src/HOL/Nominal/Examples/Weakening.thy
author urbanc
Wed, 11 Jan 2006 14:00:11 +0100
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parent 18354 715d6df89fcc
child 18654 94782c7c4247
permissions -rw-r--r--
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(* $Id$ *)
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theory weakening 
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imports "../nominal" 
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begin
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(* WEAKENING EXAMPLE*)
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section {* Simply-Typed Lambda-Calculus *}
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(*======================================*)
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atom_decl name 
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nominal_datatype lam = Var "name"
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                     | App "lam" "lam"
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                     | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
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nominal_datatype ty =
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    TVar "nat"
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  | TArr "ty" "ty" (infix "\<rightarrow>" 200)
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lemma perm_ty[simp]:
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  fixes pi ::"name prm"
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  and   \<tau>  ::"ty"
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  shows "pi\<bullet>\<tau> = \<tau>"
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by (induct \<tau> rule: ty.induct_weak, simp_all add: perm_nat_def)  
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(* valid contexts *)
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consts
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  ctxts :: "((name\<times>ty) list) set" 
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  valid :: "(name\<times>ty) list \<Rightarrow> bool"
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translations
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  "valid \<Gamma>" \<rightleftharpoons> "\<Gamma> \<in> ctxts"  
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inductive ctxts
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intros
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v1[intro]: "valid []"
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v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)"
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lemma eqvt_valid:
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  fixes   pi:: "name prm"
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  assumes a: "valid \<Gamma>"
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  shows   "valid (pi\<bullet>\<Gamma>)"
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using a
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apply(induct)
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apply(auto simp add: fresh_eqvt)
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done
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(* typing judgements *)
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consts
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  typing :: "(((name\<times>ty) list)\<times>lam\<times>ty) set" 
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syntax
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  "_typing_judge" :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80) 
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translations
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  "\<Gamma> \<turnstile> t : \<tau>" \<rightleftharpoons> "(\<Gamma>,t,\<tau>) \<in> typing"  
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inductive typing
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intros
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t1[intro]: "\<lbrakk>valid \<Gamma>; (a,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : \<tau>"
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t2[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; \<Gamma> \<turnstile> t2 : \<tau>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : \<sigma>"
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t3[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : \<tau>\<rightarrow>\<sigma>"
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lemma eqvt_typing: 
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  fixes  \<Gamma> :: "(name\<times>ty) list"
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  and    t :: "lam"
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  and    \<tau> :: "ty"
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  and    pi:: "name prm"
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  assumes a: "\<Gamma> \<turnstile> t : \<tau>"
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  shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : \<tau>"
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using a
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proof (induct)
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  case (t1 \<Gamma> \<tau> a)
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  have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
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  moreover
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  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])
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  ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> ((pi::name prm)\<bullet>Var a) : \<tau>"
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    using typing.intros by (force simp add: pt_list_set_pi[OF pt_name_inst, symmetric])
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next 
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  case (t3 \<Gamma> \<sigma> \<tau> a t)
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  moreover have "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" by (rule pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
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  ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Lam [a].t) :\<tau>\<rightarrow>\<sigma>" by force 
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qed (auto)
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lemma typing_induct[consumes 1, case_names t1 t2 t3]:
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  fixes  P :: "'a::fs_name\<Rightarrow>(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>bool"
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  and    \<Gamma> :: "(name\<times>ty) list"
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  and    t :: "lam"
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  and    \<tau> :: "ty"
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  and    x :: "'a::fs_name"
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  assumes a: "\<Gamma> \<turnstile> t : \<tau>"
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  and a1:    "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P x \<Gamma> (Var a) \<tau>"
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  and a2:    "\<And>\<Gamma> \<tau> \<sigma> t1 t2 x. 
