(* "$Id$" *)
(* *)
(* Formalisation of some typical SOS-proofs *)
(* *)
(* This work arose from challenge suggested by Adam *)
(* Chlipala suggested on the POPLmark mailing list. *)
(* *)
(* We thank Nick Benton for helping us with the *)
(* termination-proof for evaluation. *)
(* *)
(* The formalisation was done by Julien Narboux and *)
(* Christian Urban. *)
theory SOS
imports "../Nominal"
begin
atom_decl name
nominal_datatype data =
DNat
| DProd "data" "data"
| DSum "data" "data"
nominal_datatype ty =
Data "data"
| Arrow "ty" "ty" ("_\<rightarrow>_" [100,100] 100)
nominal_datatype trm =
Var "name"
| Lam "\<guillemotleft>name\<guillemotright>trm" ("Lam [_]._" [100,100] 100)
| App "trm" "trm"
| Const "nat"
| Pr "trm" "trm"
| Fst "trm"
| Snd "trm"
| InL "trm"
| InR "trm"
| Case "trm" "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Case _ of inl _ \<rightarrow> _ | inr _ \<rightarrow> _" [100,100,100,100,100] 100)
lemma in_eqvt[eqvt]:
fixes pi::"name prm"
and x::"'a::pt_name"
assumes "x\<in>X"
shows "pi\<bullet>x \<in> pi\<bullet>X"
using assms by (perm_simp add: pt_set_bij1a[OF pt_name_inst, OF at_name_inst])
lemma perm_data[simp]:
fixes D::"data"
and pi::"name prm"
shows "pi\<bullet>D = D"
by (induct D rule: data.weak_induct) (simp_all)
lemma perm_ty[simp]:
fixes T::"ty"
and pi::"name prm"
shows "pi\<bullet>T = T"
by (induct T rule: ty.weak_induct) (simp_all)
lemma fresh_ty[simp]:
fixes x::"name"
and T::"ty"
shows "x\<sharp>T"
by (simp add: fresh_def supp_def)
text {* substitution *}
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:
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: perm_bij)
lemma lookup_fresh:
fixes z::"name"
assumes "z\<sharp>\<theta>" and "z\<sharp>x"
shows "z \<sharp>lookup \<theta> x"
using assms
by (induct rule: lookup.induct) (auto simp add: fresh_list_cons)
lemma lookup_fresh':
assumes "z\<sharp>\<theta>"
shows "lookup \<theta> z = Var z"
using assms
by (induct rule: lookup.induct)
(auto simp add: fresh_list_cons fresh_prod fresh_atm)
text {* Parallel Substitution *}
consts
psubst :: "(name\<times>trm) list \<Rightarrow> trm \<Rightarrow> trm" ("_<_>" [95,95] 105)
nominal_primrec
"\<theta><(Var x)> = (lookup \<theta> x)"
"\<theta><(App e\<^isub>1 e\<^isub>2)> = App (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>)"
"x\<sharp>\<theta> \<Longrightarrow> \<theta><(Lam [x].e)> = Lam [x].(\<theta><e>)"
"\<theta><(Const n)> = Const n"
"\<theta><(Pr e\<^isub>1 e\<^isub>2)> = Pr (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>)"
"\<theta><(Fst e)> = Fst (\<theta><e>)"
"\<theta><(Snd e)> = Snd (\<theta><e>)"
"\<theta><(InL e)> = InL (\<theta><e>)"
"\<theta><(InR e)> = InR (\<theta><e>)"
"\<lbrakk>y\<noteq>x; x\<sharp>(e,e\<^isub>2,\<theta>); y\<sharp>(e,e\<^isub>1,\<theta>)\<rbrakk>
\<Longrightarrow> \<theta><Case e of inl x \<rightarrow> e\<^isub>1 | inr y \<rightarrow> e\<^isub>2> = (Case (\<theta><e>) of inl x \<rightarrow> (\<theta><e\<^isub>1>) | inr y \<rightarrow> (\<theta><e\<^isub>2>))"
apply(finite_guess add: lookup_eqvt)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess add: fs_name1 lookup_eqvt)+
done
lemma psubst_eqvt[eqvt]:
fixes pi::"name prm"
and t::"trm"
shows "pi\<bullet>(\<theta><t>) = (pi\<bullet>\<theta>)<(pi\<bullet>t)>"
by (nominal_induct t avoiding: \<theta> rule: trm.induct)
(perm_simp add: fresh_bij lookup_eqvt)+
lemma fresh_psubst:
fixes z::"name"
and t::"trm"
assumes "z\<sharp>t" and "z\<sharp>\<theta>"
shows "z\<sharp>(\<theta><t>)"
using assms
by (nominal_induct t avoiding: z \<theta> t rule: trm.induct)
(auto simp add: abs_fresh lookup_fresh)
abbreviation
subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100)
where "t[x::=t'] \<equiv> ([(x,t')])<t>"
lemma subst[simp]:
shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
and "(App t\<^isub>1 t\<^isub>2)[y::=t'] = App (t\<^isub>1[y::=t']) (t\<^isub>2[y::=t'])"
and "x\<sharp>(y,t') \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
and "(Const n)[y::=t'] = Const n"
and "(Pr e\<^isub>1 e\<^isub>2)[y::=t'] = Pr (e\<^isub>1[y::=t']) (e\<^isub>2[y::=t'])"
and "(Fst e)[y::=t'] = Fst (e[y::=t'])"
and "(Snd e)[y::=t'] = Snd (e[y::=t'])"
and "(InL e)[y::=t'] = InL (e[y::=t'])"
and "(InR e)[y::=t'] = InR (e[y::=t'])"
and "\<lbrakk>z\<noteq>x; x\<sharp>(y,e,e\<^isub>2,t'); z\<sharp>(y,e,e\<^isub>1,t')\<rbrakk>
\<Longrightarrow> (Case e of inl x \<rightarrow> e\<^isub>1 | inr z \<rightarrow> e\<^isub>2)[y::=t'] =
(Case (e[y::=t']) of inl x \<rightarrow> (e\<^isub>1[y::=t']) | inr z \<rightarrow> (e\<^isub>2[y::=t']))"
by (simp_all add: fresh_list_cons fresh_list_nil)
lemma subst_eqvt[eqvt]:
fixes pi::"name prm"
and t::"trm"
shows "pi\<bullet>(t[x::=t']) = (pi\<bullet>t)[(pi\<bullet>x)::=(pi\<bullet>t')]"
by (nominal_induct t avoiding: x t' rule: trm.induct)
(perm_simp add: fresh_bij)+
lemma fresh_subst:
fixes z::"name"
and t\<^isub>1::"trm"
and t2::"trm"
assumes "z\<sharp>t\<^isub>1" and "z\<sharp>t\<^isub>2"
shows "z\<sharp>t\<^isub>1[y::=t\<^isub>2]"
using assms
by (nominal_induct t\<^isub>1 avoiding: z y t\<^isub>2 rule: trm.induct)
(auto simp add: abs_fresh fresh_atm)
lemma fresh_subst':
fixes z::"name"
and t\<^isub>1::"trm"
and t2::"trm"
assumes "z\<sharp>[y].t\<^isub>1" and "z\<sharp>t\<^isub>2"
shows "z\<sharp>t\<^isub>1[y::=t\<^isub>2]"
using assms
by (nominal_induct t\<^isub>1 avoiding: y t\<^isub>2 z rule: trm.induct)
(auto simp add: abs_fresh fresh_nat fresh_atm)
lemma forget:
fixes x::"name"
and L::"trm"
assumes "x\<sharp>L"
shows "L[x::=P] = L"
using assms
by (nominal_induct L avoiding: x P rule: trm.induct)
(auto simp add: fresh_atm abs_fresh)
lemma psubst_empty[simp]:
shows "[]<t> = t"
by (nominal_induct t rule: trm.induct, auto simp add:fresh_list_nil)
lemma psubst_subst_psubst:
assumes h:"x\<sharp>\<theta>"
shows "\<theta><e>[x::=e'] = ((x,e')#\<theta>)<e>"
using h
apply(nominal_induct e avoiding: \<theta> x e' rule: trm.induct)
apply(auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh' fresh_psubst)
done
lemma fresh_subst_fresh:
assumes "a\<sharp>e"
shows "a\<sharp>t[a::=e]"
using assms
by (nominal_induct t avoiding: a e rule: trm.induct)
(auto simp add: fresh_atm abs_fresh fresh_nat)
text {* Typing-Judgements *}
inductive2
valid :: "(name \<times> 'a::pt_name) list \<Rightarrow> bool"
where
v_nil[intro]: "valid []"
| v_cons[intro]: "\<lbrakk>valid \<Gamma>;x\<sharp>\<Gamma>\<rbrakk> \<Longrightarrow> valid ((x,T)#\<Gamma>)"
equivariance valid
inductive_cases2
valid_cons_inv_auto[elim]:"valid ((x,T)#\<Gamma>)"
abbreviation
"sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" ("_ \<subseteq> _" [55,55] 55)
