub>1::=e'] \<Down> e''"
using assms
apply -
- apply (rule b_Case_inv_auto, auto)
- apply(drule_tac u="e'" in subst_fun_eq)
- apply(simp)
- done
+ apply(rule b_Case_inv_auto)
+ apply(auto)
+ apply(simp add: alpha)
+ apply(auto)
+ apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
+ apply(perm_simp add: fresh_prod)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
+ apply(perm_simp add: eqvt calc_atm)
+ apply(assumption)
+ apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
+ apply(perm_simp add: fresh_prod)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
+ apply(perm_simp add: eqvt calc_atm)
+ apply(assumption)
+done
lemma b_CaseR_elim[elim]:
- assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''" and "\<And> t. \<not> e \<Down> InL t"
+ assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
+ and "\<And> t. \<not> e \<Down> InL t"
+ and "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>2\<sharp>e"
obtains e' where "e \<Down> InR e'" and "e\<^isub>2[x\<^isub>2::=e'] \<Down> e''"
- using assms
+ using assms
apply -
- apply (rule b_Case_inv_auto, auto)
- apply(drule_tac u="e'" in subst_fun_eq)+
- apply(simp)
- done
+ apply(rule b_Case_inv_auto)
+ apply(auto)
+ apply(simp add: alpha)
+ apply(auto)
+ apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
+ apply(perm_simp add: fresh_prod)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
+ apply(perm_simp add: eqvt calc_atm)
+ apply(assumption)
+ apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
+ apply(perm_simp add: fresh_prod)
+ apply(drule meta_mp)
+ apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
+ apply(perm_simp add: eqvt calc_atm)
+ apply(assumption)
+done
inductive2
val :: "trm\<Rightarrow>bool"
@@ -965,7 +913,6 @@
| v_InL[intro]: "val e \<Longrightarrow> val (InL e)"
| v_InR[intro]: "val e \<Longrightarrow> val (InR e)"
-
declare trm.inject [simp add]
declare ty.inject [simp add]
declare data.inject [simp add]
@@ -986,17 +933,18 @@
declare data.inject [simp del]
lemma subject_reduction:
- assumes "e \<Down> e'" and "\<Gamma> \<turnstile> e : T"
+ assumes a: "e \<Down> e'"
+ and b: "\<Gamma> \<turnstile> e : T"
shows "\<Gamma> \<turnstile> e' : T"
- using assms
+ using a b
proof (nominal_induct avoiding: \<Gamma> arbitrary: T rule: big_induct_strong)
case (b_App x e\<^isub>1 e\<^isub>2 e\<^isub>2' e' e \<Gamma> T)
have vc: "x\<sharp>\<Gamma>" by fact
have "\<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T" by fact
- then obtain T' where
+ then obtain T' where
a1: "\<Gamma> \<turnstile> e\<^isub>1 : T'\<rightarrow>T" and
a2: "\<Gamma> \<turnstile> e\<^isub>2 : T'" by auto
- have ih1: "\<Gamma> \<turnstile> e\<^isub>1 : T'\<rightarrow>T \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T'\<rightarrow>T" by fact
+ have ih1: "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T' \<rightarrow> T" by fact
have ih2: "\<Gamma> \<turnstile> e\<^isub>2 : T' \<Longrightarrow> \<Gamma> \<turnstile> e\<^isub>2' : T'" by fact
have ih3: "\<Gamma> \<turnstile> e[x::=e\<^isub>2'] : T \<Longrightarrow> \<Gamma> \<turnstile> e' : T" by fact
have "\<Gamma> \<turnstile> Lam [x].e : T'\<rightarrow>T" using ih1 a1 by simp
@@ -1023,76 +971,57 @@