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              \<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<And>z. P z \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>)) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<And>z. P z \<Gamma> t2 \<tau>)
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              \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
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  and a3:    "\<And>a \<Gamma> \<tau> \<sigma> t x. a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma>)
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              \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
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  shows "P x \<Gamma> t \<tau>"
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proof -
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  from a have "\<And>(pi::name prm) x. P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) \<tau>"
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  proof (induct)
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    case (t1 \<Gamma> \<tau> a)
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    have j1: "valid \<Gamma>" by fact
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    have j2: "(a,\<tau>)\<in>set \<Gamma>" by fact
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    from j1 have j3: "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
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    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])  
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    hence j4: "(pi\<bullet>a,\<tau>)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_name_inst])
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   106
    show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) \<tau>" using a1 j3 j4 by simp
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  next
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    case (t2 \<Gamma> \<sigma> \<tau> t1 t2)
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    thus ?case using a2 by (simp, blast intro: eqvt_typing)
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  next
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    case (t3 \<Gamma> \<sigma> \<tau> a t)
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    have k1: "a\<sharp>\<Gamma>" by fact
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    have k2: "((a,\<tau>)#\<Gamma>)\<turnstile>t:\<sigma>" by fact
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    have k3: "\<And>(pi::name prm) (x::'a::fs_name). P x (pi \<bullet>((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma>" by fact
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    have f: "\<exists>c::name. c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)"
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      by (rule at_exists_fresh[OF at_name_inst], simp add: fs_name1)
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    then obtain c::"name" 
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      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>)"
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      by (force simp add: fresh_prod at_fresh[OF at_name_inst])
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    from k1 have k1a: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" 
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      by (simp add: pt_fresh_left[OF pt_name_inst, OF at_name_inst] 
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                    pt_rev_pi[OF pt_name_inst, OF at_name_inst])
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   123
    have l1: "(([(c,pi\<bullet>a)]@pi)\<bullet>\<Gamma>) = (pi\<bullet>\<Gamma>)" using f4 k1a 
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      by (simp only: pt2[OF pt_name_inst], rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst])
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   125
    have "\<And>x. P x (([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma>" using k3 by force
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    hence l2: "\<And>x. P x ((c, \<tau>)#(pi\<bullet>\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma>" using f1 l1
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      by (force simp add: pt2[OF pt_name_inst]  at_calc[OF at_name_inst])
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   128
    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)
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   129
    hence l3: "((c, \<tau>)#(pi\<bullet>\<Gamma>)) \<turnstile> (([(c,pi\<bullet>a)]@pi)\<bullet>t) : \<sigma>" using l1 f1 
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      by (force simp add: pt2[OF pt_name_inst]  at_calc[OF at_name_inst])
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   131
    have l4: "P x (pi\<bullet>\<Gamma>) (Lam [c].(([(c,pi\<bullet>a)]@pi)\<bullet>t)) (\<tau> \<rightarrow> \<sigma>)" using f2 f4 l2 l3 a3 by auto
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    have alpha: "(Lam [c].([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t))) = (Lam [(pi\<bullet>a)].(pi\<bullet>t))" using f1 f3
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   133
      by (simp add: lam.inject alpha)
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   134
    show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (\<tau> \<rightarrow> \<sigma>)" using l4 alpha 
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   135
      by (simp only: pt2[OF pt_name_inst], simp)
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   136
  qed
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   137
  hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) \<tau>" by blast
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   138
  thus "P x \<Gamma> t \<tau>" by simp
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   139
qed
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4c9c081a416b Initial commit of the theory "Weakening".
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   140
4c9c081a416b Initial commit of the theory "Weakening".
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(* Now it comes: The Weakening Lemma *)
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4c9c081a416b Initial commit of the theory "Weakening".
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constdefs
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  "sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80)
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  "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow>  (a,\<sigma>)\<in>set \<Gamma>2"
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   146
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   147
lemma weakening_version1: 
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   148
  assumes a: "\<Gamma>1 \<turnstile> t : \<sigma>" 
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   149
  and     b: "valid \<Gamma>2" 
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   150
  and     c: "\<Gamma>1 \<lless> \<Gamma>2"
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   151
  shows "\<Gamma>2 \<turnstile> t:\<sigma>"
18311
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   152
using a b c
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   153
apply(nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
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4c9c081a416b Initial commit of the theory "Weakening".
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apply(auto simp add: sub_def)
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   155
(* FIXME: this was completely automatic before the *)
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   156
(* change to meta-connectives :o(                  *)
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   157
apply(atomize)
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   158
apply(auto)
18105
4c9c081a416b Initial commit of the theory "Weakening".
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   159
done
4c9c081a416b Initial commit of the theory "Weakening".
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diff changeset
   160
18311
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   161
lemma weakening_version2: 
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4c9c081a416b Initial commit of the theory "Weakening".
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  fixes \<Gamma>1::"(name\<times>ty) list"
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  and   t ::"lam"
4c9c081a416b Initial commit of the theory "Weakening".
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  and   \<tau> ::"ty"
4c9c081a416b Initial commit of the theory "Weakening".
