(*                                                    *)
(* 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" ("_\<rightarrow>_" [100,100] 100)

nominal_datatype trm =
    Unit
  | Var "name" ("Var _" [100] 100)
  | Lam "\<guillemotleft>name\<guillemotright>trm" ("Lam [_]._" [100,100] 100)
  | App "trm" "trm" ("App _ _" [110,110] 100)
  | Const "nat"

type_synonym Ctxt  = "(name\<times>ty) list"
type_synonym Subst = "(name\<times>trm) list"

lemma perm_ty[simp]:
  fixes T::"ty"
  and   pi::"name prm"
  shows "pi\<bullet>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 "(\<exists> T\<^sub>1 T\<^sub>2. T=T\<^sub>1\<rightarrow>T\<^sub>2) \<or> T=TUnit \<or> 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 (T\<^sub>1\<rightarrow>T\<^sub>2) = size T\<^sub>1 + size T\<^sub>2"
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 \<open>Substitutions\<close>

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

lemma lookup_eqvt[eqvt]:
  fixes pi::"name prm"
  shows "pi\<bullet>(lookup \<theta> x) = lookup (pi\<bullet>\<theta>) (pi\<bullet>x)"
by (induct \<theta>) (auto simp add: perm_bij)

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

lemma lookup_fresh':
  assumes a: "z\<sharp>\<theta>"
  shows "lookup \<theta> z = Var z"
using a
by (induct rule: lookup.induct)
   (auto simp add: fresh_list_cons fresh_prod fresh_atm)

nominal_primrec
  psubst :: "Subst \<Rightarrow> trm \<Rightarrow> trm"  ("_<_>" [100,100] 130)
where
  "\<theta><(Var x)> = (lookup \<theta> x)"
| "\<theta><(App t\<^sub>1 t\<^sub>2)> = App \<theta><t\<^sub>1> \<theta><t\<^sub>2>"
| "x\<sharp>\<theta> \<Longrightarrow> \<theta><(Lam [x].t)> = Lam [x].(\<theta><t>)"
| "\<theta><(Const n)> = Const n"
| "\<theta><(Unit)> = Unit"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
done

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

lemma subst[simp]:
  shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
  and   "(App t\<^sub>1 t\<^sub>2)[y::=t'] = App (t\<^sub>1[y::=t']) (t\<^sub>2[y::=t'])"
  and   "x\<sharp>(y,t') \<Longrightarrow> (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\<bullet>(t[x::=t']) = (pi\<bullet>t)[(pi\<bullet>x)::=(pi\<bullet>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>t\<^sub>1"
  shows "t\<^sub>1[a::=t\<^sub>2] = ([(c,a)]\<bullet>t\<^sub>1)[c::=t\<^sub>2]"
using a
apply(nominal_induct t\<^sub>1 avoiding: a c t\<^sub>2 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>\<theta>"
  shows "z\<sharp>(\<theta><t>)"
using a
by (nominal_induct t avoiding: z \<theta> t rule: trm.strong_induct)
   (auto simp add: abs_fresh lookup_fresh)

lemma fresh_subst'':
  fixes z::"name"
  assumes "z\<sharp>t\<^sub>2"
  shows "z\<sharp>t\<^sub>1[z::=t\<^sub>2]"
using assms
by (nominal_induct t\<^sub>1 avoiding: t\<^sub>2 z rule: trm.strong_induct)
   (auto simp add: abs_fresh fresh_nat fresh_atm)

lemma fresh_subst':
  fixes z::"name"
  assumes "z\<sharp>[y].t\<^sub>1" "z\<sharp>t\<^sub>2"
  shows "z\<sharp>t\<^sub>1[y::=t\<^sub>2]"
using assms
by (nominal_induct t\<^sub>1 avoiding: y t\<^sub>2 z rule: trm.strong_induct)
   (auto simp add: abs_fresh fresh_nat fresh_atm)

lemma fresh_subst:
  fixes z::"name"
  assumes a: "z\<sharp>t\<^sub>1" "z\<sharp>t\<^sub>2"
  shows "z\<sharp>t\<^sub>1[y::=t\<^sub>2]"
using a
by (auto simp add: fresh_subst' abs_fresh)

lemma fresh_psubst_simp:
  assumes "x\<sharp>t"
  shows "((x,u)#\<theta>)<t> = \<theta><t>"
using assms
proof (nominal_induct t avoiding: x u \<theta> rule: trm.strong_induct)
  case (Lam y t x u)
  have fs: "y\<sharp>\<theta>" "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:"\<And>n T. n\<sharp>t \<Longrightarrow> ((n,T)#\<theta>)<t> = \<theta><t>" by fact
  ultimately have "((x,u)#\<theta>)<t> = \<theta><t>" by auto
  moreover have "((x,u)#\<theta>)<Lam [y].t> = Lam [y].(((x,u)#\<theta>)<t>)" using fs
    by (simp add: fresh_list_cons fresh_prod)
  moreover have " \<theta><Lam [y].t> = Lam [y]. (\<theta><t>)" using fs by simp
  ultimately show "((x,u)#\<theta>)<Lam [y].t> = \<theta><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].t\<^sub>1 = [y].t\<^sub>2"
  shows "t\<^sub>1[x::=u] = t\<^sub>2[y::=u]"
proof -
  {
    assume "x=y" and "t\<^sub>1=t\<^sub>2"
    then have ?thesis using h by simp
  }
  moreover
  {
    assume h1:"x \<noteq> y" and h2:"t\<^sub>1=[(x,y)] \<bullet> t\<^sub>2" and h3:"x \<sharp> t\<^sub>2"
    then have "([(x,y)] \<bullet> t\<^sub>2)[x::=u] = t\<^sub>2[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>\<theta>"
  shows "\<theta><t>[c::=s] = ((c,s)#\<theta>)<t>"
  using h
by (nominal_induct t avoiding: \<theta> 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>\<theta>"
  shows "\<theta><Var x> = Var x"
using a
by (induct \<theta> arbitrary: x) (auto simp add:fresh_list_cons fresh_prod fresh_atm)

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

section \<open>Typing\<close>

inductive
  valid :: "Ctxt \<Rightarrow> bool"
where
  v_nil[intro]:  "valid []"
| v_cons[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk> \<Longrightarrow> valid ((a,T)#\<Gamma>)"

equivariance valid

inductive_cases
  valid_cons_elim_auto[elim]:"valid ((x,T)#\<Gamma>)"

abbreviation
  "sub_context" :: "Ctxt \<Rightarrow> Ctxt \<Rightarrow> bool" (" _ \<subseteq> _ " [55,55] 55)
where
  "\<Gamma>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2 \<equiv> \<forall>a T. (a,T)\<in>set \<Gamma>\<^sub>1 \<longrightarrow> (a,T)\<in>set \<Gamma>\<^sub>2"

lemma valid_monotonicity[elim]:
 fixes \<Gamma> \<Gamma>' :: Ctxt
 assumes a: "\<Gamma> \<subseteq> \<Gamma>'"
 and     b: "x\<sharp>\<Gamma>'"
 shows "(x,T\<^sub>1)#\<Gamma> \<subseteq> (x,T\<^sub>1)#\<Gamma>'"
using a b by auto

lemma fresh_context:
  fixes  \<Gamma> :: "Ctxt"
  and    a :: "name"
  assumes "a\<sharp>\<Gamma>"
  shows "\<not>(\<exists>\<tau>::ty. (a,\<tau>)\<in>set \<Gamma>)"
using assms
by (induct \<Gamma>)
   (auto simp add: fresh_prod fresh_list_cons fresh_atm)

lemma type_unicity_in_context:
  assumes a: "valid \<Gamma>"
  and     b: "(x,T\<^sub>1) \<in> set \<Gamma>"
  and     c: "(x,T\<^sub>2) \<in> set \<Gamma>"
  shows "T\<^sub>1=T\<^sub>2"
using a b c
by (induct \<Gamma>)
   (auto dest!: fresh_context)

inductive
  typing :: "Ctxt\<Rightarrow>trm\<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> e\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2; \<Gamma> \<turnstile> e\<^sub>2 : T\<^sub>1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2 : T\<^sub>2"
| T_Lam[intro]:   "\<lbrakk>x\<sharp>\<Gamma>; (x,T\<^sub>1)#\<Gamma> \<turnstile> t : T\<^sub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].t : T\<^sub>1\<rightarrow>T\<^sub>2"
| T_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Const n : TBase"
| T_Unit[intro]:  "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Unit : TUnit"

equivariance typing

nominal_inductive typing
  by (simp_all add: abs_fresh)

