Theory Crary

Up to index of Isabelle/HOL/HOL-Nominal/Examples

theory Crary
imports Nominal

(*                                                    *)
(* Formalisation of the chapter on Logical Relations  *)
(* and a Case Study in Equivalence Checking           *)
(* by Karl Crary from the book on Advanced Topics in  *)
(* Types and Programming Languages, MIT Press 2005    *)

(* The formalisation was done by Julien Narboux and   *)
(* Christian Urban.                                   *)

theory Crary
  imports "../Nominal"
begin

atom_decl name 

nominal_datatype ty = 
    TBase 
  | TUnit 
  | Arrow "ty" "ty" ("_->_" [100,100] 100)

nominal_datatype trm = 
    Unit
  | Var "name" ("Var _" [100] 100)
  | Lam "«name»trm" ("Lam [_]._" [100,100] 100)
  | App "trm" "trm" ("App _ _" [110,110] 100)
  | Const "nat"

types Ctxt  = "(name×ty) list"
types Subst = "(name×trm) list" 


lemma perm_ty[simp]: 
  fixes T::"ty"
  and   pi::"name prm"
  shows "pi•T = T"
  by (induct T rule: ty.induct) (simp_all)

lemma fresh_ty[simp]:
  fixes x::"name" 
  and   T::"ty"
  shows "x\<sharp>T"
  by (simp add: fresh_def supp_def)

lemma ty_cases:
  fixes T::ty
  shows "(∃ T1 T2. T=T1->T2) ∨ T=TUnit ∨ T=TBase"
by (induct T rule:ty.induct) (auto)

instantiation ty :: size
begin

nominal_primrec size_ty
where
  "size (TBase) = 1"
| "size (TUnit) = 1"
| "size (T1->T2) = size T1 + size T2"
by (rule TrueI)+

instance ..

end

lemma ty_size_greater_zero[simp]:
  fixes T::"ty"
  shows "size T > 0"
by (nominal_induct rule: ty.strong_induct) (simp_all)

section {* Substitutions *}

fun
  lookup :: "Subst => name => trm"   
where
  "lookup [] x        = Var x"
| "lookup ((y,T)#ϑ) x = (if x=y then T else lookup ϑ x)"

lemma lookup_eqvt[eqvt]:
  fixes pi::"name prm"
  shows "pi•(lookup ϑ x) = lookup (pi•ϑ) (pi•x)"
by (induct ϑ) (auto simp add: perm_bij)

lemma lookup_fresh:
  fixes z::"name"
  assumes a: "z\<sharp>ϑ" "z\<sharp>x"
  shows "z\<sharp> lookup ϑ x"
using a
by (induct rule: lookup.induct) 
   (auto simp add: fresh_list_cons)

lemma lookup_fresh':
  assumes a: "z\<sharp>ϑ"
  shows "lookup ϑ z = Var z"
using a
by (induct rule: lookup.induct)
   (auto simp add: fresh_list_cons fresh_prod fresh_atm)
 
nominal_primrec
  psubst :: "Subst => trm => trm"  ("_<_>" [100,100] 130)
where
  "ϑ<(Var x)> = (lookup ϑ x)"
| "ϑ<(App t1 t2)> = App ϑ<t1> ϑ<t2>"
| "x\<sharp>ϑ ==> ϑ<(Lam [x].t)> = Lam [x].(ϑ<t>)"
| "ϑ<(Const n)> = Const n"
| "ϑ<(Unit)> = Unit"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
done

abbreviation 
 subst :: "trm => name => trm => trm" ("_[_::=_]" [100,100,100] 100)
where
  "t[x::=t']  ≡ ([(x,t')])<t>" 

lemma subst[simp]:
  shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
  and   "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
  and   "x\<sharp>(y,t') ==> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
  and   "Const n[y::=t'] = Const n"
  and   "Unit [y::=t'] = Unit"
  by (simp_all add: fresh_list_cons fresh_list_nil)

lemma subst_eqvt[eqvt]:
  fixes pi::"name prm" 
  shows "pi•(t[x::=t']) = (pi•t)[(pi•x)::=(pi•t')]"
  by (nominal_induct t avoiding: x t' rule: trm.strong_induct)
     (perm_simp add: fresh_bij)+

lemma subst_rename: 
  fixes c::"name"
  assumes a: "c\<sharp>t1"
  shows "t1[a::=t2] = ([(c,a)]•t1)[c::=t2]"
using a
apply(nominal_induct t1 avoiding: a c t2 rule: trm.strong_induct)
apply(simp add: trm.inject calc_atm fresh_atm abs_fresh perm_nat_def)+
done

lemma fresh_psubst: 
  fixes z::"name"
  assumes a: "z\<sharp>t" "z\<sharp>ϑ"
  shows "z\<sharp>(ϑ<t>)"
using a
by (nominal_induct t avoiding: z ϑ t rule: trm.strong_induct)
   (auto simp add: abs_fresh lookup_fresh)

lemma fresh_subst'':
  fixes z::"name"
  assumes "z\<sharp>t2"
  shows "z\<sharp>t1[z::=t2]"
using assms 
by (nominal_induct t1 avoiding: t2 z rule: trm.strong_induct)
   (auto simp add: abs_fresh fresh_nat fresh_atm)

lemma fresh_subst':
  fixes z::"name"
  assumes "z\<sharp>[y].t1" "z\<sharp>t2"
  shows "z\<sharp>t1[y::=t2]"
using assms 
by (nominal_induct t1 avoiding: y t2 z rule: trm.strong_induct)
   (auto simp add: abs_fresh fresh_nat fresh_atm)

lemma fresh_subst:
  fixes z::"name"
  assumes a: "z\<sharp>t1" "z\<sharp>t2"
  shows "z\<sharp>t1[y::=t2]"
using a 
by (auto simp add: fresh_subst' abs_fresh) 

lemma fresh_psubst_simp:
  assumes "x\<sharp>t"
  shows "((x,u)#ϑ)<t> = ϑ<t>" 
using assms
proof (nominal_induct t avoiding: x u ϑ rule: trm.strong_induct)
  case (Lam y t x u)
  have fs: "y\<sharp>ϑ" "y\<sharp>x" "y\<sharp>u" by fact+
  moreover have "x\<sharp> Lam [y].t" by fact 
  ultimately have "x\<sharp>t" by (simp add: abs_fresh fresh_atm)
  moreover have ih:"!!n T. n\<sharp>t ==> ((n,T)#ϑ)<t> = ϑ<t>" by fact
  ultimately have "((x,u)#ϑ)<t> = ϑ<t>" by auto
  moreover have "((x,u)#ϑ)<Lam [y].t> = Lam [y].(((x,u)#ϑ)<t>)" using fs 
    by (simp add: fresh_list_cons fresh_prod)
  moreover have " ϑ<Lam [y].t> = Lam [y]. (ϑ<t>)" using fs by simp
  ultimately show "((x,u)#ϑ)<Lam [y].t> = ϑ<Lam [y].t>" by auto
qed (auto simp add: fresh_atm abs_fresh)

