Added Pattern.thy to Nominal/Examples.
header {* Simply-typed lambda-calculus with let and tuple patterns *}
theory Pattern
imports Nominal
begin
no_syntax
"_Map" :: "maplets => 'a ~=> 'b" ("(1[_])")
atom_decl name
nominal_datatype ty =
Atom nat
| Arrow ty ty (infixr "\<rightarrow>" 200)
| TupleT ty ty (infixr "\<otimes>" 210)
lemma fresh_type [simp]: "(a::name) \<sharp> (T::ty)"
by (induct T rule: ty.induct) (simp_all add: fresh_nat)
lemma supp_type [simp]: "supp (T::ty) = ({} :: name set)"
by (induct T rule: ty.induct) (simp_all add: ty.supp supp_nat)
lemma perm_type: "(pi::name prm) \<bullet> (T::ty) = T"
by (induct T rule: ty.induct) (simp_all add: perm_nat_def)
nominal_datatype trm =
Var name
| Tuple trm trm ("(1'\<langle>_,/ _'\<rangle>)")
| Abs ty "\<guillemotleft>name\<guillemotright>trm"
| App trm trm (infixl "\<cdot>" 200)
| Let ty trm btrm
and btrm =
Base trm
| Bind ty "\<guillemotleft>name\<guillemotright>btrm"
abbreviation
Abs_syn :: "name \<Rightarrow> ty \<Rightarrow> trm \<Rightarrow> trm" ("(3\<lambda>_:_./ _)" [0, 0, 10] 10)
where
"\<lambda>x:T. t \<equiv> Abs T x t"
datatype pat =
PVar name ty
| PTuple pat pat ("(1'\<langle>\<langle>_,/ _'\<rangle>\<rangle>)")
(* FIXME: The following should be done automatically by the nominal package *)
overloading pat_perm \<equiv> "perm :: name prm \<Rightarrow> pat \<Rightarrow> pat" (unchecked)
begin
primrec pat_perm
where
"pat_perm pi (PVar x ty) = PVar (pi \<bullet> x) (pi \<bullet> ty)"
| "pat_perm pi \<langle>\<langle>p, q\<rangle>\<rangle> = \<langle>\<langle>pat_perm pi p, pat_perm pi q\<rangle>\<rangle>"
end
declare pat_perm.simps [eqvt]
lemma supp_PVar [simp]: "((supp (PVar x T))::name set) = supp x"
by (simp add: supp_def perm_fresh_fresh)
lemma supp_PTuple [simp]: "((supp \<langle>\<langle>p, q\<rangle>\<rangle>)::name set) = supp p \<union> supp q"
by (simp add: supp_def Collect_disj_eq del: disj_not1)
instance pat :: pt_name
proof intro_classes
case goal1
show ?case by (induct x) simp_all
next
case goal2
show ?case by (induct x) (simp_all add: pt_name2)
next
case goal3
then show ?case by (induct x) (simp_all add: pt_name3)
qed
instance pat :: fs_name
proof intro_classes
case goal1
show ?case by (induct x) (simp_all add: fin_supp)
qed
(* the following function cannot be defined using nominal_primrec, *)
(* since variable parameters are currently not allowed. *)
primrec abs_pat :: "pat \<Rightarrow> btrm \<Rightarrow> btrm" ("(3\<lambda>[_]./ _)" [0, 10] 10)
where
"(\<lambda>[PVar x T]. t) = Bind T x t"
| "(\<lambda>[\<langle>\<langle>p, q\<rangle>\<rangle>]. t) = (\<lambda>[p]. \<lambda>[q]. t)"
lemma abs_pat_eqvt [eqvt]:
"(pi :: name prm) \<bullet> (\<lambda>[p]. t) = (\<lambda>[pi \<bullet> p]. (pi \<bullet> t))"
by (induct p arbitrary: t) simp_all
lemma abs_pat_fresh [simp]:
"(x::name) \<sharp> (\<lambda>[p]. t) = (x \<in> supp p \<or> x \<sharp> t)"
by (induct p arbitrary: t) (simp_all add: abs_fresh supp_atm)
lemma abs_pat_alpha:
assumes fresh: "((pi::name prm) \<bullet> supp p::name set) \<sharp>* t"
and pi: "set pi \<subseteq> supp p \<times> pi \<bullet> supp p"
shows "(\<lambda>[p]. t) = (\<lambda>[pi \<bullet> p]. pi \<bullet> t)"
proof -
note pt_name_inst at_name_inst pi
moreover have "(supp p::name set) \<sharp>* (\<lambda>[p]. t)"
by (simp add: fresh_star_def)
moreover from fresh
have "(pi \<bullet> supp p::name set) \<sharp>* (\<lambda>[p]. t)"
by (simp add: fresh_star_def)
ultimately have "pi \<bullet> (\<lambda>[p]. t) = (\<lambda>[p]. t)"
by (rule pt_freshs_freshs)
then show ?thesis by (simp add: eqvts)
qed
primrec pat_vars :: "pat \<Rightarrow> name list"
where
"pat_vars (PVar x T) = [x]"
| "pat_vars \<langle>\<langle>p, q\<rangle>\<rangle> = pat_vars q @ pat_vars p"
lemma pat_vars_eqvt [eqvt]:
"(pi :: name prm) \<bullet> (pat_vars p) = pat_vars (pi \<bullet> p)"
by (induct p rule: pat.induct) (simp_all add: eqvts)
lemma set_pat_vars_supp: "set (pat_vars p) = supp p"
by (induct p) (auto simp add: supp_atm)
lemma distinct_eqvt [eqvt]:
"(pi :: name prm) \<bullet> (distinct (xs::name list)) = distinct (pi \<bullet> xs)"
by (induct xs) (simp_all add: eqvts)
primrec pat_type :: "pat \<Rightarrow> ty"
where
"pat_type (PVar x T) = T"
| "pat_type \<langle>\<langle>p, q\<rangle>\<rangle> = pat_type p \<otimes> pat_type q"
lemma pat_type_eqvt [eqvt]:
"(pi :: name prm) \<bullet> (pat_type p) = pat_type (pi \<bullet> p)"
by (induct p) simp_all
lemma pat_type_perm_eq: "pat_type ((pi :: name prm) \<bullet> p) = pat_type p"
by (induct p) (simp_all add: perm_type)
types ctx = "(name \<times> ty) list"
inductive
ptyping :: "pat \<Rightarrow> ty \<Rightarrow> ctx \<Rightarrow> bool" ("\<turnstile> _ : _ \<Rightarrow> _" [60, 60, 60] 60)
where
PVar: "\<turnstile> PVar x T : T \<Rightarrow> [(x, T)]"
| PTuple: "\<turnstile> p : T \<Rightarrow> \<Delta>\<^isub>1 \<Longrightarrow> \<turnstile> q : U \<Rightarrow> \<Delta>\<^isub>2 \<Longrightarrow> \<turnstile> \<langle>\<langle>p, q\<rangle>\<rangle> : T \<otimes> U \<Rightarrow> \<Delta>\<^isub>2 @ \<Delta>\<^isub>1"
lemma pat_vars_ptyping:
assumes "\<turnstile> p : T \<Rightarrow> \<Delta>"
shows "pat_vars p = map fst \<Delta>" using assms
by induct simp_all
inductive
valid :: "ctx \<Rightarrow> bool"
where
Nil [intro!]: "valid []"
| Cons [intro!]: "valid \<Gamma> \<Longrightarrow> x \<sharp> \<Gamma> \<Longrightarrow> valid ((x, T) # \<Gamma>)"
inductive_cases validE[elim!]: "valid ((x, T) # \<Gamma>)"
lemma fresh_ctxt_set_eq: "((x::name) \<sharp> (\<Gamma>::ctx)) = (x \<notin> fst ` set \<Gamma>)"