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

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

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

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

section \<open>Definitional Equivalence\<close>

inductive
  def_equiv :: "Ctxt\<Rightarrow>trm\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" ("_ \<turnstile> _ \<equiv> _ : _" [60,60] 60)
where
  Q_Refl[intro]:  "\<Gamma> \<turnstile> t : T \<Longrightarrow> \<Gamma> \<turnstile> t \<equiv> t : T"
| Q_Symm[intro]:  "\<Gamma> \<turnstile> t \<equiv> s : T \<Longrightarrow> \<Gamma> \<turnstile> s \<equiv> t : T"
| Q_Trans[intro]: "\<lbrakk>\<Gamma> \<turnstile> s \<equiv> t : T; \<Gamma> \<turnstile> t \<equiv> u : T\<rbrakk> \<Longrightarrow>  \<Gamma> \<turnstile> s \<equiv> u : T"
| Q_Abs[intro]:   "\<lbrakk>x\<sharp>\<Gamma>; (x,T\<^sub>1)#\<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x]. s\<^sub>2 \<equiv>  Lam [x]. t\<^sub>2 : T\<^sub>1 \<rightarrow> T\<^sub>2"
| Q_App[intro]:   "\<lbrakk>\<Gamma> \<turnstile> s\<^sub>1 \<equiv> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2 ; \<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>1\<rbrakk> \<Longrightarrow>  \<Gamma> \<turnstile> App s\<^sub>1 s\<^sub>2 \<equiv> App t\<^sub>1 t\<^sub>2 : T\<^sub>2"
| Q_Beta[intro]:  "\<lbrakk>x\<sharp>(\<Gamma>,s\<^sub>2,t\<^sub>2); (x,T\<^sub>1)#\<Gamma> \<turnstile> s\<^sub>1 \<equiv> t\<^sub>1 : T\<^sub>2 ; \<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>1\<rbrakk>
                   \<Longrightarrow>  \<Gamma> \<turnstile> App (Lam [x]. s\<^sub>1) s\<^sub>2 \<equiv> t\<^sub>1[x::=t\<^sub>2] : T\<^sub>2"
| Q_Ext[intro]:   "\<lbrakk>x\<sharp>(\<Gamma>,s,t); (x,T\<^sub>1)#\<Gamma> \<turnstile> App s (Var x) \<equiv> App t (Var x) : T\<^sub>2\<rbrakk>
                   \<Longrightarrow> \<Gamma> \<turnstile> s \<equiv> t : T\<^sub>1 \<rightarrow> T\<^sub>2"
| Q_Unit[intro]:  "\<lbrakk>\<Gamma> \<turnstile> s : TUnit; \<Gamma> \<turnstile> t: TUnit\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s \<equiv> t : TUnit"

equivariance def_equiv

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

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

section \<open>Weak Head Reduction\<close>

inductive
  whr_def :: "trm\<Rightarrow>trm\<Rightarrow>bool" ("_ \<leadsto> _" [80,80] 80)
where
  QAR_Beta[intro]: "App (Lam [x]. t\<^sub>1) t\<^sub>2 \<leadsto> t\<^sub>1[x::=t\<^sub>2]"
| QAR_App[intro]:  "t\<^sub>1 \<leadsto> t\<^sub>1' \<Longrightarrow> App t\<^sub>1 t\<^sub>2 \<leadsto> App t\<^sub>1' t\<^sub>2"

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 \<open>Weak Head Normalisation\<close>

abbreviation
 nf :: "trm \<Rightarrow> bool" ("_ \<leadsto>|" [100] 100)
where
  "t\<leadsto>|  \<equiv> \<not>(\<exists> u. t \<leadsto> u)"

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

declare trm.inject[simp]

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

declare trm.inject[simp del]

equivariance whn_def

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:"\<And>b. t \<Down> b \<Longrightarrow> 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 \<open>Algorithmic Term Equivalence and Algorithmic Path Equivalence\<close>

inductive
  alg_equiv :: "Ctxt\<Rightarrow>trm\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" ("_ \<turnstile> _ \<Leftrightarrow> _ : _" [60,60,60,60] 60)
and
  alg_path_equiv :: "Ctxt\<Rightarrow>trm\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" ("_ \<turnstile> _ \<leftrightarrow> _ : _" [60,60,60,60] 60)
where
  QAT_Base[intro]:  "\<lbrakk>s \<Down> p; t \<Down> q; \<Gamma> \<turnstile> p \<leftrightarrow> q : TBase\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s \<Leftrightarrow> t : TBase"
| QAT_Arrow[intro]: "\<lbrakk>x\<sharp>(\<Gamma>,s,t); (x,T\<^sub>1)#\<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2\<rbrakk>
                     \<Longrightarrow> \<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1 \<rightarrow> T\<^sub>2"
| QAT_One[intro]:   "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> s \<Leftrightarrow> t : TUnit"
| QAP_Var[intro]:   "\<lbrakk>valid \<Gamma>;(x,T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x \<leftrightarrow> Var x : T"
| QAP_App[intro]:   "\<lbrakk>\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1 \<rightarrow> T\<^sub>2; \<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App p s \<leftrightarrow> App q t : T\<^sub>2"
| QAP_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<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]:
  "\<Gamma> \<turnstile> s\<Leftrightarrow>t : TBase"
  "\<Gamma> \<turnstile> s\<Leftrightarrow>t : T\<^sub>1 \<rightarrow> T\<^sub>2"
  "\<Gamma> \<turnstile> s\<leftrightarrow>t : TBase"
  "\<Gamma> \<turnstile> s\<leftrightarrow>t : TUnit"
  "\<Gamma> \<turnstile> s\<leftrightarrow>t : T\<^sub>1 \<rightarrow> T\<^sub>2"