lemma forget: 
  fixes x::"name"
  assumes a: "x\<sharp>t" 
  shows "t[x::=t'] = t"
  using a
by (nominal_induct t avoiding: x t' rule: trm.strong_induct)
   (auto simp add: fresh_atm abs_fresh)

lemma subst_fun_eq:
  fixes u::trm
  assumes h:"[x].t1 = [y].t2"
  shows "t1[x::=u] = t2[y::=u]"
proof -
  { 
    assume "x=y" and "t1=t2"
    then have ?thesis using h by simp
  }
  moreover 
  {
    assume h1:"x ≠ y" and h2:"t1=[(x,y)] • t2" and h3:"x \<sharp> t2"
    then have "([(x,y)] • t2)[x::=u] = t2[y::=u]" by (simp add: subst_rename)
    then have ?thesis using h2 by simp 
  }
  ultimately show ?thesis using alpha h by blast
qed

lemma psubst_empty[simp]:
  shows "[]<t> = t"
by (nominal_induct t rule: trm.strong_induct) 
   (auto simp add: fresh_list_nil)

lemma psubst_subst_psubst:
  assumes h:"c\<sharp>ϑ"
  shows "ϑ<t>[c::=s] = ((c,s)#ϑ)<t>"
  using h
by (nominal_induct t avoiding: ϑ c s rule: trm.strong_induct)
   (auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh' fresh_psubst)

lemma subst_fresh_simp:
  assumes a: "x\<sharp>ϑ"
  shows "ϑ<Var x> = Var x"
using a
by (induct ϑ arbitrary: x, auto simp add:fresh_list_cons fresh_prod fresh_atm)

lemma psubst_subst_propagate: 
  assumes "x\<sharp>ϑ" 
  shows "ϑ<t[x::=u]> = ϑ<t>[x::=ϑ<u>]"
using assms
proof (nominal_induct t avoiding: x u ϑ rule: trm.strong_induct)
  case (Var n x u ϑ)
  { assume "x=n"
    moreover have "x\<sharp>ϑ" by fact 
    ultimately have "ϑ<Var n[x::=u]> = ϑ<Var n>[x::=ϑ<u>]" using subst_fresh_simp by auto
  }
  moreover 
  { assume h:"x≠n"
    then have "x\<sharp>Var n" by (auto simp add: fresh_atm) 
    moreover have "x\<sharp>ϑ" by fact
    ultimately have "x\<sharp>ϑ<Var n>" using fresh_psubst by blast
    then have " ϑ<Var n>[x::=ϑ<u>] =  ϑ<Var n>" using forget by auto
    then have "ϑ<Var n[x::=u]> = ϑ<Var n>[x::=ϑ<u>]" using h by auto
  }
  ultimately show ?case by auto  
next
  case (Lam n t x u ϑ)
  have fs:"n\<sharp>x" "n\<sharp>u" "n\<sharp>ϑ" "x\<sharp>ϑ" by fact+
  have ih:"!! y s ϑ. y\<sharp>ϑ ==> ((ϑ<(t[y::=s])>) = ((ϑ<t>)[y::=(ϑ<s>)]))" by fact
  have "ϑ <(Lam [n].t)[x::=u]> = ϑ<Lam [n]. (t [x::=u])>" using fs by auto
  then have "ϑ <(Lam [n].t)[x::=u]> = Lam [n]. ϑ<t [x::=u]>" using fs by auto
  moreover have "ϑ<t[x::=u]> = ϑ<t>[x::=ϑ<u>]" using ih fs by blast
  ultimately have "ϑ <(Lam [n].t)[x::=u]> = Lam [n].(ϑ<t>[x::=ϑ<u>])" by auto
  moreover have "Lam [n].(ϑ<t>[x::=ϑ<u>]) = (Lam [n].ϑ<t>)[x::=ϑ<u>]" using fs fresh_psubst by auto
  ultimately have "ϑ<(Lam [n].t)[x::=u]> = (Lam [n].ϑ<t>)[x::=ϑ<u>]" using fs by auto
  then show "ϑ<(Lam [n].t)[x::=u]> = ϑ<Lam [n].t>[x::=ϑ<u>]" using fs by auto
qed (auto)

section {* Typing *}

inductive
  valid :: "Ctxt => bool"
where
  v_nil[intro]:  "valid []"
| v_cons[intro]: "[|valid Γ;a\<sharp>Γ|] ==> valid ((a,T)#Γ)"

equivariance valid 

inductive_cases
  valid_cons_elim_auto[elim]:"valid ((x,T)#Γ)"

abbreviation
  "sub_context" :: "Ctxt => Ctxt => bool" (" _ ⊆ _ " [55,55] 55)
where
  1 ⊆ Γ2 ≡ ∀a T. (a,T)∈set Γ1 --> (a,T)∈set Γ2"

lemma valid_monotonicity[elim]:
 fixes Γ Γ' :: Ctxt
 assumes a: "Γ ⊆ Γ'" 
 and     b: "x\<sharp>Γ'"
 shows "(x,T1)#Γ ⊆ (x,T1)#Γ'"
using a b by auto

lemma fresh_context: 
  fixes  Γ :: "Ctxt"
  and    a :: "name"
  assumes "a\<sharp>Γ"
  shows "¬(∃τ::ty. (a,τ)∈set Γ)"
using assms 
by (induct Γ)
   (auto simp add: fresh_prod fresh_list_cons fresh_atm)

lemma type_unicity_in_context:
  assumes a: "valid Γ" 
  and     b: "(x,T1) ∈ set Γ" 
  and     c: "(x,T2) ∈ set Γ"
  shows "T1=T2"
using a b c
by (induct Γ)
   (auto dest!: fresh_context)

inductive
  typing :: "Ctxt=>trm=>ty=>bool" (" _ \<turnstile> _ : _ " [60,60,60] 60) 
where
  T_Var[intro]:   "[|valid Γ; (x,T)∈set Γ|] ==> Γ \<turnstile> Var x : T"
| T_App[intro]:   "[|Γ \<turnstile> e1 : T1->T2; Γ \<turnstile> e2 : T1|] ==> Γ \<turnstile> App e1 e2 : T2"
| T_Lam[intro]:   "[|x\<sharp>Γ; (x,T1)#Γ \<turnstile> t : T2|] ==> Γ \<turnstile> Lam [x].t : T1->T2"
| T_Const[intro]: "valid Γ ==> Γ \<turnstile> Const n : TBase"
| T_Unit[intro]:  "valid Γ ==> Γ \<turnstile> Unit : TUnit"

equivariance typing

nominal_inductive typing
  by (simp_all add: abs_fresh)

lemma typing_implies_valid:
  assumes a: "Γ \<turnstile> t : T"
  shows "valid Γ"
  using a by (induct) (auto)

declare trm.inject [simp add]
declare ty.inject  [simp add]

inductive_cases typing_inv_auto[elim]:
  "Γ \<turnstile> Lam [x].t : T"
  "Γ \<turnstile> Var x : T"
  "Γ \<turnstile> App x y : T"
  "Γ \<turnstile> Const n : T"
  "Γ \<turnstile> Unit : TUnit"
  "Γ \<turnstile> s : TUnit"