by (induct \<Gamma>) (auto simp add: fresh_list_nil fresh_list_cons fresh_prod fresh_atm)
lemma valid_distinct: "valid \<Gamma> = distinct (map fst \<Gamma>)"
by (induct \<Gamma>) (auto simp add: fresh_ctxt_set_eq [symmetric])
abbreviation
"sub_ctx" :: "ctx \<Rightarrow> ctx \<Rightarrow> bool" ("_ \<sqsubseteq> _")
where
"\<Gamma>\<^isub>1 \<sqsubseteq> \<Gamma>\<^isub>2 \<equiv> \<forall>x. x \<in> set \<Gamma>\<^isub>1 \<longrightarrow> x \<in> set \<Gamma>\<^isub>2"
abbreviation
Let_syn :: "pat \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm" ("(LET (_ =/ _)/ IN (_))" 10)
where
"LET p = t IN u \<equiv> Let (pat_type p) t (\<lambda>[p]. Base u)"
inductive typing :: "ctx \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60, 60, 60] 60)
where
Var [intro]: "valid \<Gamma> \<Longrightarrow> (x, T) \<in> set \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
| Tuple [intro]: "\<Gamma> \<turnstile> t : T \<Longrightarrow> \<Gamma> \<turnstile> u : U \<Longrightarrow> \<Gamma> \<turnstile> \<langle>t, u\<rangle> : T \<otimes> U"
| Abs [intro]: "(x, T) # \<Gamma> \<turnstile> t : U \<Longrightarrow> \<Gamma> \<turnstile> (\<lambda>x:T. t) : T \<rightarrow> U"
| App [intro]: "\<Gamma> \<turnstile> t : T \<rightarrow> U \<Longrightarrow> \<Gamma> \<turnstile> u : T \<Longrightarrow> \<Gamma> \<turnstile> t \<cdot> u : U"
| Let: "((supp p)::name set) \<sharp>* t \<Longrightarrow>
\<Gamma> \<turnstile> t : T \<Longrightarrow> \<turnstile> p : T \<Rightarrow> \<Delta> \<Longrightarrow> \<Delta> @ \<Gamma> \<turnstile> u : U \<Longrightarrow>
\<Gamma> \<turnstile> (LET p = t IN u) : U"
equivariance ptyping
equivariance valid
equivariance typing
lemma valid_typing:
assumes "\<Gamma> \<turnstile> t : T"
shows "valid \<Gamma>" using assms
by induct auto
lemma pat_var:
assumes "\<turnstile> p : T \<Rightarrow> \<Delta>"
shows "(supp p::name set) = supp \<Delta>" using assms
by induct (auto simp add: supp_list_nil supp_list_cons supp_prod supp_list_append)
lemma valid_app_fresh:
assumes "valid (\<Delta> @ \<Gamma>)" and "(x::name) \<in> supp \<Delta>"
shows "x \<sharp> \<Gamma>" using assms
by (induct \<Delta>)
(auto simp add: supp_list_nil supp_list_cons supp_prod supp_atm fresh_list_append)
lemma pat_freshs:
assumes "\<turnstile> p : T \<Rightarrow> \<Delta>"
shows "(supp p::name set) \<sharp>* c = (supp \<Delta>::name set) \<sharp>* c" using assms
by (auto simp add: fresh_star_def pat_var)
lemma valid_app_mono:
assumes "valid (\<Delta> @ \<Gamma>\<^isub>1)" and "(supp \<Delta>::name set) \<sharp>* \<Gamma>\<^isub>2" and "valid \<Gamma>\<^isub>2" and "\<Gamma>\<^isub>1 \<sqsubseteq> \<Gamma>\<^isub>2"
shows "valid (\<Delta> @ \<Gamma>\<^isub>2)" using assms
by (induct \<Delta>)
(auto simp add: supp_list_cons fresh_star_Un_elim supp_prod
fresh_list_append supp_atm fresh_star_insert_elim fresh_star_empty_elim)
nominal_inductive2 typing
avoids
Abs: "{x}"
| Let: "(supp p)::name set"
by (auto simp add: fresh_star_def abs_fresh fin_supp pat_var
dest!: valid_typing valid_app_fresh)
lemma better_T_Let [intro]:
assumes t: "\<Gamma> \<turnstile> t : T" and p: "\<turnstile> p : T \<Rightarrow> \<Delta>" and u: "\<Delta> @ \<Gamma> \<turnstile> u : U"
shows "\<Gamma> \<turnstile> (LET p = t IN u) : U"
proof -
obtain pi::"name prm" where pi: "(pi \<bullet> (supp p::name set)) \<sharp>* (t, Base u, \<Gamma>)"
and pi': "set pi \<subseteq> supp p \<times> (pi \<bullet> supp p)"
by (rule at_set_avoiding [OF at_name_inst fin_supp fin_supp])
from p u have p_fresh: "(supp p::name set) \<sharp>* \<Gamma>"
by (auto simp add: fresh_star_def pat_var dest!: valid_typing valid_app_fresh)
from pi have p_fresh': "(pi \<bullet> (supp p::name set)) \<sharp>* \<Gamma>"
by (simp add: fresh_star_prod_elim)
from pi have p_fresh'': "(pi \<bullet> (supp p::name set)) \<sharp>* Base u"
by (simp add: fresh_star_prod_elim)
from pi have "(supp (pi \<bullet> p)::name set) \<sharp>* t"
by (simp add: fresh_star_prod_elim eqvts)
moreover note t
moreover from p have "pi \<bullet> (\<turnstile> p : T \<Rightarrow> \<Delta>)" by (rule perm_boolI)
then have "\<turnstile> (pi \<bullet> p) : T \<Rightarrow> (pi \<bullet> \<Delta>)" by (simp add: eqvts perm_type)
moreover from u have "pi \<bullet> (\<Delta> @ \<Gamma> \<turnstile> u : U)" by (rule perm_boolI)
with pt_freshs_freshs [OF pt_name_inst at_name_inst pi' p_fresh p_fresh']
have "(pi \<bullet> \<Delta>) @ \<Gamma> \<turnstile> (pi \<bullet> u) : U" by (simp add: eqvts perm_type)
ultimately have "\<Gamma> \<turnstile> (LET (pi \<bullet> p) = t IN (pi \<bullet> u)) : U"
by (rule Let)
then show ?thesis by (simp add: abs_pat_alpha [OF p_fresh'' pi'] pat_type_perm_eq)
qed
lemma weakening:
assumes "\<Gamma>\<^isub>1 \<turnstile> t : T" and "valid \<Gamma>\<^isub>2" and "\<Gamma>\<^isub>1 \<sqsubseteq> \<Gamma>\<^isub>2"
shows "\<Gamma>\<^isub>2 \<turnstile> t : T" using assms
apply (nominal_induct \<Gamma>\<^isub>1 t T avoiding: \<Gamma>\<^isub>2 rule: typing.strong_induct)
apply auto
apply (drule_tac x="(x, T) # \<Gamma>\<^isub>2" in meta_spec)
apply (auto intro: valid_typing)
apply (drule_tac x="\<Gamma>\<^isub>2" in meta_spec)
apply (drule_tac x="\<Delta> @ \<Gamma>\<^isub>2" in meta_spec)
apply (auto intro: valid_typing)
apply (rule typing.Let)
apply assumption+
apply (drule meta_mp)
apply (rule valid_app_mono)
apply (rule valid_typing)
apply assumption
apply (auto simp add: pat_freshs)
done
inductive