  "\<Gamma> \<turnstile> Var x \<leftrightarrow> t : T"
  "\<Gamma> \<turnstile> Var x \<leftrightarrow> t : T'"
  "\<Gamma> \<turnstile> s \<leftrightarrow> Var x : T"
  "\<Gamma> \<turnstile> s \<leftrightarrow> Var x : T'"
  "\<Gamma> \<turnstile> Const n \<leftrightarrow> t : T"
  "\<Gamma> \<turnstile> s \<leftrightarrow> Const n : T"
  "\<Gamma> \<turnstile> App p s \<leftrightarrow> t : T"
  "\<Gamma> \<turnstile> s \<leftrightarrow> App q t : T"
  "\<Gamma> \<turnstile> Lam[x].s \<leftrightarrow> t : T"
  "\<Gamma> \<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>\<Gamma>" "x\<sharp>t" "x\<sharp>u"
  and h: "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T\<^sub>1\<rightarrow>T\<^sub>2"
  shows "(x,T\<^sub>1)#\<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2"
proof -
  obtain y where fs2: "y\<sharp>(\<Gamma>,t,u)" and "(y,T\<^sub>1)#\<Gamma> \<turnstile> App t (Var y) \<Leftrightarrow> App u (Var y) : T\<^sub>2"
    using h by auto
  then have "([(x,y)]\<bullet>((y,T\<^sub>1)#\<Gamma>)) \<turnstile> [(x,y)]\<bullet> App t (Var y) \<Leftrightarrow> [(x,y)]\<bullet> App u (Var y) : T\<^sub>2"
    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 "\<lbrakk> \<Gamma> \<turnstile> s \<Leftrightarrow> t : T ; \<Gamma> \<turnstile> s \<Leftrightarrow> u : T' \<rbrakk> \<Longrightarrow> T = T'"
  and "\<lbrakk> \<Gamma> \<turnstile> s \<leftrightarrow> t : T ; \<Gamma> \<turnstile> s \<leftrightarrow> u : T' \<rbrakk> \<Longrightarrow> T = T'"

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

lemma algorithmic_path_type_unicity:
  shows "\<Gamma> \<turnstile> s \<leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<turnstile> s \<leftrightarrow> u : T' \<Longrightarrow> T = T'"
proof (induct arbitrary:  u T'
       rule: alg_equiv_alg_path_equiv.inducts(2) [of _ _ _ _ _  "%a b c d . True"    ])
  case (QAP_Var \<Gamma> x T u T')
  have "\<Gamma> \<turnstile> Var x \<leftrightarrow> u : T'" by fact
  then have "u=Var x" and "(x,T') \<in> set \<Gamma>" by auto
  moreover have "valid \<Gamma>" "(x,T) \<in> set \<Gamma>" by fact+
  ultimately show "T=T'" using type_unicity_in_context by auto
next
  case (QAP_App \<Gamma> p q T\<^sub>1 T\<^sub>2 s t u T\<^sub>2')
  have ih:"\<And>u T. \<Gamma> \<turnstile> p \<leftrightarrow> u : T \<Longrightarrow> T\<^sub>1\<rightarrow>T\<^sub>2 = T" by fact
  have "\<Gamma> \<turnstile> App p s \<leftrightarrow> u : T\<^sub>2'" by fact
  then obtain r t T\<^sub>1' where "u = App r t"  "\<Gamma> \<turnstile> p \<leftrightarrow> r : T\<^sub>1' \<rightarrow> T\<^sub>2'" by auto
  with ih have "T\<^sub>1\<rightarrow>T\<^sub>2 = T\<^sub>1' \<rightarrow> T\<^sub>2'" by auto
  then show "T\<^sub>2=T\<^sub>2'" using ty.inject by auto
qed (auto)

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

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

lemma algorithmic_transitivity:
  shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<turnstile> t \<Leftrightarrow> u : T \<Longrightarrow> \<Gamma> \<turnstile> s \<Leftrightarrow> u : T"
  and   "\<Gamma> \<turnstile> s \<leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<turnstile> t \<leftrightarrow> u : T \<Longrightarrow> \<Gamma> \<turnstile> s \<leftrightarrow> u : T"
proof (nominal_induct \<Gamma> s t T and \<Gamma> s t T avoiding: u rule: alg_equiv_alg_path_equiv.strong_inducts)
  case (QAT_Base s p t q \<Gamma> u)
  have "\<Gamma> \<turnstile> t \<Leftrightarrow> u : TBase" by fact
  then obtain r' q' where b1: "t \<Down> q'" and b2: "u \<Down> r'" and b3: "\<Gamma> \<turnstile> q' \<leftrightarrow> r' : TBase" by auto
  have ih: "\<Gamma> \<turnstile> q \<leftrightarrow> r' : TBase \<Longrightarrow> \<Gamma> \<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 "\<Gamma> \<turnstile> p \<leftrightarrow> r' : TBase" by simp
  moreover
  have "s \<Down> p" by fact
  ultimately show "\<Gamma> \<turnstile> s \<Leftrightarrow> u : TBase" using b2 by auto
next
  case (QAT_Arrow  x \<Gamma> s t T\<^sub>1 T\<^sub>2 u)
  have ih:"(x,T\<^sub>1)#\<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2
                                   \<Longrightarrow> (x,T\<^sub>1)#\<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2" by fact
  have fs: "x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>u" by fact+
  have "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T\<^sub>1\<rightarrow>T\<^sub>2" by fact
  then have "(x,T\<^sub>1)#\<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2" using fs
    by (simp add: Q_Arrow_strong_inversion)
  with ih have "(x,T\<^sub>1)#\<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2" by simp
  then show "\<Gamma> \<turnstile> s \<Leftrightarrow> u : T\<^sub>1\<rightarrow>T\<^sub>2" using fs by (auto simp add: fresh_prod)
next
  case (QAP_App \<Gamma> p q T\<^sub>1 T\<^sub>2 s t u)
  have "\<Gamma> \<turnstile> App q t \<leftrightarrow> u : T\<^sub>2" by fact
  then obtain r T\<^sub>1' v where ha: "\<Gamma> \<turnstile> q \<leftrightarrow> r : T\<^sub>1'\<rightarrow>T\<^sub>2" and hb: "\<Gamma> \<turnstile> t \<Leftrightarrow> v : T\<^sub>1'" and eq: "u = App r v"
    by auto
  have ih1: "\<Gamma> \<turnstile> q \<leftrightarrow> r : T\<^sub>1\<rightarrow>T\<^sub>2 \<Longrightarrow> \<Gamma> \<turnstile> p \<leftrightarrow> r : T\<^sub>1\<rightarrow>T\<^sub>2" by fact
  have ih2:"\<Gamma> \<turnstile> t \<Leftrightarrow> v : T\<^sub>1 \<Longrightarrow> \<Gamma> \<turnstile> s \<Leftrightarrow> v : T\<^sub>1" by fact
  have "\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1\<rightarrow>T\<^sub>2" by fact
  then have "\<Gamma> \<turnstile> q \<leftrightarrow> p : T\<^sub>1\<rightarrow>T\<^sub>2" by (simp add: algorithmic_symmetry)
  with ha have "T\<^sub>1'\<rightarrow>T\<^sub>2 = T\<^sub>1\<rightarrow>T\<^sub>2" using algorithmic_path_type_unicity by simp
  then have "T\<^sub>1' = T\<^sub>1" by (simp add: ty.inject)
  then have "\<Gamma> \<turnstile> s \<Leftrightarrow> v : T\<^sub>1" "\<Gamma> \<turnstile> p \<leftrightarrow> r : T\<^sub>1\<rightarrow>T\<^sub>2" using ih1 ih2 ha hb by auto
  then show "\<Gamma> \<turnstile> App p s \<leftrightarrow> u : T\<^sub>2" using eq by auto
qed (auto)

