(* $Id$ *)
theory weakening
imports "../nominal"
begin
(* WEAKENING EXAMPLE*)
section {* Simply-Typed Lambda-Calculus *}
(*======================================*)
atom_decl name
nominal_datatype lam = Var "name"
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
datatype ty =
TVar "string"
| TArr "ty" "ty" (infix "\<rightarrow>" 200)
primrec
"pi\<bullet>(TVar s) = TVar s"
"pi\<bullet>(\<tau> \<rightarrow> \<sigma>) = (\<tau> \<rightarrow> \<sigma>)"
lemma perm_ty[simp]:
fixes pi ::"name prm"
and \<tau> ::"ty"
shows "pi\<bullet>\<tau> = \<tau>"
by (cases \<tau>, simp_all)
instance ty :: pt_name
apply(intro_classes)
apply(simp_all)
done
instance ty :: fs_name
apply(intro_classes)
apply(simp add: supp_def)
done
(* valid contexts *)
consts
ctxts :: "((name\<times>ty) list) set"
valid :: "(name\<times>ty) list \<Rightarrow> bool"
translations
"valid \<Gamma>" \<rightleftharpoons> "\<Gamma> \<in> ctxts"
inductive ctxts
intros
v1[intro]: "valid []"
v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)"
lemma eqvt_valid:
fixes pi:: "name prm"
assumes a: "valid \<Gamma>"
shows "valid (pi\<bullet>\<Gamma>)"
using a
apply(induct)
apply(auto simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
done
(* typing judgements *)
consts
typing :: "(((name\<times>ty) list)\<times>lam\<times>ty) set"
syntax
"_typing_judge" :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80)
translations
"\<Gamma> \<turnstile> t : \<tau>" \<rightleftharpoons> "(\<Gamma>,t,\<tau>) \<in> typing"
inductive typing
intros
t1[intro]: "\<lbrakk>valid \<Gamma>; (a,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : \<tau>"
t2[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; \<Gamma> \<turnstile> t2 : \<tau>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : \<sigma>"
t3[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : \<tau>\<rightarrow>\<sigma>"
lemma eqvt_typing:
fixes \<Gamma> :: "(name\<times>ty) list"
and t :: "lam"
and \<tau> :: "ty"
and pi:: "name prm"
assumes a: "\<Gamma> \<turnstile> t : \<tau>"
shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : \<tau>"
using a
proof (induct)
case (t1 \<Gamma> \<tau> a)
have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
moreover
have "(pi\<bullet>(a,\<tau>))\<in>((pi::name prm)\<bullet>set \<Gamma>)" by (rule pt_set_bij2[OF pt_name_inst, OF at_name_inst])
ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> ((pi::name prm)\<bullet>Var a) : \<tau>"
using typing.intros by (force simp add: pt_list_set_pi[OF pt_name_inst, symmetric])
next
case (t3 \<Gamma> \<sigma> \<tau> a t)
moreover have "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" by (rule pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Lam [a].t) :\<tau>\<rightarrow>\<sigma>" by force
qed (auto)
lemma typing_induct_weak[THEN spec, case_names t1 t2 t3]:
fixes P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a\<Rightarrow>bool"
and \<Gamma> :: "(name\<times>ty) list"
and t :: "lam"
and \<tau> :: "ty"
assumes a: "\<Gamma> \<turnstile> t : \<tau>"
and a1: "\<And>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
and a2: "\<And>x \<Gamma> \<tau> \<sigma> t1 t2.
\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<forall>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<forall>z. P \<Gamma> t2 \<tau> z)
\<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
and a3: "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t.
a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
\<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
shows "\<forall>x. P \<Gamma> t \<tau> x"
using a by (induct, simp_all add: a1 a2 a3)
lemma typing_induct_aux[rule_format]:
fixes P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a::fs_name\<Rightarrow>bool"
and \<Gamma> :: "(name\<times>ty) list"
and t :: "lam"
and \<tau> :: "ty"
assumes a: "\<Gamma> \<turnstile> t : \<tau>"
and a1: "\<And>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
and a2: "\<And>x \<Gamma> \<tau> \<sigma> t1 t2.
