theory TypeSchemes1

imports "../Nominal2"

begin

section {* Type Schemes defined as two separate nominal datatypes *}

atom_decl name 

nominal_datatype ty =

  Var "name"

| Fun "ty" "ty" ("_ \<rightarrow> _")

nominal_datatype tys =

  All xs::"name fset" ty::"ty" binds (set+) xs in ty ("All [_]._")

thm tys.distinct

thm tys.induct tys.strong_induct

thm tys.exhaust tys.strong_exhaust

thm tys.fv_defs

thm tys.bn_defs

thm tys.perm_simps

thm tys.eq_iff

thm tys.fv_bn_eqvt

thm tys.size_eqvt

thm tys.supports

thm tys.supp

thm tys.fresh

subsection {* Some Tests about Alpha-Equality *}

lemma

  shows "All [{|a, b|}].((Var a) \<rightarrow> (Var b)) = All [{|b, a|}]. ((Var a) \<rightarrow> (Var b))"

  apply(simp add: Abs_eq_iff)

  apply(rule_tac x="0::perm" in exI)

  apply(simp add: alphas fresh_star_def ty.supp supp_at_base)

  done

lemma

  shows "All [{|a, b|}].((Var a) \<rightarrow> (Var b)) = All [{|a, b|}].((Var b) \<rightarrow> (Var a))"

  apply(simp add: Abs_eq_iff)

  apply(rule_tac x="(atom a \<rightleftharpoons> atom b)" in exI)

  apply(simp add: alphas fresh_star_def supp_at_base ty.supp)

  done

lemma

  shows "All [{|a, b, c|}].((Var a) \<rightarrow> (Var b)) = All [{|a, b|}].((Var a) \<rightarrow> (Var b))"

  apply(simp add: Abs_eq_iff)

  apply(rule_tac x="0::perm" in exI)

  apply(simp add: alphas fresh_star_def ty.supp supp_at_base)

done

lemma

  assumes a: "a \<noteq> b"

  shows "\<not>(All [{|a, b|}].((Var a) \<rightarrow> (Var b)) = All [{|c|}].((Var c) \<rightarrow> (Var c)))"

  using a

  apply(simp add: Abs_eq_iff)

  apply(clarify)

  apply(simp add: alphas fresh_star_def ty.supp supp_at_base)

  apply auto

  done

subsection {* Substitution function for types and type schemes *}

type_synonym 

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

fun

  lookup :: "Subst \<Rightarrow> name \<Rightarrow> ty"

where

  "lookup [] Y = Var Y"

| "lookup ((X, T) # Ts) Y = (if X = Y then T else lookup Ts Y)"

lemma lookup_eqvt[eqvt]:

  shows "(p \<bullet> lookup Ts T) = lookup (p \<bullet> Ts) (p \<bullet> T)"

  by (induct Ts T rule: lookup.induct) (simp_all)

nominal_primrec

  subst  :: "Subst \<Rightarrow> ty \<Rightarrow> ty" ("_<_>" [100,60] 120)

where

  "\<theta><Var X> = lookup \<theta> X"

| "\<theta><T1 \<rightarrow> T2> = (\<theta><T1>) \<rightarrow> (\<theta><T2>)"

  unfolding eqvt_def subst_graph_def

  apply (rule, perm_simp, rule)

  apply(rule TrueI)

  apply(case_tac x)

  apply(rule_tac y="b" in ty.exhaust)

  apply(blast)

  apply(blast)

  apply(simp_all)

  done

termination (eqvt)

  by lexicographic_order

lemma supp_fun_app_eqvt:

  assumes e: "eqvt f"

  shows "supp (f a b) \<subseteq> supp a \<union> supp b"

  using supp_fun_app_eqvt[OF e] supp_fun_app

  by blast

lemma supp_subst:

  "supp (subst \<theta> t) \<subseteq> supp \<theta> \<union> supp t"

  apply (rule supp_fun_app_eqvt)

  unfolding eqvt_def 

  by (simp add: permute_fun_def subst.eqvt)

nominal_primrec

  substs :: "(name \<times> ty) list \<Rightarrow> tys \<Rightarrow> tys" ("_<_>" [100,60] 120)

where

  "fset (map_fset atom Xs) \<sharp>* \<theta> \<Longrightarrow> \<theta><All [Xs].T> = All [Xs].(\<theta><T>)"

  unfolding eqvt_def substs_graph_def

  apply (rule, perm_simp, rule)

  apply auto[2]

  apply (rule_tac y="b" and c="a" in tys.strong_exhaust)

  apply auto[1]

  apply(simp)

  apply(erule conjE)

  apply (erule Abs_res_fcb)

  apply (simp add: Abs_fresh_iff)

  apply(simp add: fresh_def)

  apply(simp add: supp_Abs)

  apply(rule impI)

  apply(subgoal_tac "x \<notin> supp \<theta>")

  prefer 2

  apply(auto simp add: fresh_star_def fresh_def)[1]

  apply(subgoal_tac "x \<in> supp T")

  using supp_subst

  apply(blast)

  using supp_subst

  apply(blast)