match :: "pat \<Rightarrow> trm \<Rightarrow> (name \<times> trm) list \<Rightarrow> bool" ("\<turnstile> _ \<rhd> _ \<Rightarrow> _" [50, 50, 50] 50)
where
PVar: "\<turnstile> PVar x T \<rhd> t \<Rightarrow> [(x, t)]"
| PProd: "\<turnstile> p \<rhd> t \<Rightarrow> \<theta> \<Longrightarrow> \<turnstile> q \<rhd> u \<Rightarrow> \<theta>' \<Longrightarrow> \<turnstile> \<langle>\<langle>p, q\<rangle>\<rangle> \<rhd> \<langle>t, u\<rangle> \<Rightarrow> \<theta> @ \<theta>'"
fun
lookup :: "(name \<times> trm) list \<Rightarrow> name \<Rightarrow> trm"
where
"lookup [] x = Var x"
| "lookup ((y, e) # \<theta>) x = (if x = y then e else lookup \<theta> x)"
lemma lookup_eqvt[eqvt]:
fixes pi :: "name prm"
and \<theta> :: "(name \<times> trm) list"
and X :: "name"
shows "pi \<bullet> (lookup \<theta> X) = lookup (pi \<bullet> \<theta>) (pi \<bullet> X)"
by (induct \<theta>) (auto simp add: eqvts)
nominal_primrec
psubst :: "(name \<times> trm) list \<Rightarrow> trm \<Rightarrow> trm" ("_\<lparr>_\<rparr>" [95,0] 210)
and psubstb :: "(name \<times> trm) list \<Rightarrow> btrm \<Rightarrow> btrm" ("_\<lparr>_\<rparr>\<^sub>b" [95,0] 210)
where
"\<theta>\<lparr>Var x\<rparr> = (lookup \<theta> x)"
| "\<theta>\<lparr>t \<cdot> u\<rparr> = \<theta>\<lparr>t\<rparr> \<cdot> \<theta>\<lparr>u\<rparr>"
| "\<theta>\<lparr>\<langle>t, u\<rangle>\<rparr> = \<langle>\<theta>\<lparr>t\<rparr>, \<theta>\<lparr>u\<rparr>\<rangle>"
| "\<theta>\<lparr>Let T t u\<rparr> = Let T (\<theta>\<lparr>t\<rparr>) (\<theta>\<lparr>u\<rparr>\<^sub>b)"
| "x \<sharp> \<theta> \<Longrightarrow> \<theta>\<lparr>\<lambda>x:T. t\<rparr> = (\<lambda>x:T. \<theta>\<lparr>t\<rparr>)"
| "\<theta>\<lparr>Base t\<rparr>\<^sub>b = Base (\<theta>\<lparr>t\<rparr>)"
| "x \<sharp> \<theta> \<Longrightarrow> \<theta>\<lparr>Bind T x t\<rparr>\<^sub>b = Bind T x (\<theta>\<lparr>t\<rparr>\<^sub>b)"
apply finite_guess+
apply (simp add: abs_fresh | fresh_guess)+
done
lemma lookup_fresh:
"x = y \<longrightarrow> x \<in> set (map fst \<theta>) \<Longrightarrow> \<forall>(y, t)\<in>set \<theta>. x \<sharp> t \<Longrightarrow> x \<sharp> lookup \<theta> y"
apply (induct \<theta>)
apply (simp_all add: split_paired_all fresh_atm)
apply (case_tac "x = y")
apply (auto simp add: fresh_atm)
done
lemma psubst_fresh:
assumes "x \<in> set (map fst \<theta>)" and "\<forall>(y, t)\<in>set \<theta>. x \<sharp> t"
shows "x \<sharp> \<theta>\<lparr>t\<rparr>" and "x \<sharp> \<theta>\<lparr>t'\<rparr>\<^sub>b" using assms
apply (nominal_induct t and t' avoiding: \<theta> rule: trm_btrm.strong_inducts)
apply simp
apply (rule lookup_fresh)
apply (rule impI)
apply (simp_all add: abs_fresh)
done
lemma psubst_eqvt[eqvt]:
fixes pi :: "name prm"
shows "pi \<bullet> (\<theta>\<lparr>t\<rparr>) = (pi \<bullet> \<theta>)\<lparr>pi \<bullet> t\<rparr>"
and "pi \<bullet> (\<theta>\<lparr>t'\<rparr>\<^sub>b) = (pi \<bullet> \<theta>)\<lparr>pi \<bullet> t'\<rparr>\<^sub>b"
by (nominal_induct t and t' avoiding: \<theta> rule: trm_btrm.strong_inducts)
(simp_all add: eqvts fresh_bij)
abbreviation
subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_[_\<mapsto>_]" [100,0,0] 100)
where
"t[x\<mapsto>t'] \<equiv> [(x,t')]\<lparr>t\<rparr>"
abbreviation
substb :: "btrm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> btrm" ("_[_\<mapsto>_]\<^sub>b" [100,0,0] 100)
where
"t[x\<mapsto>t']\<^sub>b \<equiv> [(x,t')]\<lparr>t\<rparr>\<^sub>b"
lemma lookup_forget:
"(supp (map fst \<theta>)::name set) \<sharp>* x \<Longrightarrow> lookup \<theta> x = Var x"
by (induct \<theta>) (auto simp add: split_paired_all fresh_star_def fresh_atm supp_list_cons supp_atm)
lemma supp_fst: "(x::name) \<in> supp (map fst (\<theta>::(name \<times> trm) list)) \<Longrightarrow> x \<in> supp \<theta>"
by (induct \<theta>) (auto simp add: supp_list_nil supp_list_cons supp_prod)
lemma psubst_forget:
"(supp (map fst \<theta>)::name set) \<sharp>* t \<Longrightarrow> \<theta>\<lparr>t\<rparr> = t"
"(supp (map fst \<theta>)::name set) \<sharp>* t' \<Longrightarrow> \<theta>\<lparr>t'\<rparr>\<^sub>b = t'"
apply (nominal_induct t and t' avoiding: \<theta> rule: trm_btrm.strong_inducts)
apply (auto simp add: fresh_star_def lookup_forget abs_fresh)
apply (drule_tac x=\<theta> in meta_spec)
apply (drule meta_mp)
apply (rule ballI)
apply (drule_tac x=x in bspec)
apply assumption
apply (drule supp_fst)
apply (auto simp add: fresh_def)
apply (drule_tac x=\<theta> in meta_spec)
apply (drule meta_mp)
apply (rule ballI)
apply (drule_tac x=x in bspec)
apply assumption
apply (drule supp_fst)
apply (auto simp add: fresh_def)
done
lemma psubst_nil: "[]\<lparr>t\<rparr> = t" "[]\<lparr>t'\<rparr>\<^sub>b = t'"
by (induct t and t' rule: trm_btrm.inducts) (simp_all add: fresh_list_nil)
lemma psubst_cons:
assumes "(supp (map fst \<theta>)::name set) \<sharp>* u"
shows "((x, u) # \<theta>)\<lparr>t\<rparr> = \<theta>\<lparr>t[x\<mapsto>u]\<rparr>" and "((x, u) # \<theta>)\<lparr>t'\<rparr>\<^sub>b = \<theta>\<lparr>t'[x\<mapsto>u]\<^sub>b\<rparr>\<^sub>b"
using assms
by (nominal_induct t and t' avoiding: x u \<theta> rule: trm_btrm.strong_inducts)
(simp_all add: fresh_list_nil fresh_list_cons psubst_forget)
lemma psubst_append:
"(supp (map fst (\<theta>\<^isub>1 @ \<theta>\<^isub>2))::name set) \<sharp>* map snd (\<theta>\<^isub>1 @ \<theta>\<^isub>2) \<Longrightarrow> (\<theta>\<^isub>1 @ \<theta>\<^isub>2)\<lparr>t\<rparr> = \<theta>\<^isub>2\<lparr>\<theta>\<^isub>1\<lparr>t\<rparr>\<rparr>"
by (induct \<theta>\<^isub>1 arbitrary: t)