then show "\<Gamma> \<turnstile> e'' : T" by (blast intro: typing_substitution)
qed (blast)+
-lemma challenge_5:
- assumes "x\<noteq>y"
- shows "App (App (Lam [x].(Lam [y].Var y)) (Const n\<^isub>1)) (Const n\<^isub>2) \<Down> (Const n\<^isub>2)"
- using assms
- by (auto intro!: big.intros simp add: forget abs_fresh fresh_atm fresh_nat)
-
-lemma challenge_6:
- shows "Fst (App (Lam [x].Pr (Var x) (Var x)) (Const n)) \<Down> Const n"
- by (auto intro!: big.intros) (simp add: fresh_nat abs_fresh)
-
-lemma challenge_4_unicity:
- assumes "e \<Down> e\<^isub>1" and "e \<Down> e\<^isub>2"
+lemma unicity_of_evaluation:
+ assumes a: "e \<Down> e\<^isub>1"
+ and b: "e \<Down> e\<^isub>2"
shows "e\<^isub>1 = e\<^isub>2"
- using assms
-proof (induct arbitrary: e\<^isub>2)
+ using a b
+proof (nominal_induct e e\<^isub>1 avoiding: e\<^isub>2 rule: big_induct_strong)
case (b_Lam x e t\<^isub>2)
have "Lam [x].e \<Down> t\<^isub>2" by fact
thus "Lam [x].e = t\<^isub>2" by (cases, simp_all add: trm.inject)
next
- case (b_Fst e e\<^isub>1 e\<^isub>2 t\<^isub>2)
- have "Fst e \<Down> t\<^isub>2" by fact
- then obtain e\<^isub>1' e\<^isub>2' where "e \<Down> Pr e\<^isub>1' e\<^isub>2'" and eq: "t\<^isub>2 = e\<^isub>1'" by auto
- then have "Pr e\<^isub>1 e\<^isub>2 = Pr e\<^isub>1' e\<^isub>2'" by auto
- thus "e\<^isub>1 = t\<^isub>2" using eq by (simp add: trm.inject)
-next
- case (b_Snd e e\<^isub>1 e\<^isub>2 t\<^isub>2)
- thus ?case by (force simp add: trm.inject)
-next
- case (b_App x e\<^isub>1 e\<^isub>2 e' e\<^isub>1' e\<^isub>2' t\<^isub>2)
- have "e\<^isub>1 \<Down> Lam [x].e\<^isub>1'" by fact
+ case (b_App x e\<^isub>1 e\<^isub>2 e\<^isub>2' e' e\<^isub>1' t\<^isub>2)
have ih1: "\<And>t. e\<^isub>1 \<Down> t \<Longrightarrow> Lam [x].e\<^isub>1' = t" by fact
- have "e\<^isub>2 \<Down> e\<^isub>2'" by fact
have ih2:"\<And>t. e\<^isub>2 \<Down> t \<Longrightarrow> e\<^isub>2' = t" by fact
- have "e\<^isub>1'[x::=e\<^isub>2'] \<Down> e'" by fact
have ih3: "\<And>t. e\<^isub>1'[x::=e\<^isub>2'] \<Down> t \<Longrightarrow> e' = t" by fact
- have f:"x\<sharp>(e\<^isub>1,e\<^isub>2,e')" by fact
+ have app: "App e\<^isub>1 e\<^isub>2 \<Down> t\<^isub>2" by fact
+ have vc: "x\<sharp>e\<^isub>1" "x\<sharp>e\<^isub>2" by fact
then have "x \<sharp> App e\<^isub>1 e\<^isub>2" by auto
- moreover
- have app:"App e\<^isub>1 e\<^isub>2 \<Down> t\<^isub>2" by fact
- ultimately have "x\<sharp>t\<^isub>2" using fresh_preserved by blast
- then have "x\<sharp>(e\<^isub>1,e\<^isub>2,t\<^isub>2)" using f by auto
- then obtain f\<^isub>1'' f\<^isub>2'' where x1:"e\<^isub>1 \<Down> Lam [x]. f\<^isub>1''" and x2:"e\<^isub>2 \<Down> f\<^isub>2''" and x3:"f\<^isub>1''[x::=f\<^isub>2''] \<Down> t\<^isub>2"
- using app by auto
- then have "Lam [x]. f\<^isub>1'' = Lam [x]. e\<^isub>1'" using ih1 by simp
- then have "f\<^isub>1'' = e\<^isub>1'" by (auto simp add: trm.inject alpha)
- moreover have "f\<^isub>2''=e\<^isub>2'" using x2 ih2 by simp
+ then have vc': "x\<sharp>t\<^isub>2" using fresh_preserved app by blast
+ from vc vc' obtain f\<^isub>1 f\<^isub>2 where x1: "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" and x2: "e\<^isub>2 \<Down> f\<^isub>2" and x3: "f\<^isub>1[x::=f\<^isub>2] \<Down> t\<^isub>2"
