--- a/src/HOL/Nominal/Examples/Weakening.thy Tue Mar 06 08:09:43 2007 +0100
+++ b/src/HOL/Nominal/Examples/Weakening.thy Tue Mar 06 15:28:22 2007 +0100
@@ -1,7 +1,7 @@
(* $Id$ *)
theory Weakening
-imports "Nominal"
+imports "../Nominal"
begin
section {* Weakening Example for the Simply-Typed Lambda-Calculus *}
@@ -20,9 +20,9 @@
lemma ty_perm[simp]:
fixes pi ::"name prm"
- and \<tau> ::"ty"
- shows "pi\<bullet>\<tau> = \<tau>"
-by (induct \<tau> rule: ty.induct_weak)
+ and T ::"ty"
+ shows "pi\<bullet>T = T"
+by (induct T rule: ty.induct_weak)
(simp_all add: perm_nat_def)
text {* valid contexts *}
@@ -37,35 +37,32 @@
assumes a: "valid \<Gamma>"
shows "valid (pi\<bullet>\<Gamma>)"
using a
-by (induct)
- (auto simp add: fresh_bij)
-
-thm eqvt
+by (induct) (auto simp add: fresh_bij)
text{* typing judgements *}
inductive2
typing :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80)
where
- t_Var[intro]: "\<lbrakk>valid \<Gamma>; (a,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : \<tau>"
- | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; \<Gamma> \<turnstile> t2 : \<tau>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : \<sigma>"
- | t_Lam[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : \<tau>\<rightarrow>\<sigma>"
+ t_Var[intro]: "\<lbrakk>valid \<Gamma>; (a,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : T"
+ | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1\<rightarrow>T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
+ | t_Lam[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,T1)#\<Gamma>) \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : T1\<rightarrow>T2"
lemma eqvt_typing[eqvt]:
fixes pi:: "name prm"
- assumes a: "\<Gamma> \<turnstile> t : \<tau>"
- shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : (pi\<bullet>\<tau>)"
+ assumes a: "\<Gamma> \<turnstile> t : T"
+ shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : (pi\<bullet>T)"
using a
proof (induct)
- case (t_Var \<Gamma> a \<tau>)
+ case (t_Var \<Gamma> a T)
have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
moreover
- have "(pi\<bullet>(a,\<tau>))\<in>(pi\<bullet>set \<Gamma>)" by (rule pt_set_bij2[OF pt_name_inst, OF at_name_inst])
- ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Var a) : (pi\<bullet>\<tau>)"
- using typing.intros by (force simp add: pt_list_set_pi[OF pt_name_inst, symmetric])
+ have "(pi\<bullet>(a,T))\<in>(pi\<bullet>set \<Gamma>)" by (rule pt_set_bij2[OF pt_name_inst, OF at_name_inst])
+ ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Var a) : (pi\<bullet>T)"
+ using typing.intros by (force simp add: set_eqvt)
next
- case (t_Lam a \<Gamma> \<tau> t \<sigma>)
+ case (t_Lam a \<Gamma> T1 t T2)
moreover have "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" by (simp add: fresh_bij)
- ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Lam [a].t) :(pi\<bullet>\<tau>\<rightarrow>\<sigma>)" by force
+ ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Lam [a].t) :(pi\<bullet>T1\<rightarrow>T2)" by force
qed (auto)
text {* the strong induction principle needs to be derived manually *}
@@ -74,51 +71,51 @@
fixes P :: "'a::fs_name\<Rightarrow>(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>bool"
and \<Gamma> :: "(name\<times>ty) list"
and t :: "lam"
- and \<tau> :: "ty"
+ and T :: "ty"
and x :: "'a::fs_name"
- assumes a: "\<Gamma> \<turnstile> t : \<tau>"
- and a1: "\<And>\<Gamma> a \<tau> x. \<lbrakk>valid \<Gamma>; (a,\<tau>) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P x \<Gamma> (Var a) \<tau>"
- and a2: "\<And>\<Gamma> \<tau> \<sigma> t1 t2 x. \<lbrakk>\<And>z. P z \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>); \<And>z. P z \<Gamma> t2 \<tau>\<rbrakk>
- \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
- and a3: "\<And>a \<Gamma> \<tau> \<sigma> t x. \<lbrakk>a\<sharp>x; a\<sharp>\<Gamma>; \<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma>\<rbrakk>
- \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
- shows "P x \<Gamma> t \<tau>"
+ assumes a: "\<Gamma> \<turnstile> t : T"
+ and a1: "\<And>\<Gamma> a T x. \<lbrakk>valid \<Gamma>; (a,T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P x \<Gamma> (Var a) T"
+ and a2: "\<And>\<Gamma> T1 T2 t1 t2 x. \<lbrakk>\<And>z. P z \<Gamma> t1 (T1\<rightarrow>T2); \<And>z. P z \<Gamma> t2 T1\<rbrakk>
+ \<Longrightarrow> P x \<Gamma> (App t1 t2) T2"
+ and a3: "\<And>a \<Gamma> T1 T2 t x. \<lbrakk>a\<sharp>x; a\<sharp>\<Gamma>; \<And>z. P z ((a,T1)#\<Gamma>) t T2\<rbrakk>
+ \<Longrightarrow> P x \<Gamma> (Lam [a].t) (T1\<rightarrow>T2)"
+ shows "P x \<Gamma> t T"
proof -