(simp_all add: psubst_nil split_paired_all supp_list_cons psubst_cons fresh_star_def
fresh_list_cons fresh_list_append supp_list_append)
lemma abs_pat_psubst [simp]:
"(supp p::name set) \<sharp>* \<theta> \<Longrightarrow> \<theta>\<lparr>\<lambda>[p]. t\<rparr>\<^sub>b = (\<lambda>[p]. \<theta>\<lparr>t\<rparr>\<^sub>b)"
by (induct p arbitrary: t) (auto simp add: fresh_star_def supp_atm)
lemma valid_insert:
assumes "valid (\<Delta> @ [(x, T)] @ \<Gamma>)"
shows "valid (\<Delta> @ \<Gamma>)" using assms
by (induct \<Delta>)
(auto simp add: fresh_list_append fresh_list_cons)
lemma fresh_set:
shows "y \<sharp> xs = (\<forall>x\<in>set xs. y \<sharp> x)"
by (induct xs) (simp_all add: fresh_list_nil fresh_list_cons)
lemma context_unique:
assumes "valid \<Gamma>"
and "(x, T) \<in> set \<Gamma>"
and "(x, U) \<in> set \<Gamma>"
shows "T = U" using assms
by induct (auto simp add: fresh_set fresh_prod fresh_atm)
lemma subst_type_aux:
assumes a: "\<Delta> @ [(x, U)] @ \<Gamma> \<turnstile> t : T"
and b: "\<Gamma> \<turnstile> u : U"
shows "\<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : T" using a b
proof (nominal_induct \<Gamma>'\<equiv>"\<Delta> @ [(x, U)] @ \<Gamma>" t T avoiding: x u \<Delta> rule: typing.strong_induct)
case (Var \<Gamma>' y T x u \<Delta>)
then have a1: "valid (\<Delta> @ [(x, U)] @ \<Gamma>)"
and a2: "(y, T) \<in> set (\<Delta> @ [(x, U)] @ \<Gamma>)"
and a3: "\<Gamma> \<turnstile> u : U" by simp_all
from a1 have a4: "valid (\<Delta> @ \<Gamma>)" by (rule valid_insert)
show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T"
proof cases
assume eq: "x = y"
from a1 a2 have "T = U" using eq by (auto intro: context_unique)
with a3 show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" using eq a4 by (auto intro: weakening)
next
assume ineq: "x \<noteq> y"
from a2 have "(y, T) \<in> set (\<Delta> @ \<Gamma>)" using ineq by simp
then show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" using ineq a4 by auto
qed
next
case (Tuple \<Gamma>' t\<^isub>1 T\<^isub>1 t\<^isub>2 T\<^isub>2)
from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>1[x\<mapsto>u] : T\<^isub>1" by (rule Tuple)
moreover from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>2[x\<mapsto>u] : T\<^isub>2" by (rule Tuple)
ultimately have "\<Delta> @ \<Gamma> \<turnstile> \<langle>t\<^isub>1[x\<mapsto>u], t\<^isub>2[x\<mapsto>u]\<rangle> : T\<^isub>1 \<otimes> T\<^isub>2" ..
then show ?case by simp
next
case (Let p t \<Gamma>' T \<Delta>' s S)
from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : T" by (rule Let)
moreover note `\<turnstile> p : T \<Rightarrow> \<Delta>'`
moreover from `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta>' @ \<Gamma>' = (\<Delta>' @ \<Delta>) @ [(x, U)] @ \<Gamma>" by simp
with `\<Gamma> \<turnstile> u : U` have "(\<Delta>' @ \<Delta>) @ \<Gamma> \<turnstile> s[x\<mapsto>u] : S" by (rule Let)
then have "\<Delta>' @ \<Delta> @ \<Gamma> \<turnstile> s[x\<mapsto>u] : S" by simp
ultimately have "\<Delta> @ \<Gamma> \<turnstile> (LET p = t[x\<mapsto>u] IN s[x\<mapsto>u]) : S"
by (rule better_T_Let)
moreover from Let have "(supp p::name set) \<sharp>* [(x, u)]"
by (simp add: fresh_star_def fresh_list_nil fresh_list_cons)
ultimately show ?case by simp
next
case (Abs y T \<Gamma>' t S)
from `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>` have "(y, T) # \<Gamma>' = ((y, T) # \<Delta>) @ [(x, U)] @ \<Gamma>"
by simp
with `\<Gamma> \<turnstile> u : U` have "((y, T) # \<Delta>) @ \<Gamma> \<turnstile> t[x\<mapsto>u] : S" by (rule Abs)
then have "(y, T) # \<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : S" by simp
then have "\<Delta> @ \<Gamma> \<turnstile> (\<lambda>y:T. t[x\<mapsto>u]) : T \<rightarrow> S"
by (rule typing.Abs)
moreover from Abs have "y \<sharp> [(x, u)]"
by (simp add: fresh_list_nil fresh_list_cons)
ultimately show ?case by simp
next
case (App \<Gamma>' t\<^isub>1 T S t\<^isub>2)
from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>1[x\<mapsto>u] : T \<rightarrow> S" by (rule App)
moreover from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>2[x\<mapsto>u] : T" by (rule App)
ultimately have "\<Delta> @ \<Gamma> \<turnstile> (t\<^isub>1[x\<mapsto>u]) \<cdot> (t\<^isub>2[x\<mapsto>u]) : S"
by (rule typing.App)
then show ?case by simp
qed
lemmas subst_type = subst_type_aux [of "[]", simplified]
lemma match_supp_fst:
assumes "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>" shows "(supp (map fst \<theta>)::name set) = supp p" using assms
by induct (simp_all add: supp_list_nil supp_list_cons supp_list_append)
lemma match_supp_snd:
assumes "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>" shows "(supp (map snd \<theta>)::name set) = supp u" using assms
by induct (simp_all add: supp_list_nil supp_list_cons supp_list_append trm.supp)
lemma match_fresh: "\<turnstile> p \<rhd> u \<Rightarrow> \<theta> \<Longrightarrow> (supp p::name set) \<sharp>* u \<Longrightarrow>
(supp (map fst \<theta>)::name set) \<sharp>* map snd \<theta>"
by (simp add: fresh_star_def fresh_def match_supp_fst match_supp_snd)
lemma match_type_aux:
assumes "\<turnstile> p : U \<Rightarrow> \<Delta>"
and "\<Gamma>\<^isub>2 \<turnstile> u : U"
and "\<Gamma>\<^isub>1 @ \<Delta> @ \<Gamma>\<^isub>2 \<turnstile> t : T"
and "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>"
and "(supp p::name set) \<sharp>* u"
shows "\<Gamma>\<^isub>1 @ \<Gamma>\<^isub>2 \<turnstile> \<theta>\<lparr>t\<rparr> : T" using assms