+ using app by (auto simp add: fresh_prod)
+ then have "Lam [x]. f\<^isub>1 = Lam [x]. e\<^isub>1'" using ih1 by simp
+ then
+ have "f\<^isub>1 = e\<^isub>1'" by (auto simp add: trm.inject alpha)
+ moreover
+ have "f\<^isub>2 = e\<^isub>2'" using x2 ih2 by simp
ultimately have "e\<^isub>1'[x::=e\<^isub>2'] \<Down> t\<^isub>2" using x3 by simp
thus ?case using ih3 by simp
next
- case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2)
- have ih1:"\<And> t. e \<Down> t \<Longrightarrow> InL e' = t" by fact
+ case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2)
+ have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact
+ have ih1:"\<And>t. e \<Down> t \<Longrightarrow> InL e' = t" by fact
have ih2:"\<And>t. e\<^isub>1[x\<^isub>1::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
- have ha:"\<And> t. (e \<Down> InR t) \<Longrightarrow> False" using ih1 by force
+ have ha: "\<not>(\<exists>t. e \<Down> InR t)" using ih1 by force
have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
- then obtain xe' where "e\<Down>InL xe'" and h:"e\<^isub>1[x\<^isub>1::=xe']\<Down>t\<^isub>2" using ha by auto
- then have "InL xe'=InL e'" using ih1 by simp
- then have "xe'=e'" by (simp add: trm.inject)
+ then obtain f' where "e \<Down> InL f'" and h: "e\<^isub>1[x\<^isub>1::=f']\<Down>t\<^isub>2" using ha fs by auto
+ then have "InL f' = InL e'" using ih1 by simp
+ then have "f' = e'" by (simp add: trm.inject)
then have "e\<^isub>1[x\<^isub>1::=e'] \<Down> t\<^isub>2" using h by simp
- then show "e''=t\<^isub>2" using ih2 by simp
+ then show "e'' = t\<^isub>2" using ih2 by simp
next
case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2 )
- have ih1:"\<And> t. e \<Down> t \<Longrightarrow> InR e' = t" by fact
- have ih2:"\<And>t. e\<^isub>2[x\<^isub>2::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
- have a:"\<And> t. (e \<Down> InL t \<Longrightarrow> False)" using ih1 by force
+ have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact
+ have ih1: "\<And>t. e \<Down> t \<Longrightarrow> InR e' = t" by fact
+ have ih2: "\<And>t. e\<^isub>2[x\<^isub>2::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
+ have ha: "\<not>(\<exists>t. e \<Down> InL t)" using ih1 by force
have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
- then obtain xe' where "e\<Down>InR xe'" and h:"e\<^isub>2[x\<^isub>2::=xe']\<Down>t\<^isub>2" using a by auto
- then have "InR xe'=InR e'" using ih1 by simp
+ then obtain f' where "e \<Down> InR f'" and h: "e\<^isub>2[x\<^isub>2::=f']\<Down>t\<^isub>2" using ha fs by auto
+ then have "InR f' = InR e'" using ih1 by simp
then have "e\<^isub>2[x\<^isub>2::=e'] \<Down> t\<^isub>2" using h by (simp add: trm.inject)
- thus "e''=t\<^isub>2" using ih2 by simp
-qed (fast)+
+ thus "e'' = t\<^isub>2" using ih2 by simp
+qed (force simp add: trm.inject)+
lemma not_val_App[simp]:
shows
@@ -1103,51 +1032,52 @@
"\<not> val (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2)"
by auto
-lemma reduces_to_value:
+lemma reduces_evaluates_to_values:
assumes h:"t \<Down> t'"
shows "val t'"
- using h by (induct, auto)
+ using h by (induct) (auto)
-lemma type_prod_down_pair:
- assumes "\<Gamma> \<turnstile> t : Data (DProd S\<^isub>1 S\<^isub>2)" and "t \<Down> t'"