where
"\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2 \<equiv> \<forall>x T. (x,T)\<in>set \<Gamma>\<^isub>1 \<longrightarrow> (x,T)\<in>set \<Gamma>\<^isub>2"
lemma type_unicity_in_context:
assumes asm1: "(x,t\<^isub>2) \<in> set ((x,t\<^isub>1)#\<Gamma>)"
and asm2: "valid ((x,t\<^isub>1)#\<Gamma>)"
shows "t\<^isub>1=t\<^isub>2"
proof -
from asm2 have "x\<sharp>\<Gamma>" by (cases, auto)
then have "(x,t\<^isub>2) \<notin> set \<Gamma>"
by (induct \<Gamma>) (auto simp add: fresh_list_cons fresh_prod fresh_atm)
then have "(x,t\<^isub>2) = (x,t\<^isub>1)" using asm1 by auto
then show "t\<^isub>1 = t\<^isub>2" by auto
qed
lemma case_distinction_on_context:
fixes \<Gamma>::"(name \<times> ty) list"
assumes asm1: "valid ((m,t)#\<Gamma>)"
and asm2: "(n,U) \<in> set ((m,T)#\<Gamma>)"
shows "(n,U) = (m,T) \<or> ((n,U) \<in> set \<Gamma> \<and> n \<noteq> m)"
proof -
from asm2 have "(n,U) \<in> set [(m,T)] \<or> (n,U) \<in> set \<Gamma>" by auto
moreover
{ assume eq: "m=n"
assume "(n,U) \<in> set \<Gamma>"
then have "\<not> n\<sharp>\<Gamma>"
by (induct \<Gamma>) (auto simp add: fresh_list_cons fresh_prod fresh_atm)
moreover have "m\<sharp>\<Gamma>" using asm1 by auto
ultimately have False using eq by auto
}
ultimately show ?thesis by auto
qed
inductive2
typing :: "(name\<times>ty) list\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
where
t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
| t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> e\<^isub>1 : T\<^isub>1\<rightarrow>T\<^isub>2; \<Gamma> \<turnstile> e\<^isub>2 : T\<^isub>1\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T\<^isub>2"
| t_Lam[intro]: "\<lbrakk>x\<sharp>\<Gamma>; (x,T\<^isub>1)#\<Gamma> \<turnstile> e : T\<^isub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T\<^isub>1\<rightarrow>T\<^isub>2"
| t_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Const n : Data(DNat)"
| t_Pr[intro]: "\<lbrakk>\<Gamma> \<turnstile> e\<^isub>1 : Data(S\<^isub>1); \<Gamma> \<turnstile> e\<^isub>2 : Data(S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Pr e\<^isub>1 e\<^isub>2 : Data (DProd S\<^isub>1 S\<^isub>2)"
| t_Fst[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd S\<^isub>1 S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Fst e : Data(S\<^isub>1)"
| t_Snd[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd S\<^isub>1 S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Snd e : Data(S\<^isub>2)"
| t_InL[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(S\<^isub>1)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InL e : Data(DSum S\<^isub>1 S\<^isub>2)"
| t_InR[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InR e : Data(DSum S\<^isub>1 S\<^isub>2)"
| t_Case[intro]: "\<lbrakk>x\<^isub>1\<sharp>(\<Gamma>,e,e\<^isub>2,x\<^isub>2); x\<^isub>2\<sharp>(\<Gamma>,e,e\<^isub>1,x\<^isub>1); \<Gamma> \<turnstile> e: Data(DSum S\<^isub>1 S\<^isub>2);
(x\<^isub>1,Data(S\<^isub>1))#\<Gamma> \<turnstile> e\<^isub>1 : T; (x\<^isub>2,Data(S\<^isub>2))#\<Gamma> \<turnstile> e\<^isub>2 : T\<rbrakk>
\<Longrightarrow> \<Gamma> \<turnstile> (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2) : T"
equivariance typing
nominal_inductive typing
by (simp_all add: abs_fresh fresh_prod fresh_atm)
lemmas typing_eqvt' = typing.eqvt[simplified]
lemma typing_implies_valid:
assumes "\<Gamma> \<turnstile> t : T"
shows "valid \<Gamma>"
using assms
by (induct) (auto)
declare trm.inject [simp add]
declare ty.inject [simp add]
declare data.inject [simp add]
inductive_cases2 typing_inv_auto[elim]:
"\<Gamma> \<turnstile> Lam [x].t : T"
"\<Gamma> \<turnstile> Var x : T"
"\<Gamma> \<turnstile> App x y : T"
"\<Gamma> \<turnstile> Const n : T"
"\<Gamma> \<turnstile> Fst x : T"
"\<Gamma> \<turnstile> Snd x : T"
"\<Gamma> \<turnstile> InL x : T"
"\<Gamma> \<turnstile> InL x : Data (DSum T\<^isub>1 T2)"
"\<Gamma> \<turnstile> InR x : T"
"\<Gamma> \<turnstile> InR x : Data (DSum T\<^isub>1 T2)"
"\<Gamma> \<turnstile> Pr x y : T"
"\<Gamma> \<turnstile> Pr e\<^isub>1 e\<^isub>2 : Data (DProd \<sigma>1 \<sigma>\<^isub>2)"
"\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T"
declare trm.inject [simp del]
declare ty.inject [simp del]
declare data.inject [simp del]
lemma t_Lam_elim[elim]:
assumes a1:"\<Gamma> \<turnstile> Lam [x].t : T"
and a2: "x\<sharp>\<Gamma>"
obtains T\<^isub>1 and T\<^isub>2 where "(x,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" and "T=T\<^isub>1\<rightarrow>T\<^isub>2"
proof -
from a1 obtain x' t' T\<^isub>1 T\<^isub>2
where b1: "x'\<sharp>\<Gamma>" and b2: "(x',T\<^isub>1)#\<Gamma> \<turnstile> t' : T\<^isub>2" and b3: "[x'].t' = [x].t" and b4: "T=T\<^isub>1\<rightarrow>T\<^isub>2"
by auto
obtain c::"name" where "c\<sharp>(\<Gamma>,x,x',t,t')" by (erule exists_fresh[OF fs_name1])
then have fs: "c\<sharp>\<Gamma>" "c\<noteq>x" "c\<noteq>x'" "c\<sharp>t" "c\<sharp>t'" by (simp_all add: fresh_atm[symmetric])
then have b5: "[(x',c)]\<bullet>t'=[(x,c)]\<bullet>t" using b3 fs by (simp add: alpha_fresh)
have "([(x,c)]\<bullet>[(x',c)]\<bullet>((x',T\<^isub>1)#\<Gamma>)) \<turnstile> ([(x,c)]\<bullet>[(x',c)]\<bullet>t') : T\<^isub>2" using b2
by (simp only: typing_eqvt')
then have "(x,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" using fs b1 a2 b5 by (perm_simp add: calc_atm)
then show ?thesis using prems b4 by simp
qed
lemma t_Case_elim[elim]:
assumes "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" and "x\<^isub>1\<sharp>\<Gamma>" and "x\<^isub>2\<sharp>\<Gamma>"
obtains \<sigma>\<^isub>1 \<sigma>\<^isub>2 where "\<Gamma> \<turnstile> e : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)"
and "(x\<^isub>1, Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1 : T"
and "(x\<^isub>2, Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2 : T"
proof -
have f:"x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" by fact+
have "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" by fact
then obtain \<sigma>\<^isub>1 \<sigma>\<^isub>2 x\<^isub>1' x\<^isub>2' e\<^isub>1' e\<^isub>2' where
h:"\<Gamma> \<turnstile> e : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" and
h1:"(x\<^isub>1',Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1' : T" and
h2:"(x\<^isub>2',Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2' : T" and
e1:"[x\<^isub>1].e\<^isub>1=[x\<^isub>1'].e\<^isub>1'" and e2:"[x\<^isub>2].e\<^isub>2=[x\<^isub>2'].e\<^isub>2'"
by auto
obtain c::name where f':"c \<sharp> (x\<^isub>1,x\<^isub>1',e\<^isub>1,e\<^isub>1',\<Gamma>)" by (erule exists_fresh[OF fs_name1])