- from a have "\<And>(pi::name prm) x. P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) (pi\<bullet>\<tau>)"
+ from a have "\<And>(pi::name prm) x. P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) (pi\<bullet>T)"
proof (induct)
- case (t_Var \<Gamma> a \<tau>)
+ case (t_Var \<Gamma> a T)
have "valid \<Gamma>" by fact
then have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt)
moreover
- have "(a,\<tau>)\<in>set \<Gamma>" by fact
- then have "pi\<bullet>(a,\<tau>)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])
- then have "(pi\<bullet>a,\<tau>)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_name_inst])
- ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) (pi\<bullet>\<tau>)" using a1 by simp
+ have "(a,T)\<in>set \<Gamma>" by fact
+ then have "pi\<bullet>(a,T)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])
+ then have "(pi\<bullet>a,T)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: set_eqvt)
+ ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) (pi\<bullet>T)" using a1 by simp
next
- case (t_App \<Gamma> t1 \<tau> \<sigma> t2)
- thus "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(App t1 t2)) (pi\<bullet>\<sigma>)" using a2
+ case (t_App \<Gamma> t1 T1 T2 t2)
+ thus "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(App t1 t2)) (pi\<bullet>T2)" using a2
by (simp only: eqvt) (blast)
next
- case (t_Lam a \<Gamma> \<tau> t \<sigma>)
+ case (t_Lam a \<Gamma> T1 t T2)
obtain c::"name" where fs: "c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)" by (rule exists_fresh[OF fs_name1])
let ?sw="[(pi\<bullet>a,c)]"
let ?pi'="?sw@pi"
have f1: "a\<sharp>\<Gamma>" by fact
have f2: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" using f1 by (simp add: fresh_bij)
have f3: "c\<sharp>?pi'\<bullet>\<Gamma>" using f1 by (auto simp add: pt_name2 fresh_left calc_atm perm_pi_simp)
- have ih1: "\<And>x. P x (?pi'\<bullet>((a,\<tau>)#\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>\<sigma>)" by fact
- then have "\<And>x. P x ((c,\<tau>)#(?pi'\<bullet>\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>\<sigma>)" by (simp add: calc_atm)
- then have "P x (?pi'\<bullet>\<Gamma>) (Lam [c].(?pi'\<bullet>t)) (\<tau>\<rightarrow>\<sigma>)" using a3 f3 fs by simp
- then have "P x (?sw\<bullet>pi\<bullet>\<Gamma>) (?sw\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) (\<tau>\<rightarrow>\<sigma>)"
+ have ih1: "\<And>x. P x (?pi'\<bullet>((a,T1)#\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>T2)" by fact
+ then have "\<And>x. P x ((c,T1)#(?pi'\<bullet>\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>T2)" by (simp add: calc_atm)
+ then have "P x (?pi'\<bullet>\<Gamma>) (Lam [c].(?pi'\<bullet>t)) (T1\<rightarrow>T2)" using a3 f3 fs by simp
+ then have "P x (?sw\<bullet>pi\<bullet>\<Gamma>) (?sw\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) (T1\<rightarrow>T2)"
by (simp del: append_Cons add: calc_atm pt_name2)
moreover have "(?sw\<bullet>(pi\<bullet>\<Gamma>)) = (pi\<bullet>\<Gamma>)"
by (rule perm_fresh_fresh) (simp_all add: fs f2)
moreover have "(?sw\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) = Lam [(pi\<bullet>a)].(pi\<bullet>t)"
by (rule perm_fresh_fresh) (simp_all add: fs f2 abs_fresh)
- ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (pi\<bullet>\<tau>\<rightarrow>\<sigma>)" by (simp)
+ ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (pi\<bullet>T1\<rightarrow>T2)" by (simp)
qed
- hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) (([]::name prm)\<bullet>\<tau>)" by blast
- thus "P x \<Gamma> t \<tau>" by simp
+ hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) (([]::name prm)\<bullet>T)" by blast
+ thus "P x \<Gamma> t T" by simp
qed
text {* definition of a subcontext *}
@@ -130,92 +127,91 @@
text {* now it comes: The Weakening Lemma *}
lemma weakening_version1:
- assumes a: "\<Gamma>1 \<turnstile> t : \<sigma>"
+ assumes a: "\<Gamma>1 \<turnstile> t : T"
and b: "valid \<Gamma>2"
and c: "\<Gamma>1 \<lless> \<Gamma>2"
- shows "\<Gamma>2 \<turnstile> t:\<sigma>"
+ shows "\<Gamma>2 \<turnstile> t : T"
using a b c
-apply(nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
-apply(auto | atomize)+
+by (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
+ (auto | atomize)+
(* FIXME: meta-quantifiers seem to be not as "automatic" as object-quantifiers *)
-done
lemma weakening_version2:
fixes \<Gamma>1::"(name\<times>ty) list"
and t ::"lam"
and \<tau> ::"ty"
- assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+ assumes a: "\<Gamma>1 \<turnstile> t:T"
and b: "valid \<Gamma>2"
and c: "\<Gamma>1 \<lless> \<Gamma>2"
- shows "\<Gamma>2 \<turnstile> t:\<sigma>"