+lemma type_prod_evaluates_to_pairs:
+ assumes a: "\<Gamma> \<turnstile> t : Data (DProd S\<^isub>1 S\<^isub>2)"
+ and b: "t \<Down> t'"
obtains t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2"
proof -
have "\<Gamma> \<turnstile> t' : Data (DProd S\<^isub>1 S\<^isub>2)" using assms subject_reduction by simp
moreover
- have "val t'" using reduces_to_value assms by simp
+ have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
-lemma type_sum_down_or:
+lemma type_sum_evaluates_to_ins:
assumes "\<Gamma> \<turnstile> t : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" and "t \<Down> t'"
- shows "(\<exists> t''. t' = InL t'') \<or> (\<exists> t''. t' = InR t'')"
+ shows "(\<exists>t''. t' = InL t'') \<or> (\<exists>t''. t' = InR t'')"
proof -
have "\<Gamma> \<turnstile> t' : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" using assms subject_reduction by simp
moreover
- have "val t'" using reduces_to_value assms by simp
+ have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain t'' where "t' = InL t'' \<or> t' = InR t''"
by (cases, auto simp add:ty.inject data.inject)
thus ?thesis by auto
qed
-lemma type_arrow_down_lam:
+lemma type_arrow_evaluates_to_lams:
assumes "\<Gamma> \<turnstile> t : \<sigma> \<rightarrow> \<tau>" and "t \<Down> t'"
obtains x t'' where "t' = Lam [x]. t''"
proof -
have "\<Gamma> \<turnstile> t' : \<sigma> \<rightarrow> \<tau>" using assms subject_reduction by simp
moreover
- have "val t'" using reduces_to_value assms by simp
+ have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain x t'' where "t' = Lam [x]. t''" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
-lemma type_nat_down_const:
+lemma type_nat_evaluates_to_consts:
assumes "\<Gamma> \<turnstile> t : Data DNat" and "t \<Down> t'"
obtains n where "t' = Const n"
proof -
have "\<Gamma> \<turnstile> t' : Data DNat " using assms subject_reduction by simp
- moreover have "val t'" using reduces_to_value assms by simp
+ moreover have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain n where "t' = Const n" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
@@ -1170,6 +1100,14 @@
by (nominal_induct S arbitrary: e rule:data.induct)
(auto)
+lemma V'_eqvt:
+ fixes pi::"name prm"
+ assumes a: "v \<in> V' S"
+ shows "(pi\<bullet>v) \<in> V' S"
+using a
+by (nominal_induct S arbitrary: v rule: data.induct)
+ (auto simp add: trm.inject)
+
consts
V :: "ty \<Rightarrow> trm set"
@@ -1179,6 +1117,27 @@
apply(rule TrueI)+
done
+lemma V_eqvt:
+ fixes pi::"name prm"
+ assumes a: "x\<in>V T"
+ shows "(pi\<bullet>x)\<in>V T"
+using a
+apply(nominal_induct T arbitrary: pi x rule: ty.induct)
+apply(auto simp add: trm.inject perm_set_def)
+apply(perm_simp add: V'_eqvt)
+apply(rule_tac x="pi\<bullet>xa" in exI)
+apply(rule_tac x="pi\<bullet>e" in exI)
+apply(simp)
+apply(auto)
+apply(drule_tac x="(rev pi)\<bullet>v" in bspec)
+apply(force)
+apply(auto)
+apply(rule_tac x="pi\<bullet>v'" in exI)
+apply(auto)
+apply(drule_tac pi="pi" in big_eqvt)
+apply(perm_simp add: eqvt)
+done
+
lemma V_arrow_elim_weak[elim] :
assumes h:"u \<in> (V (T\<^isub>1 \<rightarrow> T\<^isub>2))"
obtains a t where "u = Lam[a].t" and "\<forall> v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