have e1':"[(x\<^isub>1,c)]\<bullet>e\<^isub>1 = [(x\<^isub>1',c)]\<bullet>e\<^isub>1'" using e1 f' by (auto simp add: alpha_fresh fresh_prod fresh_atm)
have "[(x\<^isub>1',c)]\<bullet>((x\<^isub>1',Data \<sigma>\<^isub>1)# \<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1' : T" using h1 typing_eqvt' by blast
then have x:"(c,Data \<sigma>\<^isub>1)#( [(x\<^isub>1',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1': T" using f'
by (auto simp add: fresh_atm calc_atm)
have "x\<^isub>1' \<sharp> \<Gamma>" using h1 typing_implies_valid by auto
then have "(c,Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> [(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f' x e1' by (auto simp add: perm_fresh_fresh)
then have "[(x\<^isub>1,c)]\<bullet>((c,Data \<sigma>\<^isub>1)#\<Gamma>) \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using typing_eqvt' by blast
then have "([(x\<^isub>1,c)]\<bullet>(c,Data \<sigma>\<^isub>1)) #\<Gamma> \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f f'
by (auto simp add: perm_fresh_fresh)
then have "([(x\<^isub>1,c)]\<bullet>(c,Data \<sigma>\<^isub>1)) #\<Gamma> \<turnstile> e\<^isub>1 : T" by perm_simp
then have g1:"(x\<^isub>1, Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1 : T" using f' by (auto simp add: fresh_atm calc_atm fresh_prod)
(* The second part of the proof is the same *)
obtain c::name where f':"c \<sharp> (x\<^isub>2,x\<^isub>2',e\<^isub>2,e\<^isub>2',\<Gamma>)" by (erule exists_fresh[OF fs_name1])
have e2':"[(x\<^isub>2,c)]\<bullet>e\<^isub>2 = [(x\<^isub>2',c)]\<bullet>e\<^isub>2'" using e2 f' by (auto simp add: alpha_fresh fresh_prod fresh_atm)
have "[(x\<^isub>2',c)]\<bullet>((x\<^isub>2',Data \<sigma>\<^isub>2)# \<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2' : T" using h2 typing_eqvt' by blast
then have x:"(c,Data \<sigma>\<^isub>2)#([(x\<^isub>2',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2': T" using f'
by (auto simp add: fresh_atm calc_atm)
have "x\<^isub>2' \<sharp> \<Gamma>" using h2 typing_implies_valid by auto
then have "(c,Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> [(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f' x e2' by (auto simp add: perm_fresh_fresh)
then have "[(x\<^isub>2,c)]\<bullet>((c,Data \<sigma>\<^isub>2)#\<Gamma>) \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using typing_eqvt' by blast
then have "([(x\<^isub>2,c)]\<bullet>(c,Data \<sigma>\<^isub>2))#\<Gamma> \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f f'
by (auto simp add: perm_fresh_fresh)
then have "([(x\<^isub>2,c)]\<bullet>(c,Data \<sigma>\<^isub>2)) #\<Gamma> \<turnstile> e\<^isub>2 : T" by perm_simp
then have g2:"(x\<^isub>2,Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2 : T" using f' by (auto simp add: fresh_atm calc_atm fresh_prod)
show ?thesis using g1 g2 prems by auto
qed
lemma weakening:
assumes "\<Gamma>\<^isub>1 \<turnstile> e: T" and "valid \<Gamma>\<^isub>2" and "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2"
shows "\<Gamma>\<^isub>2 \<turnstile> e: T"
using assms
proof(nominal_induct \<Gamma>\<^isub>1 e T avoiding: \<Gamma>\<^isub>2 rule: typing.strong_induct)
case (t_Lam x \<Gamma>\<^isub>1 T\<^isub>1 t T\<^isub>2 \<Gamma>\<^isub>2)
have ih: "\<lbrakk>valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2); (x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2\<rbrakk> \<Longrightarrow> (x,T\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> t : T\<^isub>2" by fact
have H1: "valid \<Gamma>\<^isub>2" by fact
have H2: "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2" by fact
have fs: "x\<sharp>\<Gamma>\<^isub>2" by fact
then have "valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2)" using H1 by auto
moreover have "(x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2" using H2 by auto
ultimately have "(x,T\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> t : T\<^isub>2" using ih by simp
thus "\<Gamma>\<^isub>2 \<turnstile> Lam [x].t : T\<^isub>1\<rightarrow>T\<^isub>2" using fs by auto
next
case (t_Case x\<^isub>1 \<Gamma>\<^isub>1 e e\<^isub>2 x\<^isub>2 e\<^isub>1 S\<^isub>1 S\<^isub>2 T \<Gamma>\<^isub>2)
then have ih\<^isub>1: "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2) \<Longrightarrow> (x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>1 : T"
and ih\<^isub>2: "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2) \<Longrightarrow> (x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>2 : T"
and ih\<^isub>3: "\<Gamma>\<^isub>2 \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2)" by auto
have fs\<^isub>1: "x\<^isub>1\<sharp>\<Gamma>\<^isub>2" "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>e\<^isub>2" "x\<^isub>1\<sharp>x\<^isub>2" by fact+
have fs\<^isub>2: "x\<^isub>2\<sharp>\<Gamma>\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>e\<^isub>1" "x\<^isub>2\<sharp>x\<^isub>1" by fact+
have "valid \<Gamma>\<^isub>2" by fact
then have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2)" and "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2)" using fs\<^isub>1 fs\<^isub>2 by auto
then have "(x\<^isub>1, Data S\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>1 : T" and "(x\<^isub>2, Data S\<^isub>2)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>2 : T" using ih\<^isub>1 ih\<^isub>2 by simp_all
with ih\<^isub>3 show "\<Gamma>\<^isub>2 \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" using fs\<^isub>1 fs\<^isub>2 by auto
qed (auto)
lemma context_exchange:
assumes a: "(x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma> \<turnstile> e : T"
shows "(x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma> \<turnstile> e : T"
proof -
from a have "valid ((x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma>)" by (simp add: typing_implies_valid)
then have "x\<^isub>1\<noteq>x\<^isub>2" "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" "valid \<Gamma>"
by (auto simp: fresh_list_cons fresh_atm[symmetric])
then have "valid ((x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma>)"
by (auto simp: fresh_list_cons fresh_atm)
moreover
have "(x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma> \<subseteq> (x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma>" by auto
ultimately show "(x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma> \<turnstile> e : T" using a by (auto intro: weakening)
qed
lemma typing_var_unicity:
assumes "(x,t\<^isub>1)#\<Gamma> \<turnstile> Var x : t\<^isub>2"
shows "t\<^isub>1=t\<^isub>2"
proof -
have "(x,t\<^isub>2) \<in> set ((x,t\<^isub>1)#\<Gamma>)" and "valid ((x,t\<^isub>1)#\<Gamma>)" using assms by auto