declare trm.inject [simp del]
declare ty.inject [simp del]


section {* Definitional Equivalence *}

inductive
  def_equiv :: "Ctxt=>trm=>trm=>ty=>bool" ("_ \<turnstile> _ ≡ _ : _" [60,60] 60) 
where
  Q_Refl[intro]:  "Γ \<turnstile> t : T ==> Γ \<turnstile> t ≡ t : T"
| Q_Symm[intro]:  "Γ \<turnstile> t ≡ s : T ==> Γ \<turnstile> s ≡ t : T"
| Q_Trans[intro]: "[|Γ \<turnstile> s ≡ t : T; Γ \<turnstile> t ≡ u : T|] ==>  Γ \<turnstile> s ≡ u : T"
| Q_Abs[intro]:   "[|x\<sharp>Γ; (x,T1)#Γ \<turnstile> s2 ≡ t2 : T2|] ==> Γ \<turnstile> Lam [x]. s2 ≡  Lam [x]. t2 : T1 -> T2"
| Q_App[intro]:   "[|Γ \<turnstile> s1 ≡ t1 : T1 -> T2 ; Γ \<turnstile> s2 ≡ t2 : T1|] ==>  Γ \<turnstile> App s1 s2 ≡ App t1 t2 : T2"
| Q_Beta[intro]:  "[|x\<sharp>(Γ,s2,t2); (x,T1)#Γ \<turnstile> s1 ≡ t1 : T2 ; Γ \<turnstile> s2 ≡ t2 : T1|] 
                   ==>  Γ \<turnstile> App (Lam [x]. s1) s2 ≡ t1[x::=t2] : T2"
| Q_Ext[intro]:   "[|x\<sharp>(Γ,s,t); (x,T1)#Γ \<turnstile> App s (Var x) ≡ App t (Var x) : T2|] 
                   ==> Γ \<turnstile> s ≡ t : T1 -> T2"
| Q_Unit[intro]:  "[|Γ \<turnstile> s : TUnit; Γ \<turnstile> t: TUnit|] ==> Γ \<turnstile> s ≡ t : TUnit"

equivariance def_equiv

nominal_inductive def_equiv
  by (simp_all add: abs_fresh fresh_subst'')

lemma def_equiv_implies_valid:
  assumes a: "Γ \<turnstile> t ≡ s : T"
  shows "valid Γ"
using a by (induct) (auto elim: typing_implies_valid)

section {* Weak Head Reduction *}

inductive
  whr_def :: "trm=>trm=>bool" ("_ \<leadsto> _" [80,80] 80) 
where
  QAR_Beta[intro]: "App (Lam [x]. t1) t2 \<leadsto> t1[x::=t2]"
| QAR_App[intro]:  "t1 \<leadsto> t1' ==> App t1 t2 \<leadsto> App t1' t2"

declare trm.inject  [simp add]
declare ty.inject  [simp add]

inductive_cases whr_inv_auto[elim]: 
  "t \<leadsto> t'"
  "Lam [x].t \<leadsto> t'"
  "App (Lam [x].t12) t2 \<leadsto> t"
  "Var x \<leadsto> t"
  "Const n \<leadsto> t"
  "App p q \<leadsto> t"
  "t \<leadsto> Const n"
  "t \<leadsto> Var x"
  "t \<leadsto> App p q"

declare trm.inject  [simp del]
declare ty.inject  [simp del]

equivariance whr_def

section {* Weak Head Normalisation *}

abbreviation 
 nf :: "trm => bool" ("_ \<leadsto>|" [100] 100)
where
  "t\<leadsto>|  ≡ ¬(∃ u. t \<leadsto> u)" 

inductive
  whn_def :: "trm=>trm=>bool" ("_ \<Down> _" [80,80] 80) 
where
  QAN_Reduce[intro]: "[|s \<leadsto> t; t \<Down> u|] ==> s \<Down> u"
| QAN_Normal[intro]: "t\<leadsto>|  ==> t \<Down> t"

declare trm.inject[simp]

inductive_cases whn_inv_auto[elim]: "t \<Down> t'"

declare trm.inject[simp del]

lemma whn_eqvt[eqvt]:
  fixes pi::"name prm"
  assumes a: "t \<Down> t'"
  shows "(pi•t) \<Down> (pi•t')"
using a
apply(induct)
apply(rule QAN_Reduce)
apply(rule whr_def.eqvt)
apply(assumption)+
apply(rule QAN_Normal)
apply(auto)
apply(drule_tac pi="rev pi" in whr_def.eqvt)
apply(perm_simp)
done

lemma red_unicity : 
  assumes a: "x \<leadsto> a" 
  and     b: "x \<leadsto> b"
  shows "a=b"
  using a b
apply (induct arbitrary: b)
apply (erule whr_inv_auto(3))
apply (clarify)
apply (rule subst_fun_eq)
apply (simp)
apply (force)
apply (erule whr_inv_auto(6))
apply (blast)+
done

lemma nf_unicity :
  assumes "x \<Down> a" and "x \<Down> b"
  shows "a=b"
  using assms 
proof (induct arbitrary: b)
  case (QAN_Reduce x t a b)
  have h:"x \<leadsto> t" "t \<Down> a" by fact+
  have ih:"!!b. t \<Down> b ==> a = b" by fact
  have "x \<Down> b" by fact
  then obtain t' where "x \<leadsto> t'" and hl:"t' \<Down> b" using h by auto
  then have "t=t'" using h red_unicity by auto
  then show "a=b" using ih hl by auto
qed (auto)


section {* Algorithmic Term Equivalence and Algorithmic Path Equivalence *}

inductive
  alg_equiv :: "Ctxt=>trm=>trm=>ty=>bool" ("_ \<turnstile> _ \<Leftrightarrow> _ : _" [60,60,60,60] 60) 
and
  alg_path_equiv :: "Ctxt=>trm=>trm=>ty=>bool" ("_ \<turnstile> _ \<leftrightarrow> _ : _" [60,60,60,60] 60) 
where
  QAT_Base[intro]:  "[|s \<Down> p; t \<Down> q; Γ \<turnstile> p \<leftrightarrow> q : TBase|] ==> Γ \<turnstile> s \<Leftrightarrow> t : TBase"
| QAT_Arrow[intro]: "[|x\<sharp>(Γ,s,t); (x,T1)#Γ \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T2|] 
                     ==> Γ \<turnstile> s \<Leftrightarrow> t : T1 -> T2"
| QAT_One[intro]:   "valid Γ ==> Γ \<turnstile> s \<Leftrightarrow> t : TUnit"
| QAP_Var[intro]:   "[|valid Γ;(x,T) ∈ set Γ|] ==> Γ \<turnstile> Var x \<leftrightarrow> Var x : T"
| QAP_App[intro]:   "[|Γ \<turnstile> p \<leftrightarrow> q : T1 -> T2; Γ \<turnstile> s \<Leftrightarrow> t : T1|] ==> Γ \<turnstile> App p s \<leftrightarrow> App q t : T2"
| QAP_Const[intro]: "valid Γ ==> Γ \<turnstile> Const n \<leftrightarrow> Const n : TBase"

equivariance alg_equiv

nominal_inductive alg_equiv
  avoids QAT_Arrow: x
  by simp_all

declare trm.inject [simp add]
declare ty.inject  [simp add]

inductive_cases alg_equiv_inv_auto[elim]: 
  "Γ \<turnstile> s\<Leftrightarrow>t : TBase"
  "Γ \<turnstile> s\<Leftrightarrow>t : T1 -> T2"
  "Γ \<turnstile> s\<leftrightarrow>t : TBase"
  "Γ \<turnstile> s\<leftrightarrow>t : TUnit"
  "Γ \<turnstile> s\<leftrightarrow>t : T1 -> T2"