lemma algorithmic_weak_head_closure:
  shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T \<Longrightarrow> s' \<leadsto> s \<Longrightarrow> t' \<leadsto> t \<Longrightarrow> \<Gamma> \<turnstile> s' \<Leftrightarrow> t' : T"
apply (nominal_induct \<Gamma> 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 "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<subseteq> \<Gamma>' \<Longrightarrow> valid \<Gamma>' \<Longrightarrow> \<Gamma>' \<turnstile> s \<Leftrightarrow> t : T"
  and   "\<Gamma> \<turnstile> s \<leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<subseteq> \<Gamma>' \<Longrightarrow> valid \<Gamma>' \<Longrightarrow> \<Gamma>' \<turnstile> s \<leftrightarrow> t : T"
proof (nominal_induct \<Gamma> s t T and \<Gamma> s t T avoiding: \<Gamma>' rule: alg_equiv_alg_path_equiv.strong_inducts)
 case (QAT_Arrow x \<Gamma> s t T\<^sub>1 T\<^sub>2 \<Gamma>')
  have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>\<Gamma>'" by fact+
  have h2:"\<Gamma> \<subseteq> \<Gamma>'" by fact
  have ih:"\<And>\<Gamma>'. \<lbrakk>(x,T\<^sub>1)#\<Gamma> \<subseteq> \<Gamma>'; valid \<Gamma>'\<rbrakk>  \<Longrightarrow> \<Gamma>' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2" by fact
  have "valid \<Gamma>'" by fact
  then have "valid ((x,T\<^sub>1)#\<Gamma>')" using fs by auto
  moreover
  have sub: "(x,T\<^sub>1)#\<Gamma> \<subseteq> (x,T\<^sub>1)#\<Gamma>'" using h2 by auto
  ultimately have "(x,T\<^sub>1)#\<Gamma>' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2" using ih by simp
  then show "\<Gamma>' \<turnstile> s \<Leftrightarrow> t : T\<^sub>1\<rightarrow>T\<^sub>2" using fs by (auto simp add: fresh_prod)
qed (auto)

lemma path_equiv_implies_nf:
  assumes "\<Gamma> \<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 \<open>Logical Equivalence\<close>

function log_equiv :: "(Ctxt \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool)" ("_ \<turnstile> _ is _ : _" [60,60,60,60] 60)
where
   "\<Gamma> \<turnstile> s is t : TUnit = True"
 | "\<Gamma> \<turnstile> s is t : TBase = \<Gamma> \<turnstile> s \<Leftrightarrow> t : TBase"
 | "\<Gamma> \<turnstile> s is t : (T\<^sub>1 \<rightarrow> T\<^sub>2) =
    (\<forall>\<Gamma>' s' t'. \<Gamma>\<subseteq>\<Gamma>' \<longrightarrow> valid \<Gamma>' \<longrightarrow> \<Gamma>' \<turnstile> s' is t' : T\<^sub>1 \<longrightarrow>  (\<Gamma>' \<turnstile> (App s s') is (App t t') : T\<^sub>2))"
apply (auto simp add: ty.inject)
apply (subgoal_tac "(\<exists>T\<^sub>1 T\<^sub>2. b=T\<^sub>1 \<rightarrow> T\<^sub>2) \<or> b=TUnit \<or> b=TBase" )
apply (force)
apply (rule ty_cases)
done

termination by lexicographic_order

lemma logical_monotonicity:
 fixes \<Gamma> \<Gamma>' :: Ctxt
 assumes a1: "\<Gamma> \<turnstile> s is t : T"
 and     a2: "\<Gamma> \<subseteq> \<Gamma>'"
 and     a3: "valid \<Gamma>'"
 shows "\<Gamma>' \<turnstile> s is t : T"
using a1 a2 a3
proof (induct arbitrary: \<Gamma>' rule: log_equiv.induct)
  case (2 \<Gamma> s t \<Gamma>')
  then show "\<Gamma>' \<turnstile> s is t : TBase" using algorithmic_monotonicity by auto
next
  case (3 \<Gamma> s t T\<^sub>1 T\<^sub>2 \<Gamma>')
  have "\<Gamma> \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2"
  and  "\<Gamma> \<subseteq> \<Gamma>'"
  and  "valid \<Gamma>'" by fact+
  then show "\<Gamma>' \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2" by simp
qed (auto)

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

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

lemma logical_transitivity:
  assumes "\<Gamma> \<turnstile> s is t : T" "\<Gamma> \<turnstile> t is u : T"
  shows "\<Gamma> \<turnstile> s is u : T"
using assms
proof (nominal_induct arbitrary: \<Gamma> s t u  rule:ty.strong_induct)
  case TBase
  then show "\<Gamma> \<turnstile> s is u : TBase" by (auto elim:  algorithmic_transitivity)
next
  case (Arrow T\<^sub>1 T\<^sub>2 \<Gamma> s t u)
  have h1:"\<Gamma> \<turnstile> s is t : T\<^sub>1 \<rightarrow> T\<^sub>2" by fact
  have h2:"\<Gamma> \<turnstile> t is u : T\<^sub>1 \<rightarrow> T\<^sub>2" by fact
  have ih1:"\<And>\<Gamma> s t u. \<lbrakk>\<Gamma> \<turnstile> s is t : T\<^sub>1; \<Gamma> \<turnstile> t is u : T\<^sub>1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s is u : T\<^sub>1" by fact
  have ih2:"\<And>\<Gamma> s t u. \<lbrakk>\<Gamma> \<turnstile> s is t : T\<^sub>2; \<Gamma> \<turnstile> t is u : T\<^sub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s is u : T\<^sub>2" by fact
  {
    fix \<Gamma>' s' u'
    assume hsub:"\<Gamma> \<subseteq> \<Gamma>'" and hl:"\<Gamma>' \<turnstile> s' is u' : T\<^sub>1" and hk: "valid \<Gamma>'"
    then have "\<Gamma>' \<turnstile> u' is s' : T\<^sub>1" using logical_symmetry by blast
    then have "\<Gamma>' \<turnstile> u' is u' : T\<^sub>1" using ih1 hl by blast
    then have "\<Gamma>' \<turnstile> App t u' is App u u' : T\<^sub>2" using h2 hsub hk by auto
    moreover have "\<Gamma>' \<turnstile>  App s s' is App t u' : T\<^sub>2" using h1 hsub hl hk by auto
    ultimately have "\<Gamma>' \<turnstile>  App s s' is App u u' : T\<^sub>2" using ih2 by blast
  }
  then show "\<Gamma> \<turnstile> s is u : T\<^sub>1 \<rightarrow> T\<^sub>2" by auto
qed (auto)

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

lemma logical_weak_head_closure':
  assumes "\<Gamma> \<turnstile> s is t : T" and "s' \<leadsto> s"
  shows "\<Gamma> \<turnstile> s' is t : T"
using assms
proof (nominal_induct arbitrary: \<Gamma> s t s' rule: ty.strong_induct)
  case (TBase  \<Gamma> s t s')
  then show ?case by force
next
  case (TUnit \<Gamma> s t s')
  then show ?case by auto
next
  case (Arrow T\<^sub>1 T\<^sub>2 \<Gamma> s t s')
  have h1:"s' \<leadsto> s" by fact
  have ih:"\<And>\<Gamma> s t s'. \<lbrakk>\<Gamma> \<turnstile> s is t : T\<^sub>2; s' \<leadsto> s\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s' is t : T\<^sub>2" by fact
  have h2:"\<Gamma> \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2" by fact
  then
  have hb:"\<forall>\<Gamma>' s' t'. \<Gamma>\<subseteq>\<Gamma>' \<longrightarrow> valid \<Gamma>' \<longrightarrow> \<Gamma>' \<turnstile> s' is t' : T\<^sub>1 \<longrightarrow> (\<Gamma>' \<turnstile> (App s s') is (App t t') : T\<^sub>2)"
    by auto
  {
    fix \<Gamma>' s\<^sub>2 t\<^sub>2
    assume "\<Gamma> \<subseteq> \<Gamma>'" and "\<Gamma>' \<turnstile> s\<^sub>2 is t\<^sub>2 : T\<^sub>1" and "valid \<Gamma>'"
    then have "\<Gamma>' \<turnstile> (App s s\<^sub>2) is (App t t\<^sub>2) : T\<^sub>2" using hb by auto
    moreover have "(App s' s\<^sub>2)  \<leadsto> (App s s\<^sub>2)" using h1 by auto
    ultimately have "\<Gamma>' \<turnstile> App s' s\<^sub>2 is App t t\<^sub>2 : T\<^sub>2" using ih by auto
  }
  then show "\<Gamma> \<turnstile> s' is t : T\<^sub>1\<rightarrow>T\<^sub>2" by auto
qed