thus "t\<^isub>1=t\<^isub>2" by (simp only: type_unicity_in_context)
qed
lemma typing_substitution:
fixes \<Gamma>::"(name \<times> ty) list"
assumes "(x,T')#\<Gamma> \<turnstile> e : T"
and "\<Gamma> \<turnstile> e': T'"
shows "\<Gamma> \<turnstile> e[x::=e'] : T"
using assms
proof (nominal_induct e avoiding: \<Gamma> e' x arbitrary: T rule: trm.induct)
case (Var y \<Gamma> e' x T)
have h1: "(x,T')#\<Gamma> \<turnstile> Var y : T" by fact
have h2: "\<Gamma> \<turnstile> e' : T'" by fact
show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T"
proof (cases "x=y")
case True
assume as: "x=y"
then have "T=T'" using h1 typing_var_unicity by auto
then show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T" using as h2 by simp
next
case False
assume as: "x\<noteq>y"
have "(y,T) \<in> set ((x,T')#\<Gamma>)" using h1 by auto
then have "(y,T) \<in> set \<Gamma>" using as by auto
moreover
have "valid \<Gamma>" using h2 by (simp only: typing_implies_valid)
ultimately show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T" using as by (simp add: t_Var)
qed
next
case (Lam y t \<Gamma> e' x T)
have vc: "y\<sharp>\<Gamma>" "y\<sharp>x" "y\<sharp>e'" by fact+
have pr1: "\<Gamma> \<turnstile> e' : T'" by fact
have pr2: "(x,T')#\<Gamma> \<turnstile> Lam [y].t : T" by fact
then obtain T\<^isub>1 T\<^isub>2 where pr2': "(y,T\<^isub>1)#(x,T')#\<Gamma> \<turnstile> t : T\<^isub>2" and eq: "T = T\<^isub>1\<rightarrow>T\<^isub>2"
using vc by (auto simp add: fresh_list_cons)
then have pr2'':"(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" by (simp add: context_exchange)
have ih: "\<lbrakk>(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2; (y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> (y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" by fact
have "valid \<Gamma>" using pr1 by (simp add: typing_implies_valid)
then have "valid ((y,T\<^isub>1)#\<Gamma>)" using vc by auto
then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using pr1 by (auto intro: weakening)
then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" using ih pr2'' by simp
then have "\<Gamma> \<turnstile> Lam [y].(t[x::=e']) : T\<^isub>1\<rightarrow>T\<^isub>2" using vc by (auto intro: t_Lam)
thus "\<Gamma> \<turnstile> (Lam [y].t)[x::=e'] : T" using vc eq by simp
next
case (Case t\<^isub>1 x\<^isub>1 t\<^isub>2 x\<^isub>2 t3 \<Gamma> e' x T)
have vc: "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>1\<sharp>e'" "x\<^isub>1\<sharp>x""x\<^isub>1\<sharp>t\<^isub>1" "x\<^isub>1\<sharp>t3" "x\<^isub>2\<sharp>\<Gamma>"
"x\<^isub>2\<sharp>e'" "x\<^isub>2\<sharp>x" "x\<^isub>2\<sharp>t\<^isub>1" "x\<^isub>2\<sharp>t\<^isub>2" "x\<^isub>2\<noteq>x\<^isub>1" by fact+
have as1: "\<Gamma> \<turnstile> e' : T'" by fact
have as2: "(x,T')#\<Gamma> \<turnstile> Case t\<^isub>1 of inl x\<^isub>1 \<rightarrow> t\<^isub>2 | inr x\<^isub>2 \<rightarrow> t3 : T" by fact
then obtain S\<^isub>1 S\<^isub>2 where
h1:"(x,T')#\<Gamma> \<turnstile> t\<^isub>1 : Data (DSum S\<^isub>1 S\<^isub>2)" and
h2:"(x\<^isub>1,Data S\<^isub>1)#(x,T')#\<Gamma> \<turnstile> t\<^isub>2 : T" and
h3:"(x\<^isub>2,Data S\<^isub>2)#(x,T')#\<Gamma> \<turnstile> t3 : T"
using vc by (auto simp add: fresh_list_cons)
have ih1: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> t\<^isub>1 : Data (DSum S\<^isub>1 S\<^isub>2); \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>1[x::=e'] : Data (DSum S\<^isub>1 S\<^isub>2)"
and ih2: "\<lbrakk>(x,T')#(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2:T; (x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> e':T'\<rbrakk> \<Longrightarrow> (x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2[x::=e']:T"
and ih3: "\<lbrakk>(x,T')#(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3:T; (x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> e':T'\<rbrakk> \<Longrightarrow> (x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3[x::=e']:T"
by fact+
from h2 have h2': "(x,T')#(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2 : T" by (rule context_exchange)
from h3 have h3': "(x,T')#(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3 : T" by (rule context_exchange)
have "\<Gamma> \<turnstile> t\<^isub>1[x::=e'] : Data (DSum S\<^isub>1 S\<^isub>2)" using h1 ih1 as1 by simp
moreover
have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using h2' by (auto dest: typing_implies_valid)
then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2[x::=e'] : T" using ih2 h2' by simp
moreover
have "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using h3' by (auto dest: typing_implies_valid)
then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3[x::=e'] : T" using ih3 h3' by simp
ultimately have "\<Gamma> \<turnstile> Case (t\<^isub>1[x::=e']) of inl x\<^isub>1 \<rightarrow> (t\<^isub>2[x::=e']) | inr x\<^isub>2 \<rightarrow> (t3[x::=e']) : T"
using vc by (auto simp add: fresh_atm fresh_subst)
thus "\<Gamma> \<turnstile> (Case t\<^isub>1 of inl x\<^isub>1 \<rightarrow> t\<^isub>2 | inr x\<^isub>2 \<rightarrow> t3)[x::=e'] : T" using vc by simp
qed (simp, fast)+
text {* Big-Step Evaluation *}
inductive2
big :: "trm\<Rightarrow>trm\<Rightarrow>bool" ("_ \<Down> _" [80,80] 80)
where
b_Lam[intro]: "Lam [x].e \<Down> Lam [x].e"
| b_App[intro]: "\<lbrakk>x\<sharp>(e\<^isub>1,e\<^isub>2,e'); e\<^isub>1\<Down>Lam [x].e; e\<^isub>2\<Down>e\<^isub>2'; e[x::=e\<^isub>2']\<Down>e'\<rbrakk> \<Longrightarrow> App e\<^isub>1 e\<^isub>2 \<Down> e'"
| b_Const[intro]: "Const n \<Down> Const n"
| b_Pr[intro]: "\<lbrakk>e\<^isub>1\<Down>e\<^isub>1'; e\<^isub>2\<Down>e\<^isub>2'\<rbrakk> \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 \<Down> Pr e\<^isub>1' e\<^isub>2'"
| b_Fst[intro]: "e\<Down>Pr e\<^isub>1 e\<^isub>2 \<Longrightarrow> Fst e\<Down>e\<^isub>1"
| b_Snd[intro]: "e\<Down>Pr e\<^isub>1 e\<^isub>2 \<Longrightarrow> Snd e\<Down>e\<^isub>2"
| b_InL[intro]: "e\<Down>e' \<Longrightarrow> InL e \<Down> InL e'"
| b_InR[intro]: "e\<Down>e' \<Longrightarrow> InR e \<Down> InR e'"
| b_CaseL[intro]: "\<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1) ; e\<Down>InL e'; e\<^isub>1[x\<^isub>1::=e']\<Down>e''\<rbrakk>
\<Longrightarrow> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
| b_CaseR[intro]: "\<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1) ; e\<Down>InR e'; e\<^isub>2[x\<^isub>2::=e']\<Down>e''\<rbrakk>
\<Longrightarrow> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
equivariance big
nominal_inductive big
by (simp_all add: abs_fresh fresh_prod fresh_atm)
lemma big_eqvt':
fixes pi::"name prm"
assumes a: "(pi\<bullet>t) \<Down> (pi\<bullet>t')"
shows "t \<Down> t'"