  "Γ \<turnstile> Var x \<leftrightarrow> t : T"
  "Γ \<turnstile> Var x \<leftrightarrow> t : T'"
  "Γ \<turnstile> s \<leftrightarrow> Var x : T"
  "Γ \<turnstile> s \<leftrightarrow> Var x : T'"
  "Γ \<turnstile> Const n \<leftrightarrow> t : T"
  "Γ \<turnstile> s \<leftrightarrow> Const n : T"
  "Γ \<turnstile> App p s \<leftrightarrow> t : T"
  "Γ \<turnstile> s \<leftrightarrow> App q t : T"
  "Γ \<turnstile> Lam[x].s \<leftrightarrow> t : T"
  "Γ \<turnstile> t \<leftrightarrow> Lam[x].s : T"

declare trm.inject [simp del]
declare ty.inject [simp del]

lemma Q_Arrow_strong_inversion:
  assumes fs: "x\<sharp>Γ" "x\<sharp>t" "x\<sharp>u" 
  and h: "Γ \<turnstile> t \<Leftrightarrow> u : T1->T2"
  shows "(x,T1)#Γ \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T2"
proof -
  obtain y where fs2: "y\<sharp>(Γ,t,u)" and "(y,T1)#Γ \<turnstile> App t (Var y) \<Leftrightarrow> App u (Var y) : T2" 
    using h by auto
  then have "([(x,y)]•((y,T1)#Γ)) \<turnstile> [(x,y)]• App t (Var y) \<Leftrightarrow> [(x,y)]• App u (Var y) : T2" 
    using  alg_equiv.eqvt[simplified] by blast
  then show ?thesis using fs fs2 by (perm_simp)
qed

(*
Warning this lemma is false:

lemma algorithmic_type_unicity:
  shows "[| Γ \<turnstile> s \<Leftrightarrow> t : T ; Γ \<turnstile> s \<Leftrightarrow> u : T' |] ==> T = T'"
  and "[| Γ \<turnstile> s \<leftrightarrow> t : T ; Γ \<turnstile> s \<leftrightarrow> u : T' |] ==> T = T'"

Here is the counter example : 
Γ \<turnstile> Const n \<Leftrightarrow> Const n : Tbase and Γ \<turnstile> Const n \<Leftrightarrow> Const n : TUnit
*)

lemma algorithmic_path_type_unicity:
  shows "Γ \<turnstile> s \<leftrightarrow> t : T ==> Γ \<turnstile> s \<leftrightarrow> u : T' ==> T = T'"
proof (induct arbitrary:  u T' 
       rule: alg_equiv_alg_path_equiv.inducts(2) [of _ _ _ _ _  "%a b c d . True"    ])
  case (QAP_Var Γ x T u T')
  have "Γ \<turnstile> Var x \<leftrightarrow> u : T'" by fact
  then have "u=Var x" and "(x,T') ∈ set Γ" by auto
  moreover have "valid Γ" "(x,T) ∈ set Γ" by fact+
  ultimately show "T=T'" using type_unicity_in_context by auto
next
  case (QAP_App Γ p q T1 T2 s t u T2')
  have ih:"!!u T. Γ \<turnstile> p \<leftrightarrow> u : T ==> T1->T2 = T" by fact
  have "Γ \<turnstile> App p s \<leftrightarrow> u : T2'" by fact
  then obtain r t T1' where "u = App r t"  "Γ \<turnstile> p \<leftrightarrow> r : T1' -> T2'" by auto
  with ih have "T1->T2 = T1' -> T2'" by auto
  then show "T2=T2'" using ty.inject by auto
qed (auto)

lemma alg_path_equiv_implies_valid:
  shows  "Γ \<turnstile> s \<Leftrightarrow> t : T ==> valid Γ" 
  and    "Γ \<turnstile> s \<leftrightarrow> t : T ==> valid Γ"
by (induct rule : alg_equiv_alg_path_equiv.inducts, auto)

lemma algorithmic_symmetry:
  shows "Γ \<turnstile> s \<Leftrightarrow> t : T ==> Γ \<turnstile> t \<Leftrightarrow> s : T" 
  and   "Γ \<turnstile> s \<leftrightarrow> t : T ==> Γ \<turnstile> t \<leftrightarrow> s : T"
by (induct rule: alg_equiv_alg_path_equiv.inducts) 
   (auto simp add: fresh_prod)

lemma algorithmic_transitivity:
  shows "Γ \<turnstile> s \<Leftrightarrow> t : T ==> Γ \<turnstile> t \<Leftrightarrow> u : T ==> Γ \<turnstile> s \<Leftrightarrow> u : T"
  and   "Γ \<turnstile> s \<leftrightarrow> t : T ==> Γ \<turnstile> t \<leftrightarrow> u : T ==> Γ \<turnstile> s \<leftrightarrow> u : T"
proof (nominal_induct Γ s t T and Γ s t T avoiding: u rule: alg_equiv_alg_path_equiv.strong_inducts)
  case (QAT_Base s p t q Γ u)
  have "Γ \<turnstile> t \<Leftrightarrow> u : TBase" by fact
  then obtain r' q' where b1: "t \<Down> q'" and b2: "u \<Down> r'" and b3: "Γ \<turnstile> q' \<leftrightarrow> r' : TBase" by auto
  have ih: "Γ \<turnstile> q \<leftrightarrow> r' : TBase ==> Γ \<turnstile> p \<leftrightarrow> r' : TBase" by fact
  have "t \<Down> q" by fact
  with b1 have eq: "q=q'" by (simp add: nf_unicity)
  with ih b3 have "Γ \<turnstile> p \<leftrightarrow> r' : TBase" by simp
  moreover
  have "s \<Down> p" by fact
  ultimately show "Γ \<turnstile> s \<Leftrightarrow> u : TBase" using b2 by auto
next
  case (QAT_Arrow  x Γ s t T1 T2 u)
  have ih:"(x,T1)#Γ \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T2 
                                   ==> (x,T1)#Γ \<turnstile> App s (Var x) \<Leftrightarrow> App u (Var x) : T2" by fact
  have fs: "x\<sharp>Γ" "x\<sharp>s" "x\<sharp>t" "x\<sharp>u" by fact+
  have "Γ \<turnstile> t \<Leftrightarrow> u : T1->T2" by fact
  then have "(x,T1)#Γ \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T2" using fs 
    by (simp add: Q_Arrow_strong_inversion)
  with ih have "(x,T1)#Γ \<turnstile> App s (Var x) \<Leftrightarrow> App u (Var x) : T2" by simp
  then show "Γ \<turnstile> s \<Leftrightarrow> u : T1->T2" using fs by (auto simp add: fresh_prod)
next
  case (QAP_App Γ p q T1 T2 s t u)
  have "Γ \<turnstile> App q t \<leftrightarrow> u : T2" by fact
  then obtain r T1' v where ha: "Γ \<turnstile> q \<leftrightarrow> r : T1'->T2" and hb: "Γ \<turnstile> t \<Leftrightarrow> v : T1'" and eq: "u = App r v" 
    by auto
  have ih1: "Γ \<turnstile> q \<leftrightarrow> r : T1->T2 ==> Γ \<turnstile> p \<leftrightarrow> r : T1->T2" by fact
  have ih2:"Γ \<turnstile> t \<Leftrightarrow> v : T1 ==> Γ \<turnstile> s \<Leftrightarrow> v : T1" by fact
  have "Γ \<turnstile> p \<leftrightarrow> q : T1->T2" by fact
  then have "Γ \<turnstile> q \<leftrightarrow> p : T1->T2" by (simp add: algorithmic_symmetry)
  with ha have "T1'->T2 = T1->T2" using algorithmic_path_type_unicity by simp
  then have "T1' = T1" by (simp add: ty.inject) 
  then have "Γ \<turnstile> s \<Leftrightarrow> v : T1" "Γ \<turnstile> p \<leftrightarrow> r : T1->T2" using ih1 ih2 ha hb by auto
  then show "Γ \<turnstile> App p s \<leftrightarrow> u : T2" using eq by auto
qed (auto)