@@ -1186,27 +1145,34 @@
lemma V_arrow_elim_strong[elim]:
fixes c::"'a::fs_name"
- assumes h: "u \<in> (V (T\<^isub>1 \<rightarrow> T\<^isub>2))"
- obtains a t where "a\<sharp>c" "u = Lam[a].t" "\<forall> v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
+ assumes h: "u \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)"
+ obtains a t where "a\<sharp>c" "u = Lam[a].t" "\<forall>v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
using h
apply -
apply(erule V_arrow_elim_weak)
-apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(a,t,c)")
+apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(a,t,c)") (*A*)
apply(erule exE)
apply(drule_tac x="a'" in meta_spec)
+apply(drule_tac x="[(a,a')]\<bullet>t" in meta_spec)
+apply(drule meta_mp)
apply(simp)
-apply(drule_tac x="[(a,a')]\<bullet>t" in meta_spec)
-apply(simp add: trm.inject alpha fresh_prod fresh_atm)
+apply(drule meta_mp)
+apply(simp add: trm.inject alpha fresh_left fresh_prod calc_atm fresh_atm)
apply(perm_simp)
-apply(simp add: fresh_left calc_atm)
+apply(force)
+apply(drule meta_mp)
+apply(rule ballI)
+apply(drule_tac x="[(a,a')]\<bullet>v" in bspec)
+apply(simp add: V_eqvt)
apply(auto)
-apply(simp add: subst_rename)
-apply(subgoal_tac "[(a',a)]\<bullet>t = [(a,a')]\<bullet>t")
-apply(simp)
-apply(rule pt_name3)
-apply(rule at_ds5[OF at_name_inst])
+apply(rule_tac x="[(a,a')]\<bullet>v'" in exI)
+apply(auto)
+apply(drule_tac pi="[(a,a')]" in big_eqvt)
+apply(perm_simp add: eqvt calc_atm)
+apply(simp add: V_eqvt)
+(*A*)
apply(rule exists_fresh')
-apply(simp add: fs_name1)
+apply(simp add: fin_supp)
done
lemma V_are_values :
@@ -1306,7 +1272,7 @@
then obtain v\<^isub>1 v\<^isub>2 where "\<theta><t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V (Data T\<^isub>a)" "\<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V (Data T\<^isub>b)"
using prems by blast
thus "\<exists>v. \<theta><Pr t\<^isub>1 t\<^isub>2> \<Down> v \<and> v \<in> V T" using eq by auto
-next
+next
case (Lam x e \<Gamma> \<theta> T)
have ih:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T" by fact
have as\<^isub>1: "\<theta> Vcloses \<Gamma>" by fact
@@ -1314,15 +1280,14 @@
have fs: "x\<sharp>\<Gamma>" "x\<sharp>\<theta>" by fact
from as\<^isub>2 fs obtain T\<^isub>1 T\<^isub>2
where "(i)": "(x,T\<^isub>1)#\<Gamma> \<turnstile> e:T\<^isub>2" and "(ii)": "T = T\<^isub>1 \<rightarrow> T\<^isub>2" by auto
- from "(i)" have "(iii)": "valid ((x,T\<^isub>1)#\<Gamma>)" by (simp add: typing_valid)
+ from "(i)" have "(iii)": "valid ((x,T\<^isub>1)#\<Gamma>)" by (simp add: typing_implies_valid)
have "\<forall>v \<in> (V T\<^isub>1). \<exists>v'. (\<theta><e>)[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
proof
fix v
assume "v \<in> (V T\<^isub>1)"
with "(iii)" as\<^isub>1 have "(x,v)#\<theta> Vcloses (x,T\<^isub>1)#\<Gamma>" using monotonicity by auto
with ih "(i)" obtain v' where "((x,v)#\<theta>)<e> \<Down> v' \<and> v' \<in> V T\<^isub>2" by blast
- then have "\<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2" using fs
- by (simp add: psubst_subst_psubst)
+ then have "\<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2" using fs by (simp add: psubst_subst_psubst)