+ shows "\<Gamma>2 \<turnstile> t:T"
using a b c
-proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
- case (t_Var \<Gamma>1 a \<tau>) (* variable case *)
+proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
+ case (t_Var \<Gamma>1 a T) (* variable case *)
have "\<Gamma>1 \<lless> \<Gamma>2" by fact
moreover
have "valid \<Gamma>2" by fact
moreover
- have "(a,\<tau>)\<in> set \<Gamma>1" by fact
- ultimately show "\<Gamma>2 \<turnstile> Var a : \<tau>" by auto
+ have "(a,T)\<in> set \<Gamma>1" by fact
+ ultimately show "\<Gamma>2 \<turnstile> Var a : T" by auto
next
- case (t_Lam a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+ case (t_Lam a \<Gamma>1 T1 T2 t) (* lambda case *)
have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
- have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+ have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t:T2" by fact
have "\<Gamma>1 \<lless> \<Gamma>2" by fact
- then have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" by simp
+ then have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" by simp
moreover
have "valid \<Gamma>2" by fact
- then have "valid ((a,\<tau>)#\<Gamma>2)" using vc by (simp add: v2)
- ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp
- with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by auto
+ then have "valid ((a,T1)#\<Gamma>2)" using vc by (simp add: v2)
+ ultimately have "((a,T1)#\<Gamma>2) \<turnstile> t:T2" using ih by simp
+ with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : T1\<rightarrow>T2" by auto
qed (auto) (* app case *)
lemma weakening_version3:
- assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+ assumes a: "\<Gamma>1 \<turnstile> t : T"
and b: "valid \<Gamma>2"
and c: "\<Gamma>1 \<lless> \<Gamma>2"
- shows "\<Gamma>2 \<turnstile> t:\<sigma>"
+ shows "\<Gamma>2 \<turnstile> t : T"
using a b c
-proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
- case (t_Lam a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
+ case (t_Lam a \<Gamma>1 T1 T2 t) (* lambda case *)
have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
- have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+ have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t : T2" by fact
have "\<Gamma>1 \<lless> \<Gamma>2" by fact
- then have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" by simp
+ then have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" by simp
moreover
have "valid \<Gamma>2" by fact
- then have "valid ((a,\<tau>)#\<Gamma>2)" using vc by (simp add: v2)
- ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp
- with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by auto
+ then have "valid ((a,T1)#\<Gamma>2)" using vc by (simp add: v2)
+ ultimately have "((a,T1)#\<Gamma>2) \<turnstile> t : T2" using ih by simp
+ with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : T1 \<rightarrow> T2" by auto
qed (auto) (* app and var case *)
text{* The original induction principle for the typing relation
is not strong enough - even this simple lemma fails: *}
lemma weakening_too_weak:
- assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+ assumes a: "\<Gamma>1 \<turnstile> t : T"
and b: "valid \<Gamma>2"
and c: "\<Gamma>1 \<lless> \<Gamma>2"
- shows "\<Gamma>2 \<turnstile> t:\<sigma>"
+ shows "\<Gamma>2 \<turnstile> t : T"
using a b c
proof (induct arbitrary: \<Gamma>2)
- case (t_Var \<Gamma>1 a \<tau>) (* variable case *)
+ case (t_Var \<Gamma>1 a T) (* variable case *)
have "\<Gamma>1 \<lless> \<Gamma>2" by fact
moreover
have "valid \<Gamma>2" by fact
moreover
- have "(a,\<tau>) \<in> (set \<Gamma>1)" by fact
- ultimately show "\<Gamma>2 \<turnstile> Var a : \<tau>" by auto
+ have "(a,T) \<in> (set \<Gamma>1)" by fact
+ ultimately show "\<Gamma>2 \<turnstile> Var a : T" by auto
next
- case (t_Lam a \<Gamma>1 \<tau> t \<sigma>) (* lambda case *)
+ case (t_Lam a \<Gamma>1 T1 t T2) (* lambda case *)
(* all assumptions available in this case*)
have a0: "a\<sharp>\<Gamma>1" by fact
- have a1: "((a,\<tau>)#\<Gamma>1) \<turnstile> t : \<sigma>" by fact
+ have a1: "((a,T1)#\<Gamma>1) \<turnstile> t : T2" by fact
have a2: "\<Gamma>1 \<lless> \<Gamma>2" by fact
have a3: "valid \<Gamma>2" by fact
- have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
- have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a2 by simp
+ have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t : T2" by fact
+ have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" using a2 by simp
moreover
- have "valid ((a,\<tau>)#\<Gamma>2)" using v2 (* fails *)
+ have "valid ((a,T1)#\<Gamma>2)" using v2 (* fails *)
oops
end
\ No newline at end of file