lemma algorithmic_weak_head_closure:
  shows "Γ \<turnstile> s \<Leftrightarrow> t : T ==> s' \<leadsto> s ==> t' \<leadsto> t ==> Γ \<turnstile> s' \<Leftrightarrow> t' : T"
apply (nominal_induct Γ s t T avoiding: s' t'  
    rule: alg_equiv_alg_path_equiv.strong_inducts(1) [of _ _ _ _ "%a b c d e. True"])
apply(auto intro!: QAT_Arrow)
done

lemma algorithmic_monotonicity:
  shows "Γ \<turnstile> s \<Leftrightarrow> t : T ==> Γ ⊆ Γ' ==> valid Γ' ==> Γ' \<turnstile> s \<Leftrightarrow> t : T"
  and   "Γ \<turnstile> s \<leftrightarrow> t : T ==> Γ ⊆ Γ' ==> valid Γ' ==> Γ' \<turnstile> s \<leftrightarrow> t : T"
proof (nominal_induct Γ s t T and Γ s t T avoiding: Γ' rule: alg_equiv_alg_path_equiv.strong_inducts)
 case (QAT_Arrow x Γ s t T1 T2 Γ')
  have fs:"x\<sharp>Γ" "x\<sharp>s" "x\<sharp>t" "x\<sharp>Γ'" by fact+
  have h2:"Γ ⊆ Γ'" by fact
  have ih:"!!Γ'. [|(x,T1)#Γ ⊆ Γ'; valid Γ'|]  ==> Γ' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T2" by fact
  have "valid Γ'" by fact
  then have "valid ((x,T1)#Γ')" using fs by auto
  moreover
  have sub: "(x,T1)#Γ ⊆ (x,T1)#Γ'" using h2 by auto
  ultimately have "(x,T1)#Γ' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T2" using ih by simp
  then show "Γ' \<turnstile> s \<Leftrightarrow> t : T1->T2" using fs by (auto simp add: fresh_prod)
qed (auto)

lemma path_equiv_implies_nf:
  assumes "Γ \<turnstile> s \<leftrightarrow> t : T"
  shows "s \<leadsto>|" and "t \<leadsto>|"
using assms
by (induct rule: alg_equiv_alg_path_equiv.inducts(2)) (simp, auto)

section {* Logical Equivalence *}

function log_equiv :: "(Ctxt => trm => trm => ty => bool)" ("_ \<turnstile> _ is _ : _" [60,60,60,60] 60) 
where    
   "Γ \<turnstile> s is t : TUnit = True"
 | "Γ \<turnstile> s is t : TBase = Γ \<turnstile> s \<Leftrightarrow> t : TBase"
 | "Γ \<turnstile> s is t : (T1 -> T2) =  
    (∀Γ' s' t'. Γ⊆Γ' --> valid Γ' --> Γ' \<turnstile> s' is t' : T1 -->  (Γ' \<turnstile> (App s s') is (App t t') : T2))"
apply (auto simp add: ty.inject)
apply (subgoal_tac "(∃T1 T2. b=T1 -> T2) ∨ b=TUnit ∨ b=TBase" )
apply (force)
apply (rule ty_cases)
done
termination by lexicographic_order

lemma logical_monotonicity:
 fixes Γ Γ' :: Ctxt
 assumes a1: "Γ \<turnstile> s is t : T" 
 and     a2: "Γ ⊆ Γ'" 
 and     a3: "valid Γ'"
 shows "Γ' \<turnstile> s is t : T"
using a1 a2 a3
proof (induct arbitrary: Γ' rule: log_equiv.induct)
  case (2 Γ s t Γ')
  then show "Γ' \<turnstile> s is t : TBase" using algorithmic_monotonicity by auto
next
  case (3 Γ s t T1 T2 Γ')
  have "Γ \<turnstile> s is t : T1->T2" 
  and  "Γ ⊆ Γ'" 
  and  "valid Γ'" by fact+
  then show "Γ' \<turnstile> s is t : T1->T2" by simp
qed (auto)