using a
apply -
apply(drule_tac pi="rev pi" in big.eqvt)
apply(perm_simp)
done
lemma fresh_preserved:
fixes x::name
fixes t::trm
fixes t'::trm
assumes "e \<Down> e'" and "x\<sharp>e"
shows "x\<sharp>e'"
using assms by (induct) (auto simp add:fresh_subst')
declare trm.inject [simp add]
declare ty.inject [simp add]
declare data.inject [simp add]
inductive_cases2 b_inv_auto[elim]:
"App e\<^isub>1 e\<^isub>2 \<Down> t"
"Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t"
"Lam[x].t \<Down> t"
"Const n \<Down> t"
"Fst e \<Down> t"
"Snd e \<Down> t"
"InL e \<Down> t"
"InR e \<Down> t"
"Pr e\<^isub>1 e\<^isub>2 \<Down> t"
declare trm.inject [simp del]
declare ty.inject [simp del]
declare data.inject [simp del]
lemma b_App_elim[elim]:
assumes "App e\<^isub>1 e\<^isub>2 \<Down> e'" and "x\<sharp>(e\<^isub>1,e\<^isub>2,e')"
obtains f\<^isub>1 and f\<^isub>2 where "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" "e\<^isub>2 \<Down> f\<^isub>2" "f\<^isub>1[x::=f\<^isub>2] \<Down> e'"
using assms
apply -
apply(erule b_inv_auto)
apply(drule_tac pi="[(xa,x)]" in big.eqvt)
apply(drule_tac pi="[(xa,x)]" in big.eqvt)
apply(drule_tac pi="[(xa,x)]" in big.eqvt)
apply(perm_simp add: calc_atm eqvts)
done
lemma b_CaseL_elim[elim]:
assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
and "\<And> t. \<not> e \<Down> InR t"
and "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>1\<sharp>e"
obtains e' where "e \<Down> InL e'" and "e\<^isub>1[x\<^isub>1::=e'] \<Down> e''"
using assms
apply -
apply(rule b_inv_auto(2))
apply(auto)
apply(simp add: alpha)
apply(auto)
apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
apply(perm_simp add: fresh_prod)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
apply(perm_simp add: eqvts calc_atm)
apply(assumption)
apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
apply(perm_simp add: fresh_prod)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
apply(perm_simp add: eqvts calc_atm)
apply(assumption)
done
lemma b_CaseR_elim[elim]:
assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
and "\<And> t. \<not> e \<Down> InL t"
and "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>2\<sharp>e"
obtains e' where "e \<Down> InR e'" and "e\<^isub>2[x\<^isub>2::=e'] \<Down> e''"
using assms
apply -
apply(rule b_inv_auto(2))
apply(auto)
apply(simp add: alpha)
apply(auto)
apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
apply(perm_simp add: fresh_prod)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
apply(perm_simp add: eqvts calc_atm)
apply(assumption)
apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
apply(perm_simp add: fresh_prod)
apply(drule meta_mp)
apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
apply(perm_simp add: eqvts calc_atm)
apply(assumption)
done
inductive2
val :: "trm\<Rightarrow>bool"
where
v_Lam[intro]: "val (Lam [x].e)"
| v_Const[intro]: "val (Const n)"
| v_Pr[intro]: "\<lbrakk>val e\<^isub>1; val e\<^isub>2\<rbrakk> \<Longrightarrow> val (Pr e\<^isub>1 e\<^isub>2)"
| v_InL[intro]: "val e \<Longrightarrow> val (InL e)"
| v_InR[intro]: "val e \<Longrightarrow> val (InR e)"
declare trm.inject [simp add]
declare ty.inject [simp add]
declare data.inject [simp add]
inductive_cases2 v_inv_auto[elim]:
"val (Const n)"
"val (Pr e\<^isub>1 e\<^isub>2)"
"val (InL e)"
"val (InR e)"
"val (Fst e)"
"val (Snd e)"
"val (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2)"
"val (Var x)"
"val (Lam [x].e)"
"val (App e\<^isub>1 e\<^isub>2)"
declare trm.inject [simp del]
declare ty.inject [simp del]
declare data.inject [simp del]
lemma subject_reduction:
assumes a: "e \<Down> e'"
and b: "\<Gamma> \<turnstile> e : T"
shows "\<Gamma> \<turnstile> e' : T"
using a b
proof (nominal_induct avoiding: \<Gamma> arbitrary: T rule: big.strong_induct)
case (b_App x e\<^isub>1 e\<^isub>2 e' e e\<^isub>2' \<Gamma> T)
have vc: "x\<sharp>\<Gamma>" by fact
have "\<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T" by fact
then obtain T' where
a1: "\<Gamma> \<turnstile> e\<^isub>1 : T'\<rightarrow>T" and
a2: "\<Gamma> \<turnstile> e\<^isub>2 : T'" by auto
have ih1: "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T' \<rightarrow> T" by fact
have ih2: "\<Gamma> \<turnstile> e\<^isub>2 : T' \<Longrightarrow> \<Gamma> \<turnstile> e\<^isub>2' : T'" by fact
have ih3: "\<Gamma> \<turnstile> e[x::=e\<^isub>2'] : T \<Longrightarrow> \<Gamma> \<turnstile> e' : T" by fact
have "\<Gamma> \<turnstile> Lam [x].e : T'\<rightarrow>T" using ih1 a1 by simp
then have "((x,T')#\<Gamma>) \<turnstile> e : T" using vc by (auto simp add: ty.inject)
moreover
have "\<Gamma> \<turnstile> e\<^isub>2': T'" using ih2 a2 by simp
ultimately have "\<Gamma> \<turnstile> e[x::=e\<^isub>2'] : T" by (simp add: typing_substitution)
thus "\<Gamma> \<turnstile> e' : T" using ih3 by simp
next
case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' \<Gamma>)
have vc: "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" by fact+
have "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" by fact
then obtain S\<^isub>1 S\<^isub>2 e\<^isub>1' e\<^isub>2' where
a1: "\<Gamma> \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2)" and
a2: "((x\<^isub>1,Data S\<^isub>1)#\<Gamma>) \<turnstile> e\<^isub>1 : T" using vc by auto
have ih1:"\<Gamma> \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2) \<Longrightarrow> \<Gamma> \<turnstile> InL e' : Data (DSum S\<^isub>1 S\<^isub>2)" by fact
have ih2:"\<Gamma> \<turnstile> e\<^isub>1[x\<^isub>1::=e'] : T \<Longrightarrow> \<Gamma> \<turnstile> e'' : T " by fact
have "\<Gamma> \<turnstile> InL e' : Data (DSum S\<^isub>1 S\<^isub>2)" using ih1 a1 by simp
then have "\<Gamma> \<turnstile> e' : Data S\<^isub>1" by auto
then have "\<Gamma> \<turnstile> e\<^isub>1[x\<^isub>1::=e'] : T" using a2 by (simp add: typing_substitution)
then show "\<Gamma> \<turnstile> e'' : T" using ih2 by simp
next
case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' \<Gamma> T)
then show "\<Gamma> \<turnstile> e'' : T" by (blast intro: typing_substitution)
qed (blast)+
lemma unicity_of_evaluation:
assumes a: "e \<Down> e\<^isub>1"
and b: "e \<Down> e\<^isub>2"
shows "e\<^isub>1 = e\<^isub>2"
using a b
proof (nominal_induct e e\<^isub>1 avoiding: e\<^isub>2 rule: big.strong_induct)
case (b_Lam x e t\<^isub>2)
have "Lam [x].e \<Down> t\<^isub>2" by fact
thus "Lam [x].e = t\<^isub>2" by (cases, simp_all add: trm.inject)
next
case (b_App x e\<^isub>1 e\<^isub>2 e' e\<^isub>1' e\<^isub>2' t\<^isub>2)
have ih1: "\<And>t. e\<^isub>1 \<Down> t \<Longrightarrow> Lam [x].e\<^isub>1' = t" by fact
have ih2:"\<And>t. e\<^isub>2 \<Down> t \<Longrightarrow> e\<^isub>2' = t" by fact