proof (induct arbitrary: \<Gamma>\<^isub>1 \<Gamma>\<^isub>2 t u T \<theta>)
case (PVar x U)
from `\<Gamma>\<^isub>1 @ [(x, U)] @ \<Gamma>\<^isub>2 \<turnstile> t : T` `\<Gamma>\<^isub>2 \<turnstile> u : U`
have "\<Gamma>\<^isub>1 @ \<Gamma>\<^isub>2 \<turnstile> t[x\<mapsto>u] : T" by (rule subst_type_aux)
moreover from `\<turnstile> PVar x U \<rhd> u \<Rightarrow> \<theta>` have "\<theta> = [(x, u)]"
by cases simp_all
ultimately show ?case by simp
next
case (PTuple p S \<Delta>\<^isub>1 q U \<Delta>\<^isub>2)
from `\<turnstile> \<langle>\<langle>p, q\<rangle>\<rangle> \<rhd> u \<Rightarrow> \<theta>` obtain u\<^isub>1 u\<^isub>2 \<theta>\<^isub>1 \<theta>\<^isub>2
where u: "u = \<langle>u\<^isub>1, u\<^isub>2\<rangle>" and \<theta>: "\<theta> = \<theta>\<^isub>1 @ \<theta>\<^isub>2"
and p: "\<turnstile> p \<rhd> u\<^isub>1 \<Rightarrow> \<theta>\<^isub>1" and q: "\<turnstile> q \<rhd> u\<^isub>2 \<Rightarrow> \<theta>\<^isub>2"
by cases simp_all
with PTuple have "\<Gamma>\<^isub>2 \<turnstile> \<langle>u\<^isub>1, u\<^isub>2\<rangle> : S \<otimes> U" by simp
then obtain u\<^isub>1: "\<Gamma>\<^isub>2 \<turnstile> u\<^isub>1 : S" and u\<^isub>2: "\<Gamma>\<^isub>2 \<turnstile> u\<^isub>2 : U"
by cases (simp_all add: ty.inject trm.inject)
note u\<^isub>1
moreover from `\<Gamma>\<^isub>1 @ (\<Delta>\<^isub>2 @ \<Delta>\<^isub>1) @ \<Gamma>\<^isub>2 \<turnstile> t : T`
have "(\<Gamma>\<^isub>1 @ \<Delta>\<^isub>2) @ \<Delta>\<^isub>1 @ \<Gamma>\<^isub>2 \<turnstile> t : T" by simp
moreover note p
moreover from `supp \<langle>\<langle>p, q\<rangle>\<rangle> \<sharp>* u` and u
have "(supp p::name set) \<sharp>* u\<^isub>1" by (simp add: fresh_star_def)
ultimately have \<theta>\<^isub>1: "(\<Gamma>\<^isub>1 @ \<Delta>\<^isub>2) @ \<Gamma>\<^isub>2 \<turnstile> \<theta>\<^isub>1\<lparr>t\<rparr> : T"
by (rule PTuple)
note u\<^isub>2
moreover from \<theta>\<^isub>1
have "\<Gamma>\<^isub>1 @ \<Delta>\<^isub>2 @ \<Gamma>\<^isub>2 \<turnstile> \<theta>\<^isub>1\<lparr>t\<rparr> : T" by simp
moreover note q
moreover from `supp \<langle>\<langle>p, q\<rangle>\<rangle> \<sharp>* u` and u
have "(supp q::name set) \<sharp>* u\<^isub>2" by (simp add: fresh_star_def)
ultimately have "\<Gamma>\<^isub>1 @ \<Gamma>\<^isub>2 \<turnstile> \<theta>\<^isub>2\<lparr>\<theta>\<^isub>1\<lparr>t\<rparr>\<rparr> : T"
by (rule PTuple)
moreover from `\<turnstile> \<langle>\<langle>p, q\<rangle>\<rangle> \<rhd> u \<Rightarrow> \<theta>` `supp \<langle>\<langle>p, q\<rangle>\<rangle> \<sharp>* u`
have "(supp (map fst \<theta>)::name set) \<sharp>* map snd \<theta>"
by (rule match_fresh)
ultimately show ?case using \<theta> by (simp add: psubst_append)
qed
lemmas match_type = match_type_aux [where \<Gamma>\<^isub>1="[]", simplified]
inductive eval :: "trm \<Rightarrow> trm \<Rightarrow> bool" ("_ \<longmapsto> _" [60,60] 60)
where
TupleL: "t \<longmapsto> t' \<Longrightarrow> \<langle>t, u\<rangle> \<longmapsto> \<langle>t', u\<rangle>"
| TupleR: "u \<longmapsto> u' \<Longrightarrow> \<langle>t, u\<rangle> \<longmapsto> \<langle>t, u'\<rangle>"
| Abs: "t \<longmapsto> t' \<Longrightarrow> (\<lambda>x:T. t) \<longmapsto> (\<lambda>x:T. t')"
| AppL: "t \<longmapsto> t' \<Longrightarrow> t \<cdot> u \<longmapsto> t' \<cdot> u"
| AppR: "u \<longmapsto> u' \<Longrightarrow> t \<cdot> u \<longmapsto> t \<cdot> u'"
| Beta: "x \<sharp> u \<Longrightarrow> (\<lambda>x:T. t) \<cdot> u \<longmapsto> t[x\<mapsto>u]"
| Let: "((supp p)::name set) \<sharp>* t \<Longrightarrow> distinct (pat_vars p) \<Longrightarrow>
\<turnstile> p \<rhd> t \<Rightarrow> \<theta> \<Longrightarrow> (LET p = t IN u) \<longmapsto> \<theta>\<lparr>u\<rparr>"
equivariance match
equivariance eval
lemma match_vars:
assumes "\<turnstile> p \<rhd> t \<Rightarrow> \<theta>" and "x \<in> supp p"
shows "x \<in> set (map fst \<theta>)" using assms
by induct (auto simp add: supp_atm)
lemma match_fresh_mono:
assumes "\<turnstile> p \<rhd> t \<Rightarrow> \<theta>" and "(x::name) \<sharp> t"
shows "\<forall>(y, t)\<in>set \<theta>. x \<sharp> t" using assms
by induct auto
nominal_inductive2 eval
avoids
Abs: "{x}"
| Beta: "{x}"
| Let: "(supp p)::name set"
apply (simp_all add: fresh_star_def abs_fresh fin_supp)
apply (rule psubst_fresh)
apply simp
apply simp
apply (rule ballI)
apply (rule psubst_fresh)
apply (rule match_vars)
apply assumption+
apply (rule match_fresh_mono)
apply auto
done
lemma typing_case_Abs:
assumes ty: "\<Gamma> \<turnstile> (\<lambda>x:T. t) : S"
and fresh: "x \<sharp> \<Gamma>"
and R: "\<And>U. S = T \<rightarrow> U \<Longrightarrow> (x, T) # \<Gamma> \<turnstile> t : U \<Longrightarrow> P"
shows P using ty
proof cases
case (Abs x' T' \<Gamma>' t' U)
obtain y::name where y: "y \<sharp> (x, \<Gamma>, \<lambda>x':T'. t')"
by (rule exists_fresh) (auto intro: fin_supp)
from `(\<lambda>x:T. t) = (\<lambda>x':T'. t')` [symmetric]
have x: "x \<sharp> (\<lambda>x':T'. t')" by (simp add: abs_fresh)
have x': "x' \<sharp> (\<lambda>x':T'. t')" by (simp add: abs_fresh)
from `(x', T') # \<Gamma>' \<turnstile> t' : U` have x'': "x' \<sharp> \<Gamma>'"
by (auto dest: valid_typing)
have "(\<lambda>x:T. t) = (\<lambda>x':T'. t')" by fact
also from x x' y have "\<dots> = [(x, y)] \<bullet> [(x', y)] \<bullet> (\<lambda>x':T'. t')"
by (simp only: perm_fresh_fresh fresh_prod)
also have "\<dots> = (\<lambda>x:T'. [(x, y)] \<bullet> [(x', y)] \<bullet> t')"
by (simp add: swap_simps perm_fresh_fresh)
finally have "(\<lambda>x:T. t) = (\<lambda>x:T'. [(x, y)] \<bullet> [(x', y)] \<bullet> t')" .