abbreviation
 log_equiv_for_psubsts :: "Ctxt \<Rightarrow> Subst \<Rightarrow> Subst \<Rightarrow> Ctxt \<Rightarrow> bool"  ("_ \<turnstile> _ is _ over _" [60,60] 60)
where
  "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma> \<equiv> \<forall>x T. (x,T) \<in> set \<Gamma> \<longrightarrow> \<Gamma>' \<turnstile> \<theta><Var x> is  \<theta>'<Var x> : T"

lemma logical_pseudo_reflexivity:
  assumes "\<Gamma>' \<turnstile> t is s over \<Gamma>"
  shows "\<Gamma>' \<turnstile> s is s over \<Gamma>"
proof -
  from assms have "\<Gamma>' \<turnstile> s is t over \<Gamma>" using logical_symmetry by blast
  with assms show "\<Gamma>' \<turnstile> s is s over \<Gamma>" using logical_transitivity by blast
qed

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

lemma equiv_subst_ext :
  assumes h1: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and     h2: "\<Gamma>' \<turnstile> s is t : T"
  and     fs: "x\<sharp>\<Gamma>"
  shows "\<Gamma>' \<turnstile> (x,s)#\<theta> is (x,t)#\<theta>' over (x,T)#\<Gamma>"
using assms
proof -
  {
    fix y U
    assume "(y,U) \<in> set ((x,T)#\<Gamma>)"
    moreover
    {
      assume "(y,U) \<in> set [(x,T)]"
      with h2 have "\<Gamma>' \<turnstile> ((x,s)#\<theta>)<Var y> is ((x,t)#\<theta>')<Var y> : U" by auto
    }
    moreover
    {
      assume hl:"(y,U) \<in> set \<Gamma>"
      then have "\<not> y\<sharp>\<Gamma>" by (induct \<Gamma>) (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)#\<theta>)<Var y> = \<theta><Var y>" "((x,t)#\<theta>')<Var y> = \<theta>'<Var y>"
        using fresh_psubst_simp by blast+
      moreover have  "\<Gamma>' \<turnstile> \<theta><Var y> is \<theta>'<Var y> : U" using h1 hl by auto
      ultimately have "\<Gamma>' \<turnstile> ((x,s)#\<theta>)<Var y> is ((x,t)#\<theta>')<Var y> : U" by auto
    }
    ultimately have "\<Gamma>' \<turnstile> ((x,s)#\<theta>)<Var y> is ((x,t)#\<theta>')<Var y> : U" by auto
  }
  then show "\<Gamma>' \<turnstile> (x,s)#\<theta> is (x,t)#\<theta>' over (x,T)#\<Gamma>" by auto
qed

theorem fundamental_theorem_1:
  assumes a1: "\<Gamma> \<turnstile> t : T"
  and     a2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and     a3: "valid \<Gamma>'"
  shows "\<Gamma>' \<turnstile> \<theta><t> is \<theta>'<t> : T"
using a1 a2 a3
proof (nominal_induct \<Gamma> t T avoiding: \<theta> \<theta>' arbitrary: \<Gamma>' rule: typing.strong_induct)
  case (T_Lam x \<Gamma> T\<^sub>1 t\<^sub>2 T\<^sub>2 \<theta> \<theta>' \<Gamma>')
  have vc: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" "x\<sharp>\<Gamma>" by fact+
  have asm1: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>" by fact
  have ih:"\<And>\<theta> \<theta>' \<Gamma>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^sub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t\<^sub>2> is \<theta>'<t\<^sub>2> : T\<^sub>2" by fact
  show "\<Gamma>' \<turnstile> \<theta><Lam [x].t\<^sub>2> is \<theta>'<Lam [x].t\<^sub>2> : T\<^sub>1\<rightarrow>T\<^sub>2" using vc
  proof (simp, intro strip)
    fix \<Gamma>'' s' t'
    assume sub: "\<Gamma>' \<subseteq> \<Gamma>''"
    and    asm2: "\<Gamma>''\<turnstile> s' is t' : T\<^sub>1"
    and    val: "valid \<Gamma>''"
    from asm1 val sub have "\<Gamma>'' \<turnstile> \<theta> is \<theta>' over \<Gamma>" using logical_subst_monotonicity by blast
    with asm2 vc have "\<Gamma>'' \<turnstile> (x,s')#\<theta> is (x,t')#\<theta>' over (x,T\<^sub>1)#\<Gamma>" using equiv_subst_ext by blast
    with ih val have "\<Gamma>'' \<turnstile> ((x,s')#\<theta>)<t\<^sub>2> is ((x,t')#\<theta>')<t\<^sub>2> : T\<^sub>2" by auto
    with vc have "\<Gamma>''\<turnstile>\<theta><t\<^sub>2>[x::=s'] is \<theta>'<t\<^sub>2>[x::=t'] : T\<^sub>2" by (simp add: psubst_subst_psubst)
    moreover
    have "App (Lam [x].\<theta><t\<^sub>2>) s' \<leadsto> \<theta><t\<^sub>2>[x::=s']" by auto
    moreover
    have "App (Lam [x].\<theta>'<t\<^sub>2>) t' \<leadsto> \<theta>'<t\<^sub>2>[x::=t']" by auto
    ultimately show "\<Gamma>''\<turnstile> App (Lam [x].\<theta><t\<^sub>2>) s' is App (Lam [x].\<theta>'<t\<^sub>2>) t' : T\<^sub>2"
      using logical_weak_head_closure by auto
  qed
qed (auto)