have ih3: "\<And>t. e\<^isub>1'[x::=e\<^isub>2'] \<Down> t \<Longrightarrow> e' = t" by fact
have app: "App e\<^isub>1 e\<^isub>2 \<Down> t\<^isub>2" by fact
have vc: "x\<sharp>e\<^isub>1" "x\<sharp>e\<^isub>2" by fact+
then have "x \<sharp> App e\<^isub>1 e\<^isub>2" by auto
then have vc': "x\<sharp>t\<^isub>2" using fresh_preserved app by blast
from vc vc' obtain f\<^isub>1 f\<^isub>2 where x1: "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" and x2: "e\<^isub>2 \<Down> f\<^isub>2" and x3: "f\<^isub>1[x::=f\<^isub>2] \<Down> t\<^isub>2"
using app by (auto simp add: fresh_prod)
then have "Lam [x]. f\<^isub>1 = Lam [x]. e\<^isub>1'" using ih1 by simp
then
have "f\<^isub>1 = e\<^isub>1'" by (auto simp add: trm.inject alpha)
moreover
have "f\<^isub>2 = e\<^isub>2'" using x2 ih2 by simp
ultimately have "e\<^isub>1'[x::=e\<^isub>2'] \<Down> t\<^isub>2" using x3 by simp
thus ?case using ih3 by simp
next
case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2)
have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact+
have ih1:"\<And>t. e \<Down> t \<Longrightarrow> InL e' = t" by fact
have ih2:"\<And>t. e\<^isub>1[x\<^isub>1::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
have ha: "\<not>(\<exists>t. e \<Down> InR t)" using ih1 by force
have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
then obtain f' where "e \<Down> InL f'" and h: "e\<^isub>1[x\<^isub>1::=f']\<Down>t\<^isub>2" using ha fs by auto
then have "InL f' = InL e'" using ih1 by simp
then have "f' = e'" by (simp add: trm.inject)
then have "e\<^isub>1[x\<^isub>1::=e'] \<Down> t\<^isub>2" using h by simp
then show "e'' = t\<^isub>2" using ih2 by simp
next
case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2 )
have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact+
have ih1: "\<And>t. e \<Down> t \<Longrightarrow> InR e' = t" by fact
have ih2: "\<And>t. e\<^isub>2[x\<^isub>2::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
have ha: "\<not>(\<exists>t. e \<Down> InL t)" using ih1 by force
have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
then obtain f' where "e \<Down> InR f'" and h: "e\<^isub>2[x\<^isub>2::=f']\<Down>t\<^isub>2" using ha fs by auto
then have "InR f' = InR e'" using ih1 by simp
then have "e\<^isub>2[x\<^isub>2::=e'] \<Down> t\<^isub>2" using h by (simp add: trm.inject)
thus "e'' = t\<^isub>2" using ih2 by simp
next
case (b_Fst e e\<^isub>1 e\<^isub>2 e\<^isub>2')
have "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
have "\<And> b. e \<Down> b \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 = b" by fact
have "Fst e \<Down> e\<^isub>2'" by fact
show "e\<^isub>1 = e\<^isub>2'" using prems by (force simp add: trm.inject)
next
case (b_Snd e e\<^isub>1 e\<^isub>2 e\<^isub>2')
have "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
have "\<And> b. e \<Down> b \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 = b" by fact
have "Snd e \<Down> e\<^isub>2'" by fact
show "e\<^isub>2 = e\<^isub>2'" using prems by (force simp add: trm.inject)
qed (blast)+
lemma not_val_App[simp]:
shows
"\<not> val (App e\<^isub>1 e\<^isub>2)"
"\<not> val (Fst e)"
"\<not> val (Snd e)"
"\<not> val (Var x)"
"\<not> val (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2)"
by auto
lemma reduces_evaluates_to_values:
assumes h:"t \<Down> t'"
shows "val t'"
using h by (induct) (auto)
lemma type_prod_evaluates_to_pairs:
assumes a: "\<Gamma> \<turnstile> t : Data (DProd S\<^isub>1 S\<^isub>2)"
and b: "t \<Down> t'"
obtains t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2"
proof -
have "\<Gamma> \<turnstile> t' : Data (DProd S\<^isub>1 S\<^isub>2)" using assms subject_reduction by simp
moreover
have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
lemma type_sum_evaluates_to_ins:
assumes "\<Gamma> \<turnstile> t : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" and "t \<Down> t'"
shows "(\<exists>t''. t' = InL t'') \<or> (\<exists>t''. t' = InR t'')"
proof -
have "\<Gamma> \<turnstile> t' : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" using assms subject_reduction by simp
moreover
have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain t'' where "t' = InL t'' \<or> t' = InR t''"
by (cases, auto simp add:ty.inject data.inject)
thus ?thesis by auto
qed
lemma type_arrow_evaluates_to_lams:
assumes "\<Gamma> \<turnstile> t : \<sigma> \<rightarrow> \<tau>" and "t \<Down> t'"
obtains x t'' where "t' = Lam [x]. t''"
proof -
have "\<Gamma> \<turnstile> t' : \<sigma> \<rightarrow> \<tau>" using assms subject_reduction by simp
moreover
have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain x t'' where "t' = Lam [x]. t''" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
lemma type_nat_evaluates_to_consts:
assumes "\<Gamma> \<turnstile> t : Data DNat" and "t \<Down> t'"
obtains n where "t' = Const n"
proof -
have "\<Gamma> \<turnstile> t' : Data DNat " using assms subject_reduction by simp
moreover have "val t'" using reduces_evaluates_to_values assms by simp
ultimately obtain n where "t' = Const n" by (cases, auto simp add:ty.inject data.inject)
thus ?thesis using prems by auto
qed
consts
V' :: "data \<Rightarrow> trm set"
nominal_primrec
"V' (DNat) = {Const n | n. n \<in> (UNIV::nat set)}"
"V' (DProd S\<^isub>1 S\<^isub>2) = {Pr x y | x y. x \<in> V' S\<^isub>1 \<and> y \<in> V' S\<^isub>2}"
"V' (DSum S\<^isub>1 S\<^isub>2) = {InL x | x. x \<in> V' S\<^isub>1} \<union> {InR y | y. y \<in> V' S\<^isub>2}"
apply(rule TrueI)+
done
lemma Vprimes_are_values :
fixes S::"data"
assumes h: "e \<in> V' S"
shows "val e"
using h
by (nominal_induct S arbitrary: e rule:data.induct)
(auto)
lemma V'_eqvt:
fixes pi::"name prm"
assumes a: "v \<in> V' S"
shows "(pi\<bullet>v) \<in> V' S"
using a
by (nominal_induct S arbitrary: v rule: data.induct)
(auto simp add: trm.inject)
consts
V :: "ty \<Rightarrow> trm set"
nominal_primrec
"V (Data S) = V' S"
"V (T\<^isub>1 \<rightarrow> T\<^isub>2) = {Lam [x].e | x e. \<forall> v \<in> (V T\<^isub>1). \<exists> v'. e[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2}"
apply(rule TrueI)+
done
lemma V_eqvt:
fixes pi::"name prm"
assumes a: "x\<in>V T"
shows "(pi\<bullet>x)\<in>V T"
using a
apply(nominal_induct T arbitrary: pi x rule: ty.induct)
apply(auto simp add: trm.inject perm_set_def)
apply(perm_simp add: V'_eqvt)
apply(rule_tac x="pi\<bullet>xa" in exI)
apply(rule_tac x="pi\<bullet>e" in exI)
apply(simp)
apply(auto)
apply(drule_tac x="(rev pi)\<bullet>v" in bspec)
apply(force)
apply(auto)
apply(rule_tac x="pi\<bullet>v'" in exI)
apply(auto)
apply(drule_tac pi="pi" in big.eqvt)