  apply clarify

  apply (simp add: subst.eqvt)

  apply (subst Abs_eq_iff)

  apply (rule_tac x="0::perm" in exI)

  apply (subgoal_tac "p \<bullet> \<theta>' = \<theta>'")

  apply (simp add: alphas fresh_star_zero)

  apply (subgoal_tac "\<And>x. x \<in> supp (subst \<theta>' (p \<bullet> T)) \<Longrightarrow> x \<in> p \<bullet> atom ` fset Xs \<longleftrightarrow> x \<in> atom ` fset Xsa")

  apply blast

  apply (subgoal_tac "x \<in> supp(p \<bullet> \<theta>', p \<bullet> T)")

  apply (simp add: supp_Pair eqvts eqvts_raw)

  apply auto[1]

  apply (subgoal_tac "(atom ` fset (p \<bullet> Xs)) \<sharp>* \<theta>'")

  apply (simp add: fresh_star_def fresh_def)

  apply(drule_tac p1="p" in iffD2[OF fresh_star_permute_iff])

  apply (simp add: eqvts eqvts_raw)

  apply (simp add: fresh_star_def fresh_def)

  apply (drule subsetD[OF supp_subst])

  apply (simp add: supp_Pair)

  apply (rule perm_supp_eq)

  apply (simp add: fresh_def fresh_star_def)

  apply blast

  done

termination (eqvt)

  by (lexicographic_order)

subsection {* Generalisation of types and type-schemes*}

fun

  subst_dom_pi :: "Subst \<Rightarrow> perm \<Rightarrow> Subst" ("_|_")

where

  "[]|p = []"

| "((X,T)#\<theta>)|p = (p \<bullet> X, T)#(\<theta>|p)" 

fun

  subst_subst :: "Subst \<Rightarrow> Subst \<Rightarrow> Subst" ("_<_>"  [100,60] 120)

where

  "\<theta><[]> = []"

| "\<theta> <((X,T)#\<theta>')> = (X,\<theta><T>)#(\<theta><\<theta>'>)"

fun

  dom :: "Subst \<Rightarrow> name fset"

where

  "dom [] = {||}"

| "dom ((X,T)#\<theta>) = {|X|} |\<union>| dom \<theta>"

lemma dom_eqvt[eqvt]:

  shows "p \<bullet> (dom \<theta>) = dom (p \<bullet> \<theta>)"

by (induct \<theta>) (auto)

lemma dom_subst:

  fixes \<theta>1 \<theta>2::"Subst"

  shows "dom (\<theta>2<\<theta>1>) = (dom \<theta>1)"

by (induct \<theta>1) (auto)

lemma dom_pi:

  shows "dom (\<theta>|p) = dom (p \<bullet> \<theta>)"

by (induct \<theta>) (auto)

lemma dom_fresh_lookup:

  fixes \<theta>::"Subst"

  assumes "\<forall>Y \<in> fset (dom \<theta>). atom Y \<sharp> X"

  shows "lookup \<theta> X = Var X"

using assms

by (induct \<theta>) (auto simp add: fresh_at_base)

lemma dom_fresh_ty:

  fixes \<theta>::"Subst"

  and   T::"ty"

  assumes "\<forall>X \<in> fset (dom \<theta>). atom X \<sharp> T"

  shows "\<theta><T> = T"

using assms

by (induct T rule: ty.induct) (auto simp add: ty.fresh dom_fresh_lookup)

lemma dom_fresh_subst:

  fixes \<theta> \<theta>'::"Subst"

  assumes "\<forall>X \<in> fset (dom \<theta>). atom X \<sharp> \<theta>'"

  shows "\<theta><\<theta>'> = \<theta>'"

using assms

by (induct \<theta>') (auto simp add: fresh_Pair fresh_Cons dom_fresh_ty)

definition

  generalises :: "ty \<Rightarrow> tys \<Rightarrow> bool" ("_ \<prec>\<prec> _")

where

  "T \<prec>\<prec> S \<longleftrightarrow> (\<exists>\<theta> Xs T'. S = All [Xs].T'\<and> T = \<theta><T'> \<and> dom \<theta> = Xs)"

lemma lookup_fresh:

  fixes X::"name"

  assumes a: "atom X \<sharp> \<theta>"

  shows "lookup \<theta> X = Var X"

  using a

  by (induct \<theta>) (auto simp add: fresh_Cons fresh_Pair fresh_at_base)

lemma lookup_dom:

  fixes X::"name"

  assumes "X |\<notin>| dom \<theta>"

  shows "lookup \<theta> X = Var X"

  using assms

  by (induct \<theta>) (auto)

lemma w1: 

  shows "\<theta><\<theta>'|p> = (\<theta><\<theta>'>)|p"

  by (induct \<theta>')(auto)

lemma w2:

  assumes "name |\<in>| dom \<theta>'" 

  shows "\<theta><lookup \<theta>' name> = lookup (\<theta><\<theta>'>) name"

using assms

apply(induct \<theta>')

apply(auto simp add: notin_empty_fset)

done

lemma w3:

  assumes "name |\<in>| dom \<theta>" 

  shows "lookup \<theta> name = lookup (\<theta>|p) (p \<bullet> name)"

using assms

apply(induct \<theta>)

apply(auto simp add: notin_empty_fset)

done

lemma fresh_lookup:

  fixes X Y::"name"

  and   \<theta>::"Subst"

  assumes asms: "atom X \<sharp> Y" "atom X \<sharp> \<theta>"

  shows "atom X \<sharp> (lookup \<theta> Y)"

  using assms

  apply (induct \<theta>)

  apply (auto simp add: fresh_Cons fresh_Pair fresh_at_base ty.fresh)

  done

lemma test:

  fixes \<theta> \<theta>'::"Subst"

  and T::"ty"

  assumes "(p \<bullet> atom ` fset (dom \<theta>')) \<sharp>* \<theta>" "supp All [dom \<theta>'].T \<sharp>* p"

  shows "\<theta><\<theta>'<T>> = \<theta><\<theta>'|p><\<theta><p \<bullet> T>>"

using assms

apply(induct T rule: ty.induct)

defer

apply(auto simp add: tys.supp ty.supp fresh_star_def)[1]

apply(auto simp add: tys.supp ty.supp fresh_star_def supp_at_base)[1]

apply(case_tac "name |\<in>| dom \<theta>'")

apply(subgoal_tac "atom (p \<bullet> name) \<sharp> \<theta>")

apply(subst (2) lookup_fresh)

apply(perm_simp)

apply(auto simp add: fresh_star_def)[1]

apply(simp)

apply(simp add: w1)

apply(simp add: w2)

apply(subst w3[symmetric])

apply(simp add: dom_subst)

apply(simp)

apply(perm_simp)

apply(rotate_tac 2)

apply(drule_tac p="p" in permute_boolI)

apply(perm_simp)

apply(auto simp add: fresh_star_def)[1]

apply(metis notin_fset)

apply(simp add: lookup_dom)

apply(perm_simp)

apply(subst dom_fresh_ty)

apply(auto)[1]

apply(rule fresh_lookup)

apply(simp add: dom_subst)

apply(simp add: dom_pi)

apply(perm_simp)

apply(rotate_tac 2)

apply(drule_tac p="p" in permute_boolI)

apply(perm_simp)

apply(simp add: fresh_at_base)

apply (metis in_fset)

apply(simp add: dom_subst)

apply(simp add: dom_pi[symmetric])

apply(perm_simp)

apply(subst supp_perm_eq)

apply(simp add: supp_at_base fresh_star_def)

apply (smt Diff_iff atom_eq_iff image_iff insertI1 notin_fset)

apply(simp)

done

lemma

  shows "T \<prec>\<prec> S \<Longrightarrow> \<theta><T> \<prec>\<prec> \<theta><S>"

unfolding generalises_def

apply(erule exE)+

apply(erule conjE)+

apply(rule_tac t="S" and s="All [Xs].T'" in subst)

apply(simp)

using at_set_avoiding3

apply -

apply(drule_tac x="fset (map_fset atom Xs)" in meta_spec)

apply(drule_tac x="\<theta>" in meta_spec)

apply(drule_tac x="All [Xs].T'" in meta_spec)

apply(drule meta_mp)

apply(simp)

apply(drule meta_mp)

apply(simp add: finite_supp)

apply(drule meta_mp)

apply(simp add: finite_supp)

apply(drule meta_mp)

apply(simp add: tys.fresh fresh_star_def)

apply(erule exE)

apply(erule conjE)+

apply(rule_tac t="All[Xs].T'" and s="p \<bullet> (All [Xs].T')" in subst)

apply(rule supp_perm_eq)

apply(assumption)

apply(perm_simp)

apply(subst substs.simps)

apply(perm_simp)

apply(simp)

apply(rule_tac x="\<theta><\<theta>'|p>" in exI)

apply(rule_tac x="p \<bullet> Xs" in exI)

apply(rule_tac x="\<theta><p \<bullet> T'>" in exI)

apply(rule conjI)

apply(simp)

apply(rule conjI)

defer

apply(simp add: dom_subst)

apply(simp add: dom_pi dom_eqvt[symmetric])

apply(rule_tac t="T" and s="\<theta>'<T'>" in subst)

apply(simp)

apply(simp)

apply(rule test)

apply(perm_simp)

apply(rotate_tac 2)

apply(drule_tac p="p" in permute_boolI)

apply(perm_simp)

apply(auto simp add: fresh_star_def)

done

lemma 

  "T \<prec>\<prec>  All [Xs].T' \<longleftrightarrow> (\<exists>\<theta>. T = \<theta><T'> \<and> dom \<theta> = Xs)"

apply(auto)

defer

unfolding generalises_def

apply(auto)[1]

apply(auto)[1]

apply(drule sym)

apply(simp add: Abs_eq_iff2)

apply(simp add: alphas)

apply(auto)

apply(rule_tac x="map (\<lambda>(X, T). (p \<bullet> X, T)) \<theta>" in exI)

apply(auto)

oops

end