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   165
  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
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   166
  and     b: "valid \<Gamma>2" 
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   167
  and     c: "\<Gamma>1 \<lless> \<Gamma>2"
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   168
  shows "\<Gamma>2 \<turnstile> t:\<sigma>"
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   169
using a b c
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   170
proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct, auto)
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   171
  case (t1 \<Gamma>1 a \<tau>)  (* variable case *)
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   172
  have "\<Gamma>1 \<lless> \<Gamma>2" 
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   173
  and  "valid \<Gamma>2" 
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   174
  and  "(a,\<tau>)\<in> set \<Gamma>1" by fact+
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   175
  thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
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4c9c081a416b Initial commit of the theory "Weakening".
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   176
next
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   177
  case (t3 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
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   178
  have a1: "\<Gamma>1 \<lless> \<Gamma>2" by fact
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   179
  have a2: "valid \<Gamma>2" by fact
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   180
  have a3: "a\<sharp>\<Gamma>2" by fact
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   181
  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
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4c9c081a416b Initial commit of the theory "Weakening".
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   182
  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
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   183
  moreover
4c9c081a416b Initial commit of the theory "Weakening".
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   184
  have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
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   185
  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by force
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4c9c081a416b Initial commit of the theory "Weakening".
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   186
  with a3 show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
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qed
4c9c081a416b Initial commit of the theory "Weakening".
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   188
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   189
lemma weakening_version3: 
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   190
  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
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   191
  and     b: "valid \<Gamma>2" 
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   192
  and     c: "\<Gamma>1 \<lless> \<Gamma>2"
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   193
  shows "\<Gamma>2 \<turnstile> t:\<sigma>"
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   194
using a b c
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   195
proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
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   196
  case (t3 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
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   197
  have fc: "a\<sharp>\<Gamma>2" by fact
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   198
  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact 
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   199
  have a1: "\<Gamma>1 \<lless> \<Gamma>2" by fact
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   200
  have a2: "valid \<Gamma>2" by fact
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   201
  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 sub_def by simp 
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   202
  moreover
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   203
  have "valid ((a,\<tau>)#\<Gamma>2)" using a2 fc by force
18346
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diff changeset
   204
  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp 
18311
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   205
  with fc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
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   206
qed (auto simp add: sub_def) (* app and var case *)
18105
4c9c081a416b Initial commit of the theory "Weakening".
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   207
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   208
text{* The original induction principle for typing 
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   209
       is not strong enough - so the simple proof fails *}
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   210
lemma weakening_too_weak: 
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   211
  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
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   212
  and     b: "valid \<Gamma>2" 
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   213
  and     c: "\<Gamma>1 \<lless> \<Gamma>2"
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   214
  shows "\<Gamma>2 \<turnstile> t:\<sigma>"
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   215
using a b c
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   216
proof (induct fixing: \<Gamma>2)
18296
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   217
  case (t1 \<Gamma>1 \<tau> a) (* variable case *)
18311
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diff changeset
   218
  have "\<Gamma>1 \<lless> \<Gamma>2" 
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diff changeset
   219
  and  "valid \<Gamma>2"
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   220
  and  "(a,\<tau>) \<in> (set \<Gamma>1)" by fact+ 
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parents:
diff changeset
   221
  thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
4c9c081a416b Initial commit of the theory "Weakening".
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diff changeset
   222
next
18311
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diff changeset
   223
  case (t3 \<Gamma>1 \<sigma> \<tau> a t) (* lambda case *)
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   224
  (* all assumption in this case*)
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diff changeset
   225
  have a0: "a\<sharp>\<Gamma>1" by fact
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diff changeset
   226
  have a1: "((a,\<tau>)#\<Gamma>1) \<turnstile> t : \<sigma>" by fact
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diff changeset
   227
  have a2: "\<Gamma>1 \<lless> \<Gamma>2" by fact
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parents: 18296
diff changeset
   228
  have a3: "valid \<Gamma>2" by fact
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diff changeset
   229
  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
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parents: 18296
diff changeset
   230
  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a2 by (simp add: sub_def)
18105
4c9c081a416b Initial commit of the theory "Weakening".
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diff changeset
   231
  moreover
18311
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diff changeset
   232
  have "valid ((a,\<tau>)#\<Gamma>2)" using v2 (* fails *) 
18105
4c9c081a416b Initial commit of the theory "Weakening".
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parents:
diff changeset
   233
4c9c081a416b Initial commit of the theory "Weakening".
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parents:
diff changeset
   234
4c9c081a416b Initial commit of the theory "Weakening".
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parents:
diff changeset
   235