then show "\<exists>v'. \<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2" by auto
qed
then have "Lam[x].\<theta><e> \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)" by auto
@@ -1336,21 +1301,20 @@
have th:"\<Gamma> \<turnstile> Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2 : T" by fact
then obtain S\<^isub>1 S\<^isub>2 where
hm:"\<Gamma> \<turnstile> t' : Data (DSum S\<^isub>1 S\<^isub>2)" and
- hl:"(n\<^isub>1,Data S\<^isub>1)# \<Gamma> \<turnstile> t\<^isub>1 : T" and
- hr:"(n\<^isub>2,Data S\<^isub>2)# \<Gamma> \<turnstile> t\<^isub>2 : T" using f by auto
+ hl:"(n\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>1 : T" and
+ hr:"(n\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t\<^isub>2 : T" using f by auto
then obtain v\<^isub>0 where ht':"\<theta><t'> \<Down> v\<^isub>0" and hS:"v\<^isub>0 \<in> V (Data (DSum S\<^isub>1 S\<^isub>2))" using prems h by blast
(* We distinguish between the cases InL and InR *)
- {
- fix v\<^isub>0'
+ { fix v\<^isub>0'
assume eqc:"v\<^isub>0 = InL v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>1"
then have inc: "v\<^isub>0' \<in> V (Data S\<^isub>1)" by auto
- have "valid \<Gamma>" using th typing_valid by auto
+ have "valid \<Gamma>" using th typing_implies_valid by auto
then moreover have "valid ((n\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using f by auto
then moreover have "(n\<^isub>1,v\<^isub>0')#\<theta> Vcloses (n\<^isub>1,Data S\<^isub>1)#\<Gamma>"
using inc h monotonicity by blast
- moreover have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>1 : T\<rbrakk> \<Longrightarrow>
- \<exists>v. \<theta><t\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
- ultimately obtain v\<^isub>1 where ho:"((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using hl by blast
+ moreover
+ have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><t\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
+ ultimately obtain v\<^isub>1 where ho: "((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using hl by blast
then have r:"\<theta><t\<^isub>1>[n\<^isub>1::=v\<^isub>0'] \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using psubst_subst_psubst f by simp
then moreover have "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>1,n\<^isub>2)"
proof -
@@ -1375,11 +1339,10 @@
ultimately have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" by auto
}
moreover
- {
- fix v\<^isub>0'
+ { fix v\<^isub>0'
assume eqc:"v\<^isub>0 = InR v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>2"
then have inc:"v\<^isub>0' \<in> V (Data S\<^isub>2)" by auto
- have "valid \<Gamma>" using th typing_valid by auto
+ have "valid \<Gamma>" using th typing_implies_valid by auto
then moreover have "valid ((n\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using f by auto
then moreover have "(n\<^isub>2,v\<^isub>0')#\<theta> Vcloses (n\<^isub>2,Data S\<^isub>2)#\<Gamma>"
using inc h monotonicity by blast
@@ -1407,7 +1370,7 @@
qed
ultimately have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using ht' eqc by auto
then have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using f by auto
-}
+ }
ultimately show "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using hS V_sum by blast
qed (force)+