then have T: "T = T'" and t: "[(x, y)] \<bullet> [(x', y)] \<bullet> t' = t"
by (simp_all add: trm.inject alpha)
from Abs T have "S = T \<rightarrow> U" by simp
moreover from `(x', T') # \<Gamma>' \<turnstile> t' : U`
have "[(x, y)] \<bullet> [(x', y)] \<bullet> ((x', T') # \<Gamma>' \<turnstile> t' : U)"
by (simp add: perm_bool)
with T t y `\<Gamma> = \<Gamma>'` x'' fresh have "(x, T) # \<Gamma> \<turnstile> t : U"
by (simp add: eqvts swap_simps perm_fresh_fresh fresh_prod)
ultimately show ?thesis by (rule R)
qed simp_all
nominal_primrec ty_size :: "ty \<Rightarrow> nat"
where
"ty_size (Atom n) = 0"
| "ty_size (T \<rightarrow> U) = ty_size T + ty_size U + 1"
| "ty_size (T \<otimes> U) = ty_size T + ty_size U + 1"
by (rule TrueI)+
lemma bind_tuple_ineq:
"ty_size (pat_type p) < ty_size U \<Longrightarrow> Bind U x t \<noteq> (\<lambda>[p]. u)"
by (induct p arbitrary: U x t u) (auto simp add: btrm.inject)
lemma valid_appD: assumes "valid (\<Gamma> @ \<Delta>)"
shows "valid \<Gamma>" "valid \<Delta>" using assms
by (induct \<Gamma>'\<equiv>"\<Gamma> @ \<Delta>" arbitrary: \<Gamma> \<Delta>)
(auto simp add: Cons_eq_append_conv fresh_list_append)
lemma valid_app_freshs: assumes "valid (\<Gamma> @ \<Delta>)"
shows "(supp \<Gamma>::name set) \<sharp>* \<Delta>" "(supp \<Delta>::name set) \<sharp>* \<Gamma>" using assms
by (induct \<Gamma>'\<equiv>"\<Gamma> @ \<Delta>" arbitrary: \<Gamma> \<Delta>)
(auto simp add: Cons_eq_append_conv fresh_star_def
fresh_list_nil fresh_list_cons supp_list_nil supp_list_cons fresh_list_append
supp_prod fresh_prod supp_atm fresh_atm
dest: notE [OF iffD1 [OF fresh_def [THEN meta_eq_to_obj_eq]]])
lemma perm_mem_left: "(x::name) \<in> ((pi::name prm) \<bullet> A) \<Longrightarrow> (rev pi \<bullet> x) \<in> A"
by (drule perm_boolI [of _ "rev pi"]) (simp add: eqvts perm_pi_simp)
lemma perm_mem_right: "(rev (pi::name prm) \<bullet> (x::name)) \<in> A \<Longrightarrow> x \<in> (pi \<bullet> A)"
by (drule perm_boolI [of _ pi]) (simp add: eqvts perm_pi_simp)
lemma perm_cases:
assumes pi: "set pi \<subseteq> A \<times> A"
shows "((pi::name prm) \<bullet> B) \<subseteq> A \<union> B"
proof
fix x assume "x \<in> pi \<bullet> B"
then show "x \<in> A \<union> B" using pi
apply (induct pi arbitrary: x B rule: rev_induct)
apply simp
apply (simp add: split_paired_all supp_eqvt)
apply (drule perm_mem_left)
apply (simp add: calc_atm split: split_if_asm)
apply (auto dest: perm_mem_right)
done
qed
lemma abs_pat_alpha':
assumes eq: "(\<lambda>[p]. t) = (\<lambda>[q]. u)"
and ty: "pat_type p = pat_type q"
and pv: "distinct (pat_vars p)"
and qv: "distinct (pat_vars q)"
shows "\<exists>pi::name prm. p = pi \<bullet> q \<and> t = pi \<bullet> u \<and>
set pi \<subseteq> (supp p \<union> supp q) \<times> (supp p \<union> supp q)"
using assms
proof (induct p arbitrary: q t u \<Delta>)
case (PVar x T)
note PVar' = this
show ?case
proof (cases q)
case (PVar x' T')
with `(\<lambda>[PVar x T]. t) = (\<lambda>[q]. u)`
have "x = x' \<and> t = u \<or> x \<noteq> x' \<and> t = [(x, x')] \<bullet> u \<and> x \<sharp> u"
by (simp add: btrm.inject alpha)
then show ?thesis
proof
assume "x = x' \<and> t = u"
with PVar PVar' have "PVar x T = ([]::name prm) \<bullet> q \<and>
t = ([]::name prm) \<bullet> u \<and>
set ([]::name prm) \<subseteq> (supp (PVar x T) \<union> supp q) \<times>
(supp (PVar x T) \<union> supp q)" by simp
then show ?thesis ..
next
assume "x \<noteq> x' \<and> t = [(x, x')] \<bullet> u \<and> x \<sharp> u"
with PVar PVar' have "PVar x T = [(x, x')] \<bullet> q \<and>
t = [(x, x')] \<bullet> u \<and>
set [(x, x')] \<subseteq> (supp (PVar x T) \<union> supp q) \<times>
(supp (PVar x T) \<union> supp q)"
by (simp add: perm_swap swap_simps supp_atm perm_type)
then show ?thesis ..
qed
next
case (PTuple p\<^isub>1 p\<^isub>2)
with PVar have "ty_size (pat_type p\<^isub>1) < ty_size T" by simp
then have "Bind T x t \<noteq> (\<lambda>[p\<^isub>1]. \<lambda>[p\<^isub>2]. u)"
by (rule bind_tuple_ineq)
moreover from PTuple PVar
have "Bind T x t = (\<lambda>[p\<^isub>1]. \<lambda>[p\<^isub>2]. u)" by simp
ultimately show ?thesis ..