apply(perm_simp add: eqvts)
done
lemma V_arrow_elim_weak[elim] :
assumes h:"u \<in> (V (T\<^isub>1 \<rightarrow> T\<^isub>2))"
obtains a t where "u = Lam[a].t" and "\<forall> v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
using h by (auto)
lemma V_arrow_elim_strong[elim]:
fixes c::"'a::fs_name"
assumes h: "u \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)"
obtains a t where "a\<sharp>c" "u = Lam[a].t" "\<forall>v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
using h
apply -
apply(erule V_arrow_elim_weak)
apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(a,t,c)") (*A*)
apply(erule exE)
apply(drule_tac x="a'" in meta_spec)
apply(drule_tac x="[(a,a')]\<bullet>t" in meta_spec)
apply(drule meta_mp)
apply(simp)
apply(drule meta_mp)
apply(simp add: trm.inject alpha fresh_left fresh_prod calc_atm fresh_atm)
apply(perm_simp)
apply(force)
apply(drule meta_mp)
apply(rule ballI)
apply(drule_tac x="[(a,a')]\<bullet>v" in bspec)
apply(simp add: V_eqvt)
apply(auto)
apply(rule_tac x="[(a,a')]\<bullet>v'" in exI)
apply(auto)
apply(drule_tac pi="[(a,a')]" in big.eqvt)
apply(perm_simp add: eqvts calc_atm)
apply(simp add: V_eqvt)
(*A*)
apply(rule exists_fresh')
apply(simp add: fin_supp)
done
lemma V_are_values :
fixes T::"ty"
assumes h:"e \<in> V T"
shows "val e"
using h by (nominal_induct T arbitrary: e rule:ty.induct, auto simp add: Vprimes_are_values)
lemma values_reduce_to_themselves:
assumes h:"val v"
shows "v \<Down> v"
using h by (induct,auto)
lemma Vs_reduce_to_themselves[simp]:
assumes h:"v \<in> V T"
shows "v \<Down> v"
using h by (simp add: values_reduce_to_themselves V_are_values)
lemma V_sum:
assumes h:"x \<in> V (Data (DSum S\<^isub>1 S\<^isub>2))"
shows "(\<exists> y. x= InL y \<and> y \<in> V' S\<^isub>1) \<or> (\<exists> y. x= InR y \<and> y \<in> V' S\<^isub>2)"
using h by simp
abbreviation
mapsto :: "(name\<times>trm) list \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> bool" ("_ maps _ to _" [55,55,55] 55)
where
"\<theta> maps x to e\<equiv> (lookup \<theta> x) = e"
abbreviation
v_closes :: "(name\<times>trm) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" ("_ Vcloses _" [55,55] 55)
where
"\<theta> Vcloses \<Gamma> \<equiv> \<forall>x T. ((x,T) \<in> set \<Gamma> \<longrightarrow> (\<exists>v. \<theta> maps x to v \<and> v \<in> (V T)))"
lemma monotonicity:
fixes m::"name"
fixes \<theta>::"(name \<times> trm) list"
assumes h1: "\<theta> Vcloses \<Gamma>"
and h2: "e \<in> V T"
and h3: "valid ((x,T)#\<Gamma>)"
shows "(x,e)#\<theta> Vcloses (x,T)#\<Gamma>"
proof(intro strip)
fix x' T'
assume "(x',T') \<in> set ((x,T)#\<Gamma>)"
then have "((x',T')=(x,T)) \<or> ((x',T')\<in>set \<Gamma> \<and> x'\<noteq>x)" using h3
by (rule_tac case_distinction_on_context)
moreover
{ (* first case *)
assume "(x',T') = (x,T)"
then have "\<exists>e'. ((x,e)#\<theta>) maps x to e' \<and> e' \<in> V T'" using h2 by auto
}
moreover
{ (* second case *)
assume "(x',T') \<in> set \<Gamma>" and neq:"x' \<noteq> x"
then have "\<exists>e'. \<theta> maps x' to e' \<and> e' \<in> V T'" using h1 by auto
then have "\<exists>e'. ((x,e)#\<theta>) maps x' to e' \<and> e' \<in> V T'" using neq by auto
}
ultimately show "\<exists>e'. ((x,e)#\<theta>) maps x' to e' \<and> e' \<in> V T'" by blast
qed
lemma termination_aux:
fixes T :: "ty"
fixes \<Gamma> :: "(name \<times> ty) list"
fixes \<theta> :: "(name \<times> trm) list"
fixes e :: "trm"
assumes h1: "\<Gamma> \<turnstile> e : T"
and h2: "\<theta> Vcloses \<Gamma>"
shows "\<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T"
using h2 h1
proof(nominal_induct e avoiding: \<Gamma> \<theta> arbitrary: T rule: trm.induct)
case (App e\<^isub>1 e\<^isub>2 \<Gamma> \<theta> T)
have ih\<^isub>1:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^isub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
have ih\<^isub>2:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^isub>2 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^isub>2> \<Down> v \<and> v \<in> V T" by fact
have as\<^isub>1:"\<theta> Vcloses \<Gamma>" by fact
have as\<^isub>2: "\<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T" by fact
from as\<^isub>2 obtain T' where "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> e\<^isub>2 : T'" by auto
then obtain v\<^isub>1 v\<^isub>2 where "(i)": "\<theta><e\<^isub>1> \<Down> v\<^isub>1" "v\<^isub>1 \<in> V (T' \<rightarrow> T)"
and "(ii)":"\<theta><e\<^isub>2> \<Down> v\<^isub>2" "v\<^isub>2 \<in> V T'" using ih\<^isub>1 ih\<^isub>2 as\<^isub>1 by blast
from "(i)" obtain x e'
where "v\<^isub>1 = Lam[x].e'"
and "(iii)": "(\<forall>v \<in> (V T').\<exists> v'. e'[x::=v] \<Down> v' \<and> v' \<in> V T)"
and "(iv)": "\<theta><e\<^isub>1> \<Down> Lam [x].e'"
and fr: "x\<sharp>(\<theta>,e\<^isub>1,e\<^isub>2)" by blast
from fr have fr\<^isub>1: "x\<sharp>\<theta><e\<^isub>1>" and fr\<^isub>2: "x\<sharp>\<theta><e\<^isub>2>" by (simp_all add: fresh_psubst)
from "(ii)" "(iii)" obtain v\<^isub>3 where "(v)": "e'[x::=v\<^isub>2] \<Down> v\<^isub>3" "v\<^isub>3 \<in> V T" by auto
from fr\<^isub>2 "(ii)" have "x\<sharp>v\<^isub>2" by (simp add: fresh_preserved)
then have "x\<sharp>e'[x::=v\<^isub>2]" by (simp add: fresh_subst_fresh)
then have fr\<^isub>3: "x\<sharp>v\<^isub>3" using "(v)" by (simp add: fresh_preserved)
from fr\<^isub>1 fr\<^isub>2 fr\<^isub>3 have "x\<sharp>(\<theta><e\<^isub>1>,\<theta><e\<^isub>2>,v\<^isub>3)" by simp
with "(iv)" "(ii)" "(v)" have "App (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>) \<Down> v\<^isub>3" by auto
then show "\<exists>v. \<theta><App e\<^isub>1 e\<^isub>2> \<Down> v \<and> v \<in> V T" using "(v)" by auto
next
case (Pr t\<^isub>1 t\<^isub>2 \<Gamma> \<theta> T)
have "\<Gamma> \<turnstile> Pr t\<^isub>1 t\<^isub>2 : T" by fact
then obtain T\<^isub>a T\<^isub>b where ta:"\<Gamma> \<turnstile> t\<^isub>1 : Data T\<^isub>a" and "\<Gamma> \<turnstile> t\<^isub>2 : Data T\<^isub>b"
and eq:"T=Data (DProd T\<^isub>a T\<^isub>b)" by auto
have h:"\<theta> Vcloses \<Gamma>" by fact
then obtain v\<^isub>1 v\<^isub>2 where "\<theta><t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V (Data T\<^isub>a)" "\<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V (Data T\<^isub>b)"
using prems by blast
thus "\<exists>v. \<theta><Pr t\<^isub>1 t\<^isub>2> \<Down> v \<and> v \<in> V T" using eq by auto
next
case (Lam x e \<Gamma> \<theta> T)
have ih:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T" by fact
have as\<^isub>1: "\<theta> Vcloses \<Gamma>" by fact
have as\<^isub>2: "\<Gamma> \<turnstile> Lam [x].e : T" by fact