theorem fundamental_theorem_2:
  assumes h1: "\<Gamma> \<turnstile> s \<equiv> t : T"
  and     h2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and     h3: "valid \<Gamma>'"
  shows "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T"
using h1 h2 h3
proof (nominal_induct \<Gamma> s t T avoiding: \<Gamma>' \<theta> \<theta>' rule: def_equiv.strong_induct)
  case (Q_Refl \<Gamma> t T \<Gamma>' \<theta> \<theta>')
  have "\<Gamma> \<turnstile> t : T"
  and  "valid \<Gamma>'" by fact+
  moreover
  have "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>" by fact
  ultimately show "\<Gamma>' \<turnstile> \<theta><t> is \<theta>'<t> : T" using fundamental_theorem_1 by blast
next
  case (Q_Symm \<Gamma> t s T \<Gamma>' \<theta> \<theta>')
  have "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and "valid \<Gamma>'" by fact+
  moreover
  have ih: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t> is \<theta>'<s> : T" by fact
  ultimately show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" using logical_symmetry by blast
next
  case (Q_Trans \<Gamma> s t T u \<Gamma>' \<theta> \<theta>')
  have ih1: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" by fact
  have ih2: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t> is \<theta>'<u> : T" by fact
  have h: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and  v: "valid \<Gamma>'" by fact+
  then have "\<Gamma>' \<turnstile> \<theta>' is \<theta>' over \<Gamma>" using logical_pseudo_reflexivity by auto
  then have "\<Gamma>' \<turnstile> \<theta>'<t> is \<theta>'<u> : T" using ih2 v by auto
  moreover have "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" using ih1 h v by auto
  ultimately show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<u> : T" using logical_transitivity by blast
next
  case (Q_Abs x \<Gamma> T\<^sub>1 s\<^sub>2 t\<^sub>2 T\<^sub>2 \<Gamma>' \<theta> \<theta>')
  have fs:"x\<sharp>\<Gamma>" by fact
  have fs2: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact+
  have h2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and  h3: "valid \<Gamma>'" by fact+
  have ih:"\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^sub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s\<^sub>2> is \<theta>'<t\<^sub>2> : T\<^sub>2" by fact
  {
    fix \<Gamma>'' s' t'
    assume "\<Gamma>' \<subseteq> \<Gamma>''" and hl:"\<Gamma>''\<turnstile> s' is t' : T\<^sub>1" and hk: "valid \<Gamma>''"
    then have "\<Gamma>'' \<turnstile> \<theta> is \<theta>' over \<Gamma>" using h2 logical_subst_monotonicity by blast
    then have "\<Gamma>'' \<turnstile> (x,s')#\<theta> is (x,t')#\<theta>' over (x,T\<^sub>1)#\<Gamma>" using equiv_subst_ext hl fs by blast
    then have "\<Gamma>'' \<turnstile> ((x,s')#\<theta>)<s\<^sub>2> is ((x,t')#\<theta>')<t\<^sub>2> : T\<^sub>2" using ih hk by blast
    then have "\<Gamma>''\<turnstile> \<theta><s\<^sub>2>[x::=s'] is \<theta>'<t\<^sub>2>[x::=t'] : T\<^sub>2" using fs2 psubst_subst_psubst by auto
    moreover have "App (Lam [x]. \<theta><s\<^sub>2>) s' \<leadsto>  \<theta><s\<^sub>2>[x::=s']"
              and "App (Lam [x].\<theta>'<t\<^sub>2>) t' \<leadsto> \<theta>'<t\<^sub>2>[x::=t']" by auto
    ultimately have "\<Gamma>'' \<turnstile> App (Lam [x]. \<theta><s\<^sub>2>) s' is App (Lam [x].\<theta>'<t\<^sub>2>) t' : T\<^sub>2"
      using logical_weak_head_closure by auto
  }
  moreover have "valid \<Gamma>'" by fact
  ultimately have "\<Gamma>' \<turnstile> Lam [x].\<theta><s\<^sub>2> is Lam [x].\<theta>'<t\<^sub>2> : T\<^sub>1\<rightarrow>T\<^sub>2" by auto
  then show "\<Gamma>' \<turnstile> \<theta><Lam [x].s\<^sub>2> is \<theta>'<Lam [x].t\<^sub>2> : T\<^sub>1\<rightarrow>T\<^sub>2" using fs2 by auto
next
  case (Q_App \<Gamma> s\<^sub>1 t\<^sub>1 T\<^sub>1 T\<^sub>2 s\<^sub>2 t\<^sub>2 \<Gamma>' \<theta> \<theta>')
  then show "\<Gamma>' \<turnstile> \<theta><App s\<^sub>1 s\<^sub>2> is \<theta>'<App t\<^sub>1 t\<^sub>2> : T\<^sub>2" by auto
next
  case (Q_Beta x \<Gamma> s\<^sub>2 t\<^sub>2 T\<^sub>1 s12 t12 T\<^sub>2 \<Gamma>' \<theta> \<theta>')
  have h: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and  h': "valid \<Gamma>'" by fact+
  have fs: "x\<sharp>\<Gamma>" by fact
  have fs2: " x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact+
  have ih1: "\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s\<^sub>2> is \<theta>'<t\<^sub>2> : T\<^sub>1" by fact
  have ih2: "\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^sub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s12> is \<theta>'<t12> : T\<^sub>2" by fact
  have "\<Gamma>' \<turnstile> \<theta><s\<^sub>2> is \<theta>'<t\<^sub>2> : T\<^sub>1" using ih1 h' h by auto
  then have "\<Gamma>' \<turnstile> (x,\<theta><s\<^sub>2>)#\<theta> is (x,\<theta>'<t\<^sub>2>)#\<theta>' over (x,T\<^sub>1)#\<Gamma>" using equiv_subst_ext h fs by blast
  then have "\<Gamma>' \<turnstile> ((x,\<theta><s\<^sub>2>)#\<theta>)<s12> is ((x,\<theta>'<t\<^sub>2>)#\<theta>')<t12> : T\<^sub>2" using ih2 h' by auto
  then have "\<Gamma>' \<turnstile> \<theta><s12>[x::=\<theta><s\<^sub>2>] is \<theta>'<t12>[x::=\<theta>'<t\<^sub>2>] : T\<^sub>2" using fs2 psubst_subst_psubst by auto
  then have "\<Gamma>' \<turnstile> \<theta><s12>[x::=\<theta><s\<^sub>2>] is \<theta>'<t12[x::=t\<^sub>2]> : T\<^sub>2" using fs2 psubst_subst_propagate by auto
  moreover have "App (Lam [x].\<theta><s12>) (\<theta><s\<^sub>2>) \<leadsto> \<theta><s12>[x::=\<theta><s\<^sub>2>]" by auto
  ultimately have "\<Gamma>' \<turnstile> App (Lam [x].\<theta><s12>) (\<theta><s\<^sub>2>) is \<theta>'<t12[x::=t\<^sub>2]> : T\<^sub>2"
    using logical_weak_head_closure' by auto
  then show "\<Gamma>' \<turnstile> \<theta><App (Lam [x].s12) s\<^sub>2> is \<theta>'<t12[x::=t\<^sub>2]> : T\<^sub>2" using fs2 by simp
next
  case (Q_Ext x \<Gamma> s t T\<^sub>1 T\<^sub>2 \<Gamma>' \<theta> \<theta>')
  have h2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
  and  h2': "valid \<Gamma>'" by fact+
  have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" by fact+
  have ih:"\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^sub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk>
                          \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><App s (Var x)> is \<theta>'<App t (Var x)> : T\<^sub>2" by fact
   {
    fix \<Gamma>'' s' t'
    assume hsub: "\<Gamma>' \<subseteq> \<Gamma>''" and hl: "\<Gamma>''\<turnstile> s' is t' : T\<^sub>1" and hk: "valid \<Gamma>''"
    then have "\<Gamma>'' \<turnstile> \<theta> is \<theta>' over \<Gamma>" using h2 logical_subst_monotonicity by blast
    then have "\<Gamma>'' \<turnstile> (x,s')#\<theta> is (x,t')#\<theta>' over (x,T\<^sub>1)#\<Gamma>" using equiv_subst_ext hl fs by blast
    then have "\<Gamma>'' \<turnstile> ((x,s')#\<theta>)<App s (Var x)>  is ((x,t')#\<theta>')<App t (Var x)> : T\<^sub>2" using ih hk by blast
    then
    have "\<Gamma>'' \<turnstile> App (((x,s')#\<theta>)<s>) (((x,s')#\<theta>)<(Var x)>) is App (((x,t')#\<theta>')<t>) (((x,t')#\<theta>')<(Var x)>) : T\<^sub>2"
      by auto
    then have "\<Gamma>'' \<turnstile> App ((x,s')#\<theta>)<s> s'  is App ((x,t')#\<theta>')<t> t' : T\<^sub>2" by auto
    then have "\<Gamma>'' \<turnstile> App (\<theta><s>) s' is App (\<theta>'<t>) t' : T\<^sub>2" using fs fresh_psubst_simp by auto
  }
  moreover have "valid \<Gamma>'" by fact
  ultimately show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T\<^sub>1\<rightarrow>T\<^sub>2" by auto
next
  case (Q_Unit \<Gamma> s t \<Gamma>' \<theta> \<theta>')
  then show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : TUnit" by auto
qed