\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<And>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<And>z. P \<Gamma> t2 \<tau> z)
\<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
and a3: "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t.
a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
\<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
shows "\<forall>(pi::name prm) (x::'a::fs_name). P (pi\<bullet>\<Gamma>) (pi\<bullet>t) \<tau> x"
using a
proof (induct)
case (t1 \<Gamma> \<tau> a)
have j1: "valid \<Gamma>" by fact
have j2: "(a,\<tau>)\<in>set \<Gamma>" by fact
show ?case
proof (intro strip, simp)
fix pi::"name prm" and x::"'a::fs_name"
from j1 have j3: "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
from j2 have "pi\<bullet>(a,\<tau>)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])
hence j4: "(pi\<bullet>a,\<tau>)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_name_inst])
show "P (pi\<bullet>\<Gamma>) (Var (pi\<bullet>a)) \<tau> x" using a1 j3 j4 by force
qed
next
case (t2 \<Gamma> \<sigma> \<tau> t1 t2)
thus ?case using a2 by (simp, blast intro: eqvt_typing)
next
case (t3 \<Gamma> \<sigma> \<tau> a t)
have k1: "a\<sharp>\<Gamma>" by fact
have k2: "((a,\<tau>)#\<Gamma>)\<turnstile>t:\<sigma>" by fact
have k3: "\<forall>(pi::name prm) (x::'a::fs_name). P (pi \<bullet> ((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma> x" by fact
show ?case
proof (intro strip, simp)
fix pi::"name prm" and x::"'a::fs_name"
have f: "\<exists>c::name. c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)"
by (rule at_exists_fresh[OF at_name_inst], simp add: fs_name1)
then obtain c::"name"
where f1: "c\<noteq>(pi\<bullet>a)" and f2: "c\<sharp>x" and f3: "c\<sharp>(pi\<bullet>t)" and f4: "c\<sharp>(pi\<bullet>\<Gamma>)"
by (force simp add: fresh_prod at_fresh[OF at_name_inst])
from k1 have k1a: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)"
by (simp add: pt_fresh_left[OF pt_name_inst, OF at_name_inst]
pt_rev_pi[OF pt_name_inst, OF at_name_inst])
have l1: "(([(c,pi\<bullet>a)]@pi)\<bullet>\<Gamma>) = (pi\<bullet>\<Gamma>)" using f4 k1a
by (simp only: pt2[OF pt_name_inst], rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst])
have "\<forall>x. P (([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma> x" using k3 by force
hence l2: "\<forall>x. P ((c, \<tau>)#(pi\<bullet>\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma> x" using f1 l1
by (force simp add: pt2[OF pt_name_inst] at_calc[OF at_name_inst] split: if_splits)
have "(([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) \<turnstile> (([(c,pi\<bullet>a)]@pi)\<bullet>t) : \<sigma>" using k2 by (rule eqvt_typing)
hence l3: "((c, \<tau>)#(pi\<bullet>\<Gamma>)) \<turnstile> (([(c,pi\<bullet>a)]@pi)\<bullet>t) : \<sigma>" using l1 f1
by (force simp add: pt2[OF pt_name_inst] at_calc[OF at_name_inst] split: if_splits)
have l4: "P (pi\<bullet>\<Gamma>) (Lam [c].(([(c,pi\<bullet>a)]@pi)\<bullet>t)) (\<tau> \<rightarrow> \<sigma>) x" using f2 f4 l2 l3 a3 by auto
have alpha: "(Lam [c].([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t))) = (Lam [(pi\<bullet>a)].(pi\<bullet>t))" using f1 f3
by (simp add: lam.inject alpha)
show "P (pi\<bullet>\<Gamma>) (Lam [(pi\<bullet>a)].(pi\<bullet>t)) (\<tau> \<rightarrow> \<sigma>) x" using l4 alpha
by (simp only: pt2[OF pt_name_inst])
qed
qed
lemma typing_induct[case_names t1 t2 t3]:
fixes P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a::fs_name\<Rightarrow>bool"
and \<Gamma> :: "(name\<times>ty) list"
and t :: "lam"
and \<tau> :: "ty"
and x :: "'a::fs_name"
assumes a: "\<Gamma> \<turnstile> t : \<tau>"
and a1: "\<And>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
and a2: "\<And>x \<Gamma> \<tau> \<sigma> t1 t2.