qed
next
case (PTuple p\<^isub>1 p\<^isub>2)
note PTuple' = this
show ?case
proof (cases q)
case (PVar x T)
with PTuple have "ty_size (pat_type p\<^isub>1) < ty_size T" by auto
then have "Bind T x u \<noteq> (\<lambda>[p\<^isub>1]. \<lambda>[p\<^isub>2]. t)"
by (rule bind_tuple_ineq)
moreover from PTuple PVar
have "Bind T x u = (\<lambda>[p\<^isub>1]. \<lambda>[p\<^isub>2]. t)" by simp
ultimately show ?thesis ..
next
case (PTuple p\<^isub>1' p\<^isub>2')
with PTuple' have "(\<lambda>[p\<^isub>1]. \<lambda>[p\<^isub>2]. t) = (\<lambda>[p\<^isub>1']. \<lambda>[p\<^isub>2']. u)" by simp
moreover from PTuple PTuple' have "pat_type p\<^isub>1 = pat_type p\<^isub>1'"
by (simp add: ty.inject)
moreover from PTuple' have "distinct (pat_vars p\<^isub>1)" by simp
moreover from PTuple PTuple' have "distinct (pat_vars p\<^isub>1')" by simp
ultimately have "\<exists>pi::name prm. p\<^isub>1 = pi \<bullet> p\<^isub>1' \<and>
(\<lambda>[p\<^isub>2]. t) = pi \<bullet> (\<lambda>[p\<^isub>2']. u) \<and>
set pi \<subseteq> (supp p\<^isub>1 \<union> supp p\<^isub>1') \<times> (supp p\<^isub>1 \<union> supp p\<^isub>1')"
by (rule PTuple')
then obtain pi::"name prm" where
"p\<^isub>1 = pi \<bullet> p\<^isub>1'" "(\<lambda>[p\<^isub>2]. t) = pi \<bullet> (\<lambda>[p\<^isub>2']. u)" and
pi: "set pi \<subseteq> (supp p\<^isub>1 \<union> supp p\<^isub>1') \<times> (supp p\<^isub>1 \<union> supp p\<^isub>1')" by auto
from `(\<lambda>[p\<^isub>2]. t) = pi \<bullet> (\<lambda>[p\<^isub>2']. u)`
have "(\<lambda>[p\<^isub>2]. t) = (\<lambda>[pi \<bullet> p\<^isub>2']. pi \<bullet> u)"
by (simp add: eqvts)
moreover from PTuple PTuple' have "pat_type p\<^isub>2 = pat_type (pi \<bullet> p\<^isub>2')"
by (simp add: ty.inject pat_type_perm_eq)
moreover from PTuple' have "distinct (pat_vars p\<^isub>2)" by simp
moreover from PTuple PTuple' have "distinct (pat_vars (pi \<bullet> p\<^isub>2'))"
by (simp add: pat_vars_eqvt [symmetric] distinct_eqvt [symmetric])
ultimately have "\<exists>pi'::name prm. p\<^isub>2 = pi' \<bullet> pi \<bullet> p\<^isub>2' \<and>
t = pi' \<bullet> pi \<bullet> u \<and>
set pi' \<subseteq> (supp p\<^isub>2 \<union> supp (pi \<bullet> p\<^isub>2')) \<times> (supp p\<^isub>2 \<union> supp (pi \<bullet> p\<^isub>2'))"
by (rule PTuple')
then obtain pi'::"name prm" where
"p\<^isub>2 = pi' \<bullet> pi \<bullet> p\<^isub>2'" "t = pi' \<bullet> pi \<bullet> u" and
pi': "set pi' \<subseteq> (supp p\<^isub>2 \<union> supp (pi \<bullet> p\<^isub>2')) \<times>
(supp p\<^isub>2 \<union> supp (pi \<bullet> p\<^isub>2'))" by auto
from PTuple PTuple' have "pi \<bullet> distinct (pat_vars \<langle>\<langle>p\<^isub>1', p\<^isub>2'\<rangle>\<rangle>)" by simp
then have "distinct (pat_vars \<langle>\<langle>pi \<bullet> p\<^isub>1', pi \<bullet> p\<^isub>2'\<rangle>\<rangle>)" by (simp only: eqvts)
with `p\<^isub>1 = pi \<bullet> p\<^isub>1'` PTuple'
have fresh: "(supp p\<^isub>2 \<union> supp (pi \<bullet> p\<^isub>2') :: name set) \<sharp>* p\<^isub>1"
by (auto simp add: set_pat_vars_supp fresh_star_def fresh_def eqvts)
from `p\<^isub>1 = pi \<bullet> p\<^isub>1'` have "pi' \<bullet> (p\<^isub>1 = pi \<bullet> p\<^isub>1')" by (rule perm_boolI)
with pt_freshs_freshs [OF pt_name_inst at_name_inst pi' fresh fresh]
have "p\<^isub>1 = pi' \<bullet> pi \<bullet> p\<^isub>1'" by (simp add: eqvts)
with `p\<^isub>2 = pi' \<bullet> pi \<bullet> p\<^isub>2'` have "\<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> = (pi' @ pi) \<bullet> \<langle>\<langle>p\<^isub>1', p\<^isub>2'\<rangle>\<rangle>"
by (simp add: pt_name2)
moreover
have "((supp p\<^isub>2 \<union> (pi \<bullet> supp p\<^isub>2')) \<times> (supp p\<^isub>2 \<union> (pi \<bullet> supp p\<^isub>2'))::(name \<times> name) set) \<subseteq>
(supp p\<^isub>2 \<union> (supp p\<^isub>1 \<union> supp p\<^isub>1' \<union> supp p\<^isub>2')) \<times> (supp p\<^isub>2 \<union> (supp p\<^isub>1 \<union> supp p\<^isub>1' \<union> supp p\<^isub>2'))"
by (rule subset_refl Sigma_mono Un_mono perm_cases [OF pi])+
with pi' have "set pi' \<subseteq> \<dots>" by (simp add: supp_eqvt [symmetric])
with pi have "set (pi' @ pi) \<subseteq> (supp \<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> \<union> supp \<langle>\<langle>p\<^isub>1', p\<^isub>2'\<rangle>\<rangle>) \<times>
(supp \<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> \<union> supp \<langle>\<langle>p\<^isub>1', p\<^isub>2'\<rangle>\<rangle>)"
by (simp add: Sigma_Un_distrib1 Sigma_Un_distrib2 Un_ac) blast
moreover note `t = pi' \<bullet> pi \<bullet> u`
ultimately have "\<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> = (pi' @ pi) \<bullet> q \<and> t = (pi' @ pi) \<bullet> u \<and>
set (pi' @ pi) \<subseteq> (supp \<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> \<union> supp q) \<times>
(supp \<langle>\<langle>p\<^isub>1, p\<^isub>2\<rangle>\<rangle> \<union> supp q)" using PTuple
by (simp add: pt_name2)
then show ?thesis ..
