src/HOL/Nominal/Examples/Pattern.thy

+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