lemma main_lemma:
  shows "Γ \<turnstile> s is t : T ==> valid Γ ==> Γ \<turnstile> s \<Leftrightarrow> t : T" 
    and "Γ \<turnstile> p \<leftrightarrow> q : T ==> Γ \<turnstile> p is q : T"
proof (nominal_induct T arbitrary: Γ s t p q rule: ty.strong_induct)
  case (Arrow T1 T2)
  { 
    case (1 Γ s t)
    have ih1:"!!Γ s t. [|Γ \<turnstile> s is t : T2; valid Γ|] ==> Γ \<turnstile> s \<Leftrightarrow> t : T2" by fact
    have ih2:"!!Γ s t. Γ \<turnstile> s \<leftrightarrow> t : T1 ==> Γ \<turnstile> s is t : T1" by fact
    have h:"Γ \<turnstile> s is t : T1->T2" by fact
    obtain x::name where fs:"x\<sharp>(Γ,s,t)" by (erule exists_fresh[OF fs_name1])
    have "valid Γ" by fact
    then have v: "valid ((x,T1)#Γ)" using fs by auto
    then have "(x,T1)#Γ \<turnstile> Var x \<leftrightarrow> Var x : T1" by auto
    then have "(x,T1)#Γ \<turnstile> Var x is Var x : T1" using ih2 by auto
    then have "(x,T1)#Γ \<turnstile> App s (Var x) is App t (Var x) : T2" using h v by auto
    then have "(x,T1)#Γ \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T2" using ih1 v by auto
    then show "Γ \<turnstile> s \<Leftrightarrow> t : T1->T2" using fs by (auto simp add: fresh_prod)
  next
    case (2 Γ p q)
    have h: "Γ \<turnstile> p \<leftrightarrow> q : T1->T2" by fact
    have ih1:"!!Γ s t. Γ \<turnstile> s \<leftrightarrow> t : T2 ==> Γ \<turnstile> s is t : T2" by fact
    have ih2:"!!Γ s t. [|Γ \<turnstile> s is t : T1; valid Γ|] ==> Γ \<turnstile> s \<Leftrightarrow> t : T1" by fact
    {
      fix Γ' s t
      assume "Γ ⊆ Γ'" and hl:"Γ' \<turnstile> s is t : T1" and hk: "valid Γ'"
      then have "Γ' \<turnstile> p \<leftrightarrow> q : T1 -> T2" using h algorithmic_monotonicity by auto
      moreover have "Γ' \<turnstile> s \<Leftrightarrow> t : T1" using ih2 hl hk by auto
      ultimately have "Γ' \<turnstile> App p s \<leftrightarrow> App q t : T2" by auto
      then have "Γ' \<turnstile> App p s is App q t : T2" using ih1 by auto
    }
    then show "Γ \<turnstile> p is q : T1->T2"  by simp
  }
next
  case TBase
  { case 2
    have h:"Γ \<turnstile> s \<leftrightarrow> t : TBase" by fact
    then have "s \<leadsto>|" and "t \<leadsto>|" using path_equiv_implies_nf by auto
    then have "s \<Down> s" and "t \<Down> t" by auto
    then have "Γ \<turnstile> s \<Leftrightarrow> t : TBase" using h by auto
    then show "Γ \<turnstile> s is t : TBase" by auto
  }
qed (auto elim: alg_path_equiv_implies_valid) 

corollary corollary_main:
  assumes a: "Γ \<turnstile> s \<leftrightarrow> t : T"
  shows "Γ \<turnstile> s \<Leftrightarrow> t : T"
using a main_lemma alg_path_equiv_implies_valid by blast

lemma logical_symmetry:
  assumes a: "Γ \<turnstile> s is t : T"
  shows "Γ \<turnstile> t is s : T"
using a 
by (nominal_induct arbitrary: Γ s t rule: ty.strong_induct) 
   (auto simp add: algorithmic_symmetry)

lemma logical_transitivity:
  assumes "Γ \<turnstile> s is t : T" "Γ \<turnstile> t is u : T" 
  shows "Γ \<turnstile> s is u : T"
using assms
proof (nominal_induct arbitrary: Γ s t u  rule:ty.strong_induct)
  case TBase
  then show "Γ \<turnstile> s is u : TBase" by (auto elim:  algorithmic_transitivity)
next 
  case (Arrow T1 T2 Γ s t u)
  have h1:"Γ \<turnstile> s is t : T1 -> T2" by fact
  have h2:"Γ \<turnstile> t is u : T1 -> T2" by fact
  have ih1:"!!Γ s t u. [|Γ \<turnstile> s is t : T1; Γ \<turnstile> t is u : T1|] ==> Γ \<turnstile> s is u : T1" by fact
  have ih2:"!!Γ s t u. [|Γ \<turnstile> s is t : T2; Γ \<turnstile> t is u : T2|] ==> Γ \<turnstile> s is u : T2" by fact
  {
    fix Γ' s' u'
    assume hsub:"Γ ⊆ Γ'" and hl:"Γ' \<turnstile> s' is u' : T1" and hk: "valid Γ'"
    then have "Γ' \<turnstile> u' is s' : T1" using logical_symmetry by blast
    then have "Γ' \<turnstile> u' is u' : T1" using ih1 hl by blast
    then have "Γ' \<turnstile> App t u' is App u u' : T2" using h2 hsub hk by auto
    moreover have "Γ' \<turnstile>  App s s' is App t u' : T2" using h1 hsub hl hk by auto
    ultimately have "Γ' \<turnstile>  App s s' is App u u' : T2" using ih2 by blast
  }
  then show "Γ \<turnstile> s is u : T1 -> T2" by auto
qed (auto)

lemma logical_weak_head_closure:
  assumes a: "Γ \<turnstile> s is t : T" 
  and     b: "s' \<leadsto> s" 
  and     c: "t' \<leadsto> t"
  shows "Γ \<turnstile> s' is t' : T"
using a b c algorithmic_weak_head_closure 
by (nominal_induct arbitrary: Γ s t s' t' rule: ty.strong_induct) 
   (auto, blast)

lemma logical_weak_head_closure':
  assumes "Γ \<turnstile> s is t : T" and "s' \<leadsto> s" 
  shows "Γ \<turnstile> s' is t : T"
using assms
proof (nominal_induct arbitrary: Γ s t s' rule: ty.strong_induct)
  case (TBase  Γ s t s')
  then show ?case by force
next
  case (TUnit Γ s t s')
  then show ?case by auto
next 
  case (Arrow T1 T2 Γ s t s')
  have h1:"s' \<leadsto> s" by fact
  have ih:"!!Γ s t s'. [|Γ \<turnstile> s is t : T2; s' \<leadsto> s|] ==> Γ \<turnstile> s' is t : T2" by fact
  have h2:"Γ \<turnstile> s is t : T1->T2" by fact
  then 
  have hb:"∀Γ' s' t'. Γ⊆Γ' --> valid Γ' --> Γ' \<turnstile> s' is t' : T1 --> (Γ' \<turnstile> (App s s') is (App t t') : T2)" 
    by auto
  {
    fix Γ' s2 t2
    assume "Γ ⊆ Γ'" and "Γ' \<turnstile> s2 is t2 : T1" and "valid Γ'"
    then have "Γ' \<turnstile> (App s s2) is (App t t2) : T2" using hb by auto
    moreover have "(App s' s2)  \<leadsto> (App s s2)" using h1 by auto  
    ultimately have "Γ' \<turnstile> App s' s2 is App t t2 : T2" using ih by auto
  }
  then show "Γ \<turnstile> s' is t : T1->T2" by auto
qed 

abbreviation 
 log_equiv_for_psubsts :: "Ctxt => Subst => Subst => Ctxt => bool"  ("_ \<turnstile> _ is _ over _" [60,60] 60) 
where
  "Γ' \<turnstile> ϑ is ϑ' over Γ ≡ ∀x T. (x,T) ∈ set Γ --> Γ' \<turnstile> ϑ<Var x> is  ϑ'<Var x> : T"

lemma logical_pseudo_reflexivity:
  assumes "Γ' \<turnstile> t is s over Γ"
  shows "Γ' \<turnstile> s is s over Γ" 
proof -
  have "Γ' \<turnstile> t is s over Γ" by fact
  moreover then have "Γ' \<turnstile> s is t over Γ" using logical_symmetry by blast
  ultimately show "Γ' \<turnstile> s is s over Γ" using logical_transitivity by blast
qed