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

text \<open>We leave soundness as an exercise - just like Crary in the ATS book :-) \\
 @{prop[mode=IfThen] "\<lbrakk>\<Gamma> \<turnstile> s \<Leftrightarrow> t : T; \<Gamma> \<turnstile> t : T; \<Gamma> \<turnstile> s : T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s \<equiv> t : T"} \\
 @{prop "\<lbrakk>\<Gamma> \<turnstile> s \<leftrightarrow> t : T; \<Gamma> \<turnstile> t : T; \<Gamma> \<turnstile> s : T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s \<equiv> t : T"}
\<close>

end


theory Heap
  imports "../00Utils/Environment" "../00Utils/Utils"
begin

type_synonym ptr = nat

datatype 'a heap = H "nat \<Rightarrow> 'a" nat

definition hempty :: "'a heap" where
  "hempty = H (\<lambda>x. undefined) 0"

primrec halloc :: "'a heap \<Rightarrow> 'a \<Rightarrow> 'a heap \<times> ptr" where
  "halloc (H h hp) a = (H (h(hp := a)) (Suc hp), hp)"

primrec hlookup :: "'a heap \<Rightarrow> ptr \<Rightarrow> 'a" where
  "hlookup (H h hp) x = h x"

primrec hcontains :: "'a heap \<Rightarrow> ptr \<Rightarrow> bool" where
  "hcontains (H h hp) x = (x < hp)"

primrec heap_all :: "(ptr \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a heap \<Rightarrow> bool" where
  "heap_all p (H h hp) = (\<forall>x < hp. p x (h x))"

primrec hmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a heap \<Rightarrow> 'b heap" where
  "hmap f (H h hp) = H (\<lambda>x. if x < hp then f (h x) else undefined) hp"

lemma [simp]: "halloc h a = (h', x) \<Longrightarrow> hcontains h y \<Longrightarrow> hlookup h' y = hlookup h y"
  by (induction h) (auto split: if_splits)

lemma [elim]: "hcontains h (Suc x) \<Longrightarrow> hcontains h x"
  by (induction h)  simp_all

lemma [elim]: "hcontains h x \<Longrightarrow> y \<le> x \<Longrightarrow> hcontains h y"
  by (induction h)  simp_all

lemma [simp]: "hcontains h x \<Longrightarrow> halloc h a = (h', y) \<Longrightarrow> hcontains h' x"
  by (induction h) auto

lemma [simp]: "halloc h a = (h', x) \<Longrightarrow> hcontains h' x"
  by (induction h) auto

lemma heap_lookup_all: "hlookup h x = a \<Longrightarrow> heap_all p h \<Longrightarrow> hcontains h x \<Longrightarrow> p x a"
  by (induction h) auto

lemma [elim]: "heap_all p h \<Longrightarrow> halloc h a = (h', x) \<Longrightarrow> p x a \<Longrightarrow> heap_all p h'"
  by (induction h) auto

lemma heap_all_impl [elim]: "(\<And>x y. p x y \<Longrightarrow> q x y) \<Longrightarrow> heap_all p h \<Longrightarrow> heap_all q h"
  by (induction h) simp_all

lemma [simp]: "heap_all f hempty"
  by (simp add: hempty_def)

lemma [simp]: "halloc h a = (h', x) \<Longrightarrow> hcontains h y \<Longrightarrow> y < x"
  by (induction h) auto

lemma [simp]: "halloc h a = (h', v) \<Longrightarrow> hcontains h x \<Longrightarrow> v \<noteq> x"
  by (induction h) auto

lemma [simp]: "hmap f hempty = hempty"
  by (simp add: hempty_def)

lemma halloc_map [simp]: "halloc h a = (h', v) \<Longrightarrow> halloc (hmap f h) (f a) = (hmap f h', v)"
  by (induction h) auto+

lemma halloc_map_inv [simp]: "halloc (hmap f h) (f a) = (h', v) \<Longrightarrow>
    \<exists>h''. halloc h a = (h'', v) \<and> h' = hmap f h''"
  by (induction h) auto

lemma [simp]: "hcontains h x \<Longrightarrow> hlookup (hmap f h) x = f (hlookup h x)"
  by (induction h) simp_all

lemma hmap_eq: "(\<And>x. hcontains h x \<Longrightarrow> f (hlookup h x) = g (hlookup h x)) \<Longrightarrow> hmap f h = hmap g h"
proof (induction h)
  case (H h hp)
  have "(\<lambda>x. if x < hp then f (h x) else undefined) = (\<lambda>x. if x < hp then g (h x) else undefined)"
  proof
    fix x
    from H show "(if x < hp then f (h x) else undefined) = (if x < hp then g (h x) else undefined)"
      by (simp split: if_splits)
  qed
  thus ?case by simp
qed

primrec stack_contains :: "'a heap \<Rightarrow> ptr list \<Rightarrow> bool" where
  "stack_contains h [] = True"
| "stack_contains h (v # vs) = (hcontains h v \<and> stack_contains h vs)"

primrec env_contains :: "'a heap \<Rightarrow> ptr list list \<Rightarrow> bool" where
  "env_contains h [] = True"
| "env_contains h (v # vs) = (stack_contains h v \<and> env_contains h vs)"

lemma [simp]: "halloc h a = (h', x) \<Longrightarrow> stack_contains h vs \<Longrightarrow> stack_contains h' vs"
  by (induction vs) auto

lemma [simp]: "halloc h a = (h', x) \<Longrightarrow> env_contains h vs \<Longrightarrow> env_contains h' vs"
  by (induction vs) simp_all

lemma [simp]: "lookup vs x = Some v \<Longrightarrow> stack_contains h vs \<Longrightarrow> hcontains h v"
  by (induction vs x rule: lookup.induct) simp_all

lemma [elim]: "stack_contains h env \<Longrightarrow> x \<in> set env \<Longrightarrow> hcontains h x"
  by (induction env) auto

fun halloc_list' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> (nat \<Rightarrow> 'a)" where
  "halloc_list' h hp [] = h"
| "halloc_list' h hp (x # xs) = halloc_list' (h(hp := x)) (Suc hp) xs"

primrec halloc_list :: "'a heap \<Rightarrow> 'a list \<Rightarrow> 'a heap \<times> ptr" where
  "halloc_list (H h hp) as = (H (halloc_list' h hp as) (hp + length as), hp)"