qed
qed
lemma typing_case_Let:
assumes ty: "\<Gamma> \<turnstile> (LET p = t IN u) : U"
and fresh: "(supp p::name set) \<sharp>* \<Gamma>"
and distinct: "distinct (pat_vars p)"
and R: "\<And>T \<Delta>. \<Gamma> \<turnstile> t : T \<Longrightarrow> \<turnstile> p : T \<Rightarrow> \<Delta> \<Longrightarrow> \<Delta> @ \<Gamma> \<turnstile> u : U \<Longrightarrow> P"
shows P using ty
proof cases
case (Let p' t' \<Gamma>' T \<Delta> u' U')
then have "(supp \<Delta>::name set) \<sharp>* \<Gamma>"
by (auto intro: valid_typing valid_app_freshs)
with Let have "(supp p'::name set) \<sharp>* \<Gamma>"
by (simp add: pat_var)
with fresh have fresh': "(supp p \<union> supp p' :: name set) \<sharp>* \<Gamma>"
by (auto simp add: fresh_star_def)
from Let have "(\<lambda>[p]. Base u) = (\<lambda>[p']. Base u')"
by (simp add: trm.inject)
moreover from Let have "pat_type p = pat_type p'"
by (simp add: trm.inject)
moreover note distinct
moreover from `\<Delta> @ \<Gamma>' \<turnstile> u' : U'` have "valid (\<Delta> @ \<Gamma>')"
by (rule valid_typing)
then have "valid \<Delta>" by (rule valid_appD)
with `\<turnstile> p' : T \<Rightarrow> \<Delta>` have "distinct (pat_vars p')"
by (simp add: valid_distinct pat_vars_ptyping)
ultimately have "\<exists>pi::name prm. p = pi \<bullet> p' \<and> Base u = pi \<bullet> Base u' \<and>
set pi \<subseteq> (supp p \<union> supp p') \<times> (supp p \<union> supp p')"
by (rule abs_pat_alpha')
then obtain pi::"name prm" where pi: "p = pi \<bullet> p'" "u = pi \<bullet> u'"
and pi': "set pi \<subseteq> (supp p \<union> supp p') \<times> (supp p \<union> supp p')"
by (auto simp add: btrm.inject)
from Let have "\<Gamma> \<turnstile> t : T" by (simp add: trm.inject)
moreover from `\<turnstile> p' : T \<Rightarrow> \<Delta>` have "\<turnstile> (pi \<bullet> p') : (pi \<bullet> T) \<Rightarrow> (pi \<bullet> \<Delta>)"
by (simp add: ptyping.eqvt)
with pi have "\<turnstile> p : T \<Rightarrow> (pi \<bullet> \<Delta>)" by (simp add: perm_type)
moreover from Let
have "(pi \<bullet> \<Delta>) @ (pi \<bullet> \<Gamma>) \<turnstile> (pi \<bullet> u') : (pi \<bullet> U)"
by (simp add: append_eqvt [symmetric] typing.eqvt)
with pi have "(pi \<bullet> \<Delta>) @ \<Gamma> \<turnstile> u : U"
by (simp add: perm_type pt_freshs_freshs
[OF pt_name_inst at_name_inst pi' fresh' fresh'])
ultimately show ?thesis by (rule R)
qed simp_all
lemma preservation:
assumes "t \<longmapsto> t'" and "\<Gamma> \<turnstile> t : T"
shows "\<Gamma> \<turnstile> t' : T" using assms
proof (nominal_induct avoiding: \<Gamma> T rule: eval.strong_induct)
case (TupleL t t' u)
from `\<Gamma> \<turnstile> \<langle>t, u\<rangle> : T` obtain T\<^isub>1 T\<^isub>2
where "T = T\<^isub>1 \<otimes> T\<^isub>2" "\<Gamma> \<turnstile> t : T\<^isub>1" "\<Gamma> \<turnstile> u : T\<^isub>2"
by cases (simp_all add: trm.inject)
from `\<Gamma> \<turnstile> t : T\<^isub>1` have "\<Gamma> \<turnstile> t' : T\<^isub>1" by (rule TupleL)
then have "\<Gamma> \<turnstile> \<langle>t', u\<rangle> : T\<^isub>1 \<otimes> T\<^isub>2" using `\<Gamma> \<turnstile> u : T\<^isub>2`
by (rule Tuple)
with `T = T\<^isub>1 \<otimes> T\<^isub>2` show ?case by simp
next
case (TupleR u u' t)
from `\<Gamma> \<turnstile> \<langle>t, u\<rangle> : T` obtain T\<^isub>1 T\<^isub>2
where "T = T\<^isub>1 \<otimes> T\<^isub>2" "\<Gamma> \<turnstile> t : T\<^isub>1" "\<Gamma> \<turnstile> u : T\<^isub>2"
by cases (simp_all add: trm.inject)
from `\<Gamma> \<turnstile> u : T\<^isub>2` have "\<Gamma> \<turnstile> u' : T\<^isub>2" by (rule TupleR)
with `\<Gamma> \<turnstile> t : T\<^isub>1` have "\<Gamma> \<turnstile> \<langle>t, u'\<rangle> : T\<^isub>1 \<otimes> T\<^isub>2"
by (rule Tuple)
with `T = T\<^isub>1 \<otimes> T\<^isub>2` show ?case by simp
next
case (Abs t t' x S)
from `\<Gamma> \<turnstile> (\<lambda>x:S. t) : T` `x \<sharp> \<Gamma>` obtain U where
T: "T = S \<rightarrow> U" and U: "(x, S) # \<Gamma> \<turnstile> t : U"
by (rule typing_case_Abs)
from U have "(x, S) # \<Gamma> \<turnstile> t' : U" by (rule Abs)
then have "\<Gamma> \<turnstile> (\<lambda>x:S. t') : S \<rightarrow> U"
by (rule typing.Abs)
with T show ?case by simp
next
case (Beta x u S t)
from `\<Gamma> \<turnstile> (\<lambda>x:S. t) \<cdot> u : T` `x \<sharp> \<Gamma>`
obtain "(x, S) # \<Gamma> \<turnstile> t : T" and "\<Gamma> \<turnstile> u : S"
by cases (auto simp add: trm.inject ty.inject elim: typing_case_Abs)
then show ?case by (rule subst_type)
next
case (Let p t \<theta> u)
from `\<Gamma> \<turnstile> (LET p = t IN u) : T` `supp p \<sharp>* \<Gamma>` `distinct (pat_vars p)`
obtain U \<Delta> where "\<turnstile> p : U \<Rightarrow> \<Delta>" "\<Gamma> \<turnstile> t : U" "\<Delta> @ \<Gamma> \<turnstile> u : T"
by (rule typing_case_Let)
then show ?case using `\<turnstile> p \<rhd> t \<Rightarrow> \<theta>` `supp p \<sharp>* t`
by (rule match_type)
next
case (AppL t t' u)
from `\<Gamma> \<turnstile> t \<cdot> u : T` obtain U where
t: "\<Gamma> \<turnstile> t : U \<rightarrow> T" and u: "\<Gamma> \<turnstile> u : U"
by cases (auto simp add: trm.inject)
from t have "\<Gamma> \<turnstile> t' : U \<rightarrow> T" by (rule AppL)
then show ?case using u by (rule typing.App)
next
case (AppR u u' t)
from `\<Gamma> \<turnstile> t \<cdot> u : T` obtain U where
t: "\<Gamma> \<turnstile> t : U \<rightarrow> T" and u: "\<Gamma> \<turnstile> u : U"
by cases (auto simp add: trm.inject)
from u have "\<Gamma> \<turnstile> u' : U" by (rule AppR)
with t show ?case by (rule typing.App)
qed
end