lemma logical_subst_monotonicity :
  fixes Γ Γ' Γ'' :: Ctxt
  assumes a: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and     b: "Γ' ⊆ Γ''"
  and     c: "valid Γ''"
  shows "Γ'' \<turnstile> ϑ is ϑ' over Γ"
using a b c logical_monotonicity by blast

lemma equiv_subst_ext :
  assumes h1: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and     h2: "Γ' \<turnstile> s is t : T" 
  and     fs: "x\<sharp>Γ"
  shows "Γ' \<turnstile> (x,s)#ϑ is (x,t)#ϑ' over (x,T)#Γ"
using assms
proof -
  {
    fix y U
    assume "(y,U) ∈ set ((x,T)#Γ)"
    moreover
    { 
      assume "(y,U) ∈ set [(x,T)]"
      with h2 have "Γ' \<turnstile> ((x,s)#ϑ)<Var y> is ((x,t)#ϑ')<Var y> : U" by auto 
    }
    moreover
    {
      assume hl:"(y,U) ∈ set Γ"
      then have "¬ y\<sharp>Γ" by (induct Γ) (auto simp add: fresh_list_cons fresh_atm fresh_prod)
      then have hf:"x\<sharp> Var y" using fs by (auto simp add: fresh_atm)
      then have "((x,s)#ϑ)<Var y> = ϑ<Var y>" "((x,t)#ϑ')<Var y> = ϑ'<Var y>" 
        using fresh_psubst_simp by blast+ 
      moreover have  "Γ' \<turnstile> ϑ<Var y> is ϑ'<Var y> : U" using h1 hl by auto
      ultimately have "Γ' \<turnstile> ((x,s)#ϑ)<Var y> is ((x,t)#ϑ')<Var y> : U" by auto
    }
    ultimately have "Γ' \<turnstile> ((x,s)#ϑ)<Var y> is ((x,t)#ϑ')<Var y> : U" by auto
  }
  then show "Γ' \<turnstile> (x,s)#ϑ is (x,t)#ϑ' over (x,T)#Γ" by auto
qed

theorem fundamental_theorem_1:
  assumes a1: "Γ \<turnstile> t : T" 
  and     a2: "Γ' \<turnstile> ϑ is ϑ' over Γ"
  and     a3: "valid Γ'" 
  shows "Γ' \<turnstile> ϑ<t> is ϑ'<t> : T"   
using a1 a2 a3
proof (nominal_induct Γ t T avoiding: ϑ ϑ' arbitrary: Γ' rule: typing.strong_induct)
  case (T_Lam x Γ T1 t2 T2 ϑ ϑ' Γ')
  have vc: "x\<sharp>ϑ" "x\<sharp>ϑ'" "x\<sharp>Γ" by fact+
  have asm1: "Γ' \<turnstile> ϑ is ϑ' over Γ" by fact
  have ih:"!!ϑ ϑ' Γ'. [|Γ' \<turnstile> ϑ is ϑ' over (x,T1)#Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<t2> is ϑ'<t2> : T2" by fact
  show "Γ' \<turnstile> ϑ<Lam [x].t2> is ϑ'<Lam [x].t2> : T1->T2" using vc
  proof (simp, intro strip)
    fix Γ'' s' t'
    assume sub: "Γ' ⊆ Γ''" 
    and    asm2: "Γ''\<turnstile> s' is t' : T1" 
    and    val: "valid Γ''"
    from asm1 val sub have "Γ'' \<turnstile> ϑ is ϑ' over Γ" using logical_subst_monotonicity by blast
    with asm2 vc have "Γ'' \<turnstile> (x,s')#ϑ is (x,t')#ϑ' over (x,T1)#Γ" using equiv_subst_ext by blast
    with ih val have "Γ'' \<turnstile> ((x,s')#ϑ)<t2> is ((x,t')#ϑ')<t2> : T2" by auto
    with vc have "Γ''\<turnstile>ϑ<t2>[x::=s'] is ϑ'<t2>[x::=t'] : T2" by (simp add: psubst_subst_psubst)
    moreover 
    have "App (Lam [x].ϑ<t2>) s' \<leadsto> ϑ<t2>[x::=s']" by auto
    moreover 
    have "App (Lam [x].ϑ'<t2>) t' \<leadsto> ϑ'<t2>[x::=t']" by auto
    ultimately show "Γ''\<turnstile> App (Lam [x].ϑ<t2>) s' is App (Lam [x].ϑ'<t2>) t' : T2" 
      using logical_weak_head_closure by auto
  qed
qed (auto)