function hsplay' :: "('a \<Rightarrow> 'b list) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow>
    (nat \<Rightarrow> 'b) \<times> nat" where
  "hpa \<le> n \<Longrightarrow> hsplay' f ha hpa hb hpb n = (hb, hpb)"
| "hpa > n \<Longrightarrow> hsplay' f ha hpa hb hpb n =
    hsplay' f ha hpa (halloc_list' hb hpb (f (ha n))) (hpb + length (f (ha n))) (Suc n)"
  by atomize_elim auto
termination
  by (relation "measure (\<lambda>(_, _, hpa, _, _, n). hpa - n)") simp_all

primrec hsplay :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a heap \<Rightarrow> 'b heap" where
  "hsplay f (H h hp) = (case hsplay' f h hp (\<lambda>x. undefined) 0 0 of (h', hp') \<Rightarrow> H h' hp')"

lemma [simp]: "y < hp \<Longrightarrow> halloc_list' h hp as y = h y"
  by (induction as arbitrary: h hp) simp_all

lemma [simp]: "halloc_list h as = (h', x) \<Longrightarrow> hcontains h y \<Longrightarrow> hlookup h' y = hlookup h y"
  by (induction h) auto

lemma [simp]: "k < length as \<Longrightarrow> halloc_list' h hp as (k + hp) = as ! k"
proof (induction as arbitrary: h hp k)
  case (Cons a as)
  thus ?case
  proof (induction k)
    case (Suc k)
    hence "halloc_list' (h(hp := a)) (Suc hp) as (k + (Suc hp)) = as ! k" by fastforce
    thus ?case by simp
  qed simp_all
qed simp_all

lemma [simp]: "halloc_list h as = (h', p) \<Longrightarrow> k < length as \<Longrightarrow>
    hlookup h' (k + p) = as ! k"
  by (induction h) (auto split: prod.splits)

lemma [simp]: "(k::nat) < m \<Longrightarrow> x < n \<Longrightarrow> k + m * x < m * n"
proof (induction x arbitrary: n)
  case 0
  thus ?case by (induction n) simp_all
next
  case (Suc x)
  thus ?case by (induction n) simp_all
qed

lemma hlookup_hsplay' [simp]: "(\<And>a. length (f a) = m \<and> f a ! k = g a) \<Longrightarrow> p < hpa \<Longrightarrow> k < m \<Longrightarrow>
  (\<And>x. x < n \<Longrightarrow> hb (k + m * x) = g (ha x)) \<Longrightarrow> hpb = m * n \<Longrightarrow>
    hsplay' f ha hpa hb hpb n = (h', hp') \<Longrightarrow> h' (k + m * p) = g (ha p)"
proof (induction f ha hpa hb hpb n rule: hsplay'.induct)
  case (2 hpa n f ha hb hpb)
  hence "\<And>x. x < Suc n \<Longrightarrow> halloc_list' hb hpb (f (ha n)) (k + m * x) = g (ha x)"
  proof -
    fix x
    assume X: "x < Suc n"
    with 2 show "?thesis x"
    proof (cases "x < n")
      case False
      with X have "x = n" by simp
      with 2 show ?thesis by simp
    qed simp_all
  qed
  with 2 show ?case by simp
qed simp_all

lemma hlookup_hsplay [simp]: "(\<And>a. length (f a) = m \<and> f a ! k = g a) \<Longrightarrow> hcontains h p \<Longrightarrow> k < m \<Longrightarrow>
    hlookup (hsplay f h) (k + m * p) = g (hlookup h p)"
proof (induction h)
  case (H h hp)
  moreover obtain h' hp' where "hsplay' f h hp (\<lambda>x. undefined) 0 0 = (h', hp')" by fastforce
  moreover with H have "p < hp \<Longrightarrow> 0 = m * 0 \<Longrightarrow> h' (k + m * p) = g (h p)"
    using hlookup_hsplay' by blast
  ultimately show ?case by simp
qed

lemma [simp]: "hsplay f hempty = hempty"
  by (simp add: hempty_def)

lemma halloc_list_hsplay' [simp]: "n \<le> hpa \<Longrightarrow> hsplay' f ha hpa hb hpb n = (h, hp) \<Longrightarrow>
  hsplay' f (ha(hpa := a)) (Suc hpa) hb hpb n = (h', hp') \<Longrightarrow>
    h' = halloc_list' h hp (f a) \<and> hp' = hp + length (f a)"
  by (induction f ha hpa hb hpb n rule: hsplay'.induct) simp_all

lemma hsplay'_index [simp]: "hsplay' f ha hpa hb hpb n = (h, hp) \<Longrightarrow> (\<And>x. length (f x) = k) \<Longrightarrow>
    hp = hpb + k * (hpa - n)"
proof (induction f ha hpa hb hpb n rule: hsplay'.induct)
  case (2 hpa n f ha hb hpb)
  moreover have "n < hpa \<Longrightarrow> k + k * (hpa - Suc n) = k * (hpa - n)"
    by (metis Suc_diff_Suc mult_Suc_right)
  ultimately show ?case by simp
qed simp_all

lemma halloc_list_hsplay [simp]: "halloc h a = (h', p) \<Longrightarrow> (\<And>x. length (f x) = k) \<Longrightarrow>
  halloc_list (hsplay f h) (f a) = (hsplay f h', k * p)"
proof (induction h)
  case (H h hp)
  hence "hp = p" by simp
  moreover with H have "h' = H (h(p := a)) (Suc p)" by simp
  moreover obtain hh' hp' where HH: "hsplay' f h p (\<lambda>x. undefined) 0 0 = (hh', hp')" by fastforce
  moreover obtain hh'' hp'' where HH': "hsplay' f (h(p := a)) (Suc p) (\<lambda>x. undefined) 0 0 =
    (hh'', hp'')" by fastforce
  moreover with HH have "hh'' = halloc_list' hh' hp' (f a) \<and> hp'' = hp' + length (f a)"
    using halloc_list_hsplay' by fast
  moreover from H HH have "hp' = 0 + k * (p - 0)" by (metis hsplay'_index)
  ultimately show ?case by fastforce
qed

lemma [simp]: "hsplay' f h hp m mp n = (h', hp') \<Longrightarrow> mp = k * n \<Longrightarrow> (\<And>a. length (f a) = k) \<Longrightarrow>
    n \<le> hp \<Longrightarrow> hp' = k * hp"
  by (induction f h hp m mp n rule: hsplay'.induct) simp_all

lemma [simp]: "hcontains h x \<Longrightarrow> hsplay f h = H h' hp' \<Longrightarrow> (\<And>a. length (f a) = k) \<Longrightarrow>
  1 < k \<Longrightarrow> Suc (k * x) < hp'"
proof (induction h)
  case (H h hp)
  moreover hence "hp' = k * hp" by (simp split: prod.splits)
  ultimately show ?case by simp
qed

primrec listify' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a list" where
  "listify' h 0 = []"
| "listify' h (Suc x) = h x # listify' h x"

primrec listify :: "'a heap \<Rightarrow> 'a list" where
  "listify (H h hp) = listify' h hp"

lemma [dest]: "listify' h x = [] \<Longrightarrow> x = 0"
  by (induction x) simp_all

lemma [dest]: "listify' h x = a # as \<Longrightarrow> \<exists>y. x = Suc y \<and> h y = a \<and> listify' h y = as"
  by (induction x) simp_all

lemma [simp]: "hp \<le> x \<Longrightarrow> listify' (h(x := v)) hp = listify' h hp"
  by (induction hp) simp_all

lemma [simp]: "hp \<le> x \<Longrightarrow> listify' (h(Suc x := v) \<circ> Suc) hp = listify' (h \<circ> Suc) hp"
  by (induction hp) auto

end