\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<forall>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<forall>z. P \<Gamma> t2 \<tau> z)
\<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
and a3: "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t.
a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
\<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
shows "P \<Gamma> t \<tau> x"
using a a1 a2 a3 typing_induct_aux[of "\<Gamma>" "t" "\<tau>" "P" "[]" "x", simplified] by force
(* Now it comes: The Weakening Lemma *)
constdefs
"sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80)
"\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow> (a,\<sigma>)\<in>set \<Gamma>2"
lemma weakening_version1[rule_format]:
assumes a: "\<Gamma>1 \<turnstile> t : \<sigma>"
shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
using a
apply(nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
apply(auto simp add: sub_def)
done
lemma weakening_version2[rule_format]:
fixes \<Gamma>1::"(name\<times>ty) list"
and t ::"lam"
and \<tau> ::"ty"
assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
using a
proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct, auto)
case (t1 \<Gamma>2 \<Gamma>1 a \<tau>) (* variable case *)
assume "\<Gamma>1 \<lless> \<Gamma>2"
and "valid \<Gamma>2"
and "(a,\<tau>)\<in> set \<Gamma>1"
thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
next
case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
and a2: "valid \<Gamma>2"
and a3: "a\<sharp>\<Gamma>2"
have i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
moreover
have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using i by force
with a3 show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
qed
lemma weakening_version3[rule_format]:
fixes \<Gamma>1::"(name\<times>ty) list"
and t ::"lam"
and \<tau> ::"ty"
assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
using a
proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
case (t1 \<Gamma>2 \<Gamma>1 a \<tau>) (* variable case *)
thus "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
next
case (t2 \<Gamma>2 \<Gamma>1 \<tau> \<sigma> t1 t2) (* variable case *)
thus "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> App t1 t2 : \<sigma>" by force
next
case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
have a3: "a\<sharp>\<Gamma>2"
and i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
show "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>"
proof (intro strip)
assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
and a2: "valid \<Gamma>2"
have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
moreover
have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using i by force
with a3 show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
qed
qed
lemma weakening_version4[rule_format]:
assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
using a
proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
have fc: "a\<sharp>\<Gamma>2"
and ih: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>" by fact
show "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>"
proof (intro strip)
assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
and a2: "valid \<Gamma>2"
have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 sub_def by simp
moreover
have "valid ((a,\<tau>)#\<Gamma>2)" using a2 fc by force
ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by force
with fc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
qed
qed (auto simp add: sub_def) (* lam and var case *)
(* original induction principle is not strong *)
(* enough - so the simple proof fails *)
lemma weakening_too_weak[rule_format]:
assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
using a
proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct_weak, auto)
case (t1 \<Gamma>2 \<Gamma>1 a \<tau>) (* variable case *)
assume "\<Gamma>1 \<lless> \<Gamma>2"
and "valid \<Gamma>2"
and "(a,\<tau>)\<in> set \<Gamma>1"
thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
next
case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
and a2: "valid \<Gamma>2"
and i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow> \<Gamma>3 \<turnstile> t:\<sigma>"
have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
moreover
have "valid ((a,\<tau>)#\<Gamma>2)" using v2 (* fails *)