theorem fundamental_theorem_2:
  assumes h1: "Γ \<turnstile> s ≡ t : T" 
  and     h2: "Γ' \<turnstile> ϑ is ϑ' over Γ"
  and     h3: "valid Γ'"
  shows "Γ' \<turnstile> ϑ<s> is ϑ'<t> : T"
using h1 h2 h3
proof (nominal_induct Γ s t T avoiding: Γ' ϑ ϑ' rule: def_equiv.strong_induct)
  case (Q_Refl Γ t T Γ' ϑ ϑ')
  have "Γ \<turnstile> t : T" 
  and  "valid Γ'" by fact+
  moreover 
  have "Γ' \<turnstile> ϑ is ϑ' over Γ" by fact
  ultimately show "Γ' \<turnstile> ϑ<t> is ϑ'<t> : T" using fundamental_theorem_1 by blast
next
  case (Q_Symm Γ t s T Γ' ϑ ϑ')
  have "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and "valid Γ'" by fact+
  moreover 
  have ih: "!! Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<t> is ϑ'<s> : T" by fact
  ultimately show "Γ' \<turnstile> ϑ<s> is ϑ'<t> : T" using logical_symmetry by blast
next
  case (Q_Trans Γ s t T u Γ' ϑ ϑ')
  have ih1: "!! Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<s> is ϑ'<t> : T" by fact
  have ih2: "!! Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<t> is ϑ'<u> : T" by fact
  have h: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and  v: "valid Γ'" by fact+
  then have "Γ' \<turnstile> ϑ' is ϑ' over Γ" using logical_pseudo_reflexivity by auto
  then have "Γ' \<turnstile> ϑ'<t> is ϑ'<u> : T" using ih2 v by auto
  moreover have "Γ' \<turnstile> ϑ<s> is ϑ'<t> : T" using ih1 h v by auto
  ultimately show "Γ' \<turnstile> ϑ<s> is ϑ'<u> : T" using logical_transitivity by blast
next
  case (Q_Abs x Γ T1 s2 t2 T2 Γ' ϑ ϑ')
  have fs:"x\<sharp>Γ" by fact
  have fs2: "x\<sharp>ϑ" "x\<sharp>ϑ'" by fact+
  have h2: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and  h3: "valid Γ'" by fact+
  have ih:"!!Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over (x,T1)#Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<s2> is ϑ'<t2> : T2" by fact
  {
    fix Γ'' s' t'
    assume "Γ' ⊆ Γ''" and hl:"Γ''\<turnstile> s' is t' : T1" and hk: "valid Γ''"
    then have "Γ'' \<turnstile> ϑ is ϑ' over Γ" using h2 logical_subst_monotonicity by blast
    then have "Γ'' \<turnstile> (x,s')#ϑ is (x,t')#ϑ' over (x,T1)#Γ" using equiv_subst_ext hl fs by blast
    then have "Γ'' \<turnstile> ((x,s')#ϑ)<s2> is ((x,t')#ϑ')<t2> : T2" using ih hk by blast
    then have "Γ''\<turnstile> ϑ<s2>[x::=s'] is ϑ'<t2>[x::=t'] : T2" using fs2 psubst_subst_psubst by auto
    moreover have "App (Lam [x]. ϑ<s2>) s' \<leadsto>  ϑ<s2>[x::=s']" 
              and "App (Lam [x].ϑ'<t2>) t' \<leadsto> ϑ'<t2>[x::=t']" by auto
    ultimately have "Γ'' \<turnstile> App (Lam [x]. ϑ<s2>) s' is App (Lam [x].ϑ'<t2>) t' : T2" 
      using logical_weak_head_closure by auto
  }
  moreover have "valid Γ'" by fact
  ultimately have "Γ' \<turnstile> Lam [x].ϑ<s2> is Lam [x].ϑ'<t2> : T1->T2" by auto
  then show "Γ' \<turnstile> ϑ<Lam [x].s2> is ϑ'<Lam [x].t2> : T1->T2" using fs2 by auto
next
  case (Q_App Γ s1 t1 T1 T2 s2 t2 Γ' ϑ ϑ')
  then show "Γ' \<turnstile> ϑ<App s1 s2> is ϑ'<App t1 t2> : T2" by auto 
next
  case (Q_Beta x Γ s2 t2 T1 s12 t12 T2 Γ' ϑ ϑ')
  have h: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and  h': "valid Γ'" by fact+
  have fs: "x\<sharp>Γ" by fact
  have fs2: " x\<sharp>ϑ" "x\<sharp>ϑ'" by fact+
  have ih1: "!!Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<s2> is ϑ'<t2> : T1" by fact
  have ih2: "!!Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over (x,T1)#Γ; valid Γ'|] ==> Γ' \<turnstile> ϑ<s12> is ϑ'<t12> : T2" by fact
  have "Γ' \<turnstile> ϑ<s2> is ϑ'<t2> : T1" using ih1 h' h by auto
  then have "Γ' \<turnstile> (x,ϑ<s2>)#ϑ is (x,ϑ'<t2>)#ϑ' over (x,T1)#Γ" using equiv_subst_ext h fs by blast
  then have "Γ' \<turnstile> ((x,ϑ<s2>)#ϑ)<s12> is ((x,ϑ'<t2>)#ϑ')<t12> : T2" using ih2 h' by auto
  then have "Γ' \<turnstile> ϑ<s12>[x::=ϑ<s2>] is ϑ'<t12>[x::=ϑ'<t2>] : T2" using fs2 psubst_subst_psubst by auto
  then have "Γ' \<turnstile> ϑ<s12>[x::=ϑ<s2>] is ϑ'<t12[x::=t2]> : T2" using fs2 psubst_subst_propagate by auto
  moreover have "App (Lam [x].ϑ<s12>) (ϑ<s2>) \<leadsto> ϑ<s12>[x::=ϑ<s2>]" by auto 
  ultimately have "Γ' \<turnstile> App (Lam [x].ϑ<s12>) (ϑ<s2>) is ϑ'<t12[x::=t2]> : T2" 
    using logical_weak_head_closure' by auto
  then show "Γ' \<turnstile> ϑ<App (Lam [x].s12) s2> is ϑ'<t12[x::=t2]> : T2" using fs2 by simp
next
  case (Q_Ext x Γ s t T1 T2 Γ' ϑ ϑ')
  have h2: "Γ' \<turnstile> ϑ is ϑ' over Γ" 
  and  h2': "valid Γ'" by fact+
  have fs:"x\<sharp>Γ" "x\<sharp>s" "x\<sharp>t" by fact+
  have ih:"!!Γ' ϑ ϑ'. [|Γ' \<turnstile> ϑ is ϑ' over (x,T1)#Γ; valid Γ'|] 
                          ==> Γ' \<turnstile> ϑ<App s (Var x)> is ϑ'<App t (Var x)> : T2" by fact
   {
    fix Γ'' s' t'
    assume hsub: "Γ' ⊆ Γ''" and hl: "Γ''\<turnstile> s' is t' : T1" and hk: "valid Γ''"
    then have "Γ'' \<turnstile> ϑ is ϑ' over Γ" using h2 logical_subst_monotonicity by blast
    then have "Γ'' \<turnstile> (x,s')#ϑ is (x,t')#ϑ' over (x,T1)#Γ" using equiv_subst_ext hl fs by blast
    then have "Γ'' \<turnstile> ((x,s')#ϑ)<App s (Var x)>  is ((x,t')#ϑ')<App t (Var x)> : T2" using ih hk by blast
    then 
    have "Γ'' \<turnstile> App (((x,s')#ϑ)<s>) (((x,s')#ϑ)<(Var x)>) is App (((x,t')#ϑ')<t>) (((x,t')#ϑ')<(Var x)>) : T2"
      by auto
    then have "Γ'' \<turnstile> App ((x,s')#ϑ)<s> s'  is App ((x,t')#ϑ')<t> t' : T2" by auto
    then have "Γ'' \<turnstile> App (ϑ<s>) s' is App (ϑ'<t>) t' : T2" using fs fresh_psubst_simp by auto
  }
  moreover have "valid Γ'" by fact
  ultimately show "Γ' \<turnstile> ϑ<s> is ϑ'<t> : T1->T2" by auto
next
  case (Q_Unit Γ s t Γ' ϑ ϑ')  
  then show "Γ' \<turnstile> ϑ<s> is ϑ'<t> : TUnit" by auto
qed


theorem completeness:
  assumes asm: "Γ \<turnstile> s ≡ t : T"
  shows   "Γ \<turnstile> s \<Leftrightarrow> t : T"
proof -
  have val: "valid Γ" using def_equiv_implies_valid asm by simp
  moreover
  {
    fix x T
    assume "(x,T) ∈ set Γ" "valid Γ"
    then have "Γ \<turnstile> Var x is Var x : T" using main_lemma(2) by blast
  }
  ultimately have "Γ \<turnstile> [] is [] over Γ" by auto 
  then have "Γ \<turnstile> []<s> is []<t> : T" using fundamental_theorem_2 val asm by blast
  then have "Γ \<turnstile> s is t : T" by simp
  then show  "Γ \<turnstile> s \<Leftrightarrow> t : T" using main_lemma(1) val by simp
qed

text {* We leave soundness as an exercise - like in the book :-) \\ 
 @{prop[mode=IfThen] "[|Γ \<turnstile> s \<Leftrightarrow> t : T; Γ \<turnstile> t : T; Γ \<turnstile> s : T|] ==> Γ \<turnstile> s ≡ t : T"} \\
 @{prop "[|Γ \<turnstile> s \<leftrightarrow> t : T; Γ \<turnstile> t : T; Γ \<turnstile> s : T|] ==> Γ \<turnstile> s ≡ t : T"} 
*}

end