theory Weakening

imports "../Nominal2" 

begin

section {* The Weakening property in the simply-typed lambda-calculus *}

atom_decl name

nominal_datatype lam =

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam" bind x in l ("Lam [_]. _" [100,100] 100)

text {* Typing *}

nominal_datatype ty =

  TVar string

| TFun ty ty ("_ \<rightarrow> _" [100,100] 100) 

lemma fresh_ty:

  fixes x::"name"

  and   T::"ty"

  shows "atom x \<sharp> T"

  by (nominal_induct T rule: ty.strong_induct)

     (simp_all add: ty.fresh pure_fresh)

text {* Valid typing contexts *}

inductive

  valid :: "(name \<times> ty) list \<Rightarrow> bool"

where

  v_Nil[intro]: "valid []"

| v_Cons[intro]: "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T) # Gamma)"

equivariance valid

text {* Typing judgements *}

inductive

  typing :: "(name \<times> ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60) 

where

    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : 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>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x]. t : T1 \<rightarrow> T2"

equivariance typing

text {* Strong induction principle for typing judgements *}

nominal_inductive typing

  avoids t_Lam: "x"

  by (simp_all add: fresh_star_def fresh_ty lam.fresh)

abbreviation

  "sub_context" :: "(name \<times> ty) list \<Rightarrow> (name \<times> ty) list \<Rightarrow> bool" (infixr "\<subseteq>" 60) 

where

  "\<Gamma>1 \<subseteq> \<Gamma>2 \<equiv> \<forall>x T. (x, T) \<in> set \<Gamma>1 \<longrightarrow> (x, T) \<in> set \<Gamma>2"

text {* The proof *}

lemma weakening_version1: 

  fixes \<Gamma>1 \<Gamma>2::"(name \<times> ty) list"

  and   t ::"lam"

  and   \<tau> ::"ty"

  assumes a: "\<Gamma>1 \<turnstile> t : T"

  and     b: "valid \<Gamma>2" 

  and     c: "\<Gamma>1 \<subseteq> \<Gamma>2"

  shows "\<Gamma>2 \<turnstile> t : T"

using a b c

proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)

  case (t_Var \<Gamma>1 x T)  (* variable case *)

  have "\<Gamma>1 \<subseteq> \<Gamma>2" by fact 

  moreover  

  have "valid \<Gamma>2" by fact 

  moreover 

  have "(x, T) \<in> set \<Gamma>1" by fact

  ultimately show "\<Gamma>2 \<turnstile> Var x : T" by auto

next

  case (t_Lam x \<Gamma>1 T1 t T2) (* lambda case *)

  have vc: "atom x \<sharp> \<Gamma>2" by fact   (* variable convention *)

  have ih: "\<lbrakk>valid ((x, T1) # \<Gamma>2); (x, T1) # \<Gamma>1 \<subseteq> (x, T1) # \<Gamma>2\<rbrakk> \<Longrightarrow> (x, T1) # \<Gamma>2 \<turnstile> t : T2" by fact

  have "\<Gamma>1 \<subseteq> \<Gamma>2" by fact

  then have "(x, T1) # \<Gamma>1 \<subseteq> (x, T1) # \<Gamma>2" by simp

  moreover

  have "valid \<Gamma>2" by fact

  then have "valid ((x, T1) # \<Gamma>2)" using vc by auto

  ultimately have "(x, T1) # \<Gamma>2 \<turnstile> t : T2" using ih by simp

  with vc show "\<Gamma>2 \<turnstile> Lam [x]. t : T1 \<rightarrow> T2" by auto

qed (auto) (* app case *)

lemma weakening_version2: 

  fixes \<Gamma>1 \<Gamma>2::"(name \<times> ty) list"

  assumes a: "\<Gamma>1 \<turnstile> t : T" 

  and     b: "valid \<Gamma>2" 

  and     c: "\<Gamma>1 \<subseteq> \<Gamma>2"

  shows "\<Gamma>2 \<turnstile> t : T"

using a b c

by (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)

   (auto | atomize)+

text {* A version with finite sets as typing contexts *}

inductive

  valid2 :: "(name \<times> ty) fset \<Rightarrow> bool"

where

  v2_Empty[intro]: "valid2 {||}"

| v2_Insert[intro]: "\<lbrakk>(x, T) |\<notin>| Gamma; valid2 Gamma\<rbrakk> \<Longrightarrow> valid2 (insert_fset (x, T) Gamma)"

equivariance valid2

inductive

  typing2 :: "(name \<times> ty) fset \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ 2\<turnstile> _ : _" [60,60,60] 60) 

where

    t2_Var[intro]: "\<lbrakk>valid2 \<Gamma>; (x, T) |\<in>| \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> 2\<turnstile> Var x : T"

  | t2_App[intro]: "\<lbrakk>\<Gamma> 2\<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> 2\<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> 2\<turnstile> App t1 t2 : T2"

  | t2_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; insert_fset (x, T1) \<Gamma> 2\<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> 2\<turnstile> Lam [x]. t : T1 \<rightarrow> T2"

equivariance typing2

nominal_inductive typing2

  avoids t2_Lam: "x"

  by (simp_all add: fresh_star_def fresh_ty lam.fresh)

lemma weakening_version3: 

  fixes \<Gamma>::"(name \<times> ty) fset"

  assumes a: "\<Gamma> 2\<turnstile> t : T" 

  and     b: "(x, T') |\<notin>| \<Gamma>"

  shows "{|(x, T')|} |\<union>| \<Gamma> 2\<turnstile> t : T"

using a b

apply(nominal_induct \<Gamma> t T avoiding: x rule: typing2.strong_induct)

apply(auto)[2]

apply(rule t2_Lam)

apply(simp add: fresh_insert_fset fresh_Pair fresh_ty)

apply(simp)

apply(drule_tac x="xa" in meta_spec)

apply(drule meta_mp)

apply(simp add: fresh_at_base)

apply(simp add: insert_fset_left_comm)

done

lemma weakening_version4: 

  fixes \<Gamma>::"(name \<times> ty) fset"

  assumes a: "\<Gamma> 2\<turnstile> t : T" 

  and     b: "(x, T') |\<notin>| \<Gamma>"

  shows "{|(x, T')|} |\<union>| \<Gamma> 2\<turnstile> t : T"

using a b

apply(induct \<Gamma> t T arbitrary: x)

apply(auto)[2]

apply(rule t2_Lam)

oops

end