theory Lambda
imports "../Nominal2" 
begin


atom_decl name

nominal_datatype lam =
  Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam"  bind x in l

thm lam.distinct
thm lam.induct
thm lam.exhaust lam.strong_exhaust
thm lam.fv_defs
thm lam.bn_defs
thm lam.perm_simps
thm lam.eq_iff
thm lam.fv_bn_eqvt
thm lam.size_eqvt


section {* Typing *}

nominal_datatype ty =
  TVar string
| TFun ty ty ("_ \<rightarrow> _") 

lemma ty_fresh:
  fixes x::"name"
  and   T::"ty"
  shows "atom x \<sharp> T"
apply (nominal_induct T rule: ty.strong_induct)
apply (simp_all add: ty.fresh pure_fresh)
done

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)"

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"

thm typing.intros
thm typing.induct

equivariance valid
equivariance typing

nominal_inductive typing
  avoids t_Lam: "x"
  by (simp_all add: fresh_star_def ty_fresh lam.fresh)


thm typing.strong_induct

abbreviation
  "sub_context" :: "(name \<times> ty) list \<Rightarrow> (name \<times> ty) list \<Rightarrow> bool" ("_ \<subseteq> _" [60,60] 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 {* Now it comes: The Weakening Lemma *}

text {*
  The first version is, after setting up the induction, 
  completely automatic except for use of atomize. *}

lemma weakening_version2: 
  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 (simp add: v_Cons)
  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_version1: 
  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
apply (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
apply (auto | atomize)+
done


inductive
  Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
where
  AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"


lemma Acc_eqvt [eqvt]:
  fixes p::"perm"
  assumes a: "Acc R x"
  shows "Acc (p \<bullet> R) (p \<bullet> x)"
using a
apply(induct)
apply(rule AccI)
apply(rotate_tac 1)
apply(drule_tac x="-p \<bullet> y" in meta_spec)
apply(simp)
apply(drule meta_mp)
apply(rule_tac p="p" in permute_boolE)
apply(perm_simp add: permute_minus_cancel)
apply(assumption)
apply(assumption)
done
 

nominal_inductive Acc .

thm Acc.strong_induct


end