have fs: "x\<sharp>\<Gamma>" "x\<sharp>\<theta>" by fact+
from as\<^isub>2 fs obtain T\<^isub>1 T\<^isub>2
where "(i)": "(x,T\<^isub>1)#\<Gamma> \<turnstile> e:T\<^isub>2" and "(ii)": "T = T\<^isub>1 \<rightarrow> T\<^isub>2" by auto
from "(i)" have "(iii)": "valid ((x,T\<^isub>1)#\<Gamma>)" by (simp add: typing_implies_valid)
have "\<forall>v \<in> (V T\<^isub>1). \<exists>v'. (\<theta><e>)[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
proof
fix v
assume "v \<in> (V T\<^isub>1)"
with "(iii)" as\<^isub>1 have "(x,v)#\<theta> Vcloses (x,T\<^isub>1)#\<Gamma>" using monotonicity by auto
with ih "(i)" obtain v' where "((x,v)#\<theta>)<e> \<Down> v' \<and> v' \<in> V T\<^isub>2" by blast
then have "\<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2" using fs by (simp add: psubst_subst_psubst)
then show "\<exists>v'. \<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2" by auto
qed
then have "Lam[x].\<theta><e> \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)" by auto
then have "\<theta><Lam [x].e> \<Down> Lam[x].\<theta><e> \<and> Lam[x].\<theta><e> \<in> V (T\<^isub>1\<rightarrow>T\<^isub>2)" using fs by auto
thus "\<exists>v. \<theta><Lam [x].e> \<Down> v \<and> v \<in> V T" using "(ii)" by auto
next
case (Case t' n\<^isub>1 t\<^isub>1 n\<^isub>2 t\<^isub>2 \<Gamma> \<theta> T)
have f: "n\<^isub>1\<sharp>\<Gamma>" "n\<^isub>1\<sharp>\<theta>" "n\<^isub>2\<sharp>\<Gamma>" "n\<^isub>2\<sharp>\<theta>" "n\<^isub>2\<noteq>n\<^isub>1" "n\<^isub>1\<sharp>t'"
"n\<^isub>1\<sharp>t\<^isub>2" "n\<^isub>2\<sharp>t'" "n\<^isub>2\<sharp>t\<^isub>1" by fact+
have h:"\<theta> Vcloses \<Gamma>" by fact
have th:"\<Gamma> \<turnstile> Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2 : T" by fact
then obtain S\<^isub>1 S\<^isub>2 where
hm:"\<Gamma> \<turnstile> t' : Data (DSum S\<^isub>1 S\<^isub>2)" and
hl:"(n\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>1 : T" and
hr:"(n\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t\<^isub>2 : T" using f by auto
then obtain v\<^isub>0 where ht':"\<theta><t'> \<Down> v\<^isub>0" and hS:"v\<^isub>0 \<in> V (Data (DSum S\<^isub>1 S\<^isub>2))" using prems h by blast
(* We distinguish between the cases InL and InR *)
{ fix v\<^isub>0'
assume eqc:"v\<^isub>0 = InL v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>1"
then have inc: "v\<^isub>0' \<in> V (Data S\<^isub>1)" by auto
have "valid \<Gamma>" using th typing_implies_valid by auto
then moreover have "valid ((n\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using f by auto
then moreover have "(n\<^isub>1,v\<^isub>0')#\<theta> Vcloses (n\<^isub>1,Data S\<^isub>1)#\<Gamma>"
using inc h monotonicity by blast
moreover
have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><t\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
ultimately obtain v\<^isub>1 where ho: "((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using hl by blast
then have r:"\<theta><t\<^isub>1>[n\<^isub>1::=v\<^isub>0'] \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using psubst_subst_psubst f by simp
then moreover have "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>1,n\<^isub>2)"
proof -
have "n\<^isub>1\<sharp>v\<^isub>0" using ht' fresh_preserved fresh_psubst f by auto
then have "n\<^isub>1\<sharp>v\<^isub>0'" using eqc by auto
then have "n\<^isub>1\<sharp>v\<^isub>1" using f r fresh_preserved fresh_subst_fresh by blast
thus "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>1,n\<^isub>2)" using f by (simp add: fresh_atm fresh_psubst)
qed
moreover have "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>1,n\<^isub>1)"
proof -
have "n\<^isub>2\<sharp>v\<^isub>0" using ht' fresh_preserved fresh_psubst f by auto
then have "n\<^isub>2\<sharp>v\<^isub>0'" using eqc by auto
then have "n\<^isub>2\<sharp>((n\<^isub>1,v\<^isub>0')#\<theta>)" using f fresh_list_cons fresh_atm by force
then have "n\<^isub>2\<sharp>((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1>" using f fresh_psubst by auto
moreover then have "n\<^isub>2 \<sharp> v\<^isub>1" using fresh_preserved ho by auto
ultimately show "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>1,n\<^isub>1)" using f by (simp add: fresh_psubst fresh_atm)
qed
ultimately have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using ht' eqc by auto
moreover
have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> = \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2>"
using f by auto
ultimately have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" by auto
}
moreover
{ fix v\<^isub>0'
assume eqc:"v\<^isub>0 = InR v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>2"
then have inc:"v\<^isub>0' \<in> V (Data S\<^isub>2)" by auto
have "valid \<Gamma>" using th typing_implies_valid by auto
then moreover have "valid ((n\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using f by auto
then moreover have "(n\<^isub>2,v\<^isub>0')#\<theta> Vcloses (n\<^isub>2,Data S\<^isub>2)#\<Gamma>"
using inc h monotonicity by blast
moreover have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>2 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><t\<^isub>2> \<Down> v \<and> v \<in> V T" by fact
ultimately obtain v\<^isub>2 where ho:"((n\<^isub>2,v\<^isub>0')#\<theta>)<t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using hr by blast
then have r:"\<theta><t\<^isub>2>[n\<^isub>2::=v\<^isub>0'] \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using psubst_subst_psubst f by simp
moreover have "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>2,n\<^isub>2)"
proof -
have "n\<^isub>1\<sharp>\<theta><t'>" using fresh_psubst f by simp
then have "n\<^isub>1\<sharp>v\<^isub>0" using ht' fresh_preserved by auto
then have "n\<^isub>1\<sharp>v\<^isub>0'" using eqc by auto
then have "n\<^isub>1\<sharp>((n\<^isub>2,v\<^isub>0')#\<theta>)" using f fresh_list_cons fresh_atm by force
then have "n\<^isub>1\<sharp>((n\<^isub>2,v\<^isub>0')#\<theta>)<t\<^isub>2>" using f fresh_psubst by auto
moreover then have "n\<^isub>1\<sharp>v\<^isub>2" using fresh_preserved ho by auto
ultimately show "n\<^isub>1 \<sharp> (\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>2,n\<^isub>2)" using f by (simp add: fresh_psubst fresh_atm)
qed
moreover have "n\<^isub>2 \<sharp> (\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>2,n\<^isub>1)"
proof -
have "n\<^isub>2\<sharp>\<theta><t'>" using fresh_psubst f by simp
then have "n\<^isub>2\<sharp>v\<^isub>0" using ht' fresh_preserved by auto
then have "n\<^isub>2\<sharp>v\<^isub>0'" using eqc by auto
then have "n\<^isub>2\<sharp>\<theta><t\<^isub>2>[n\<^isub>2::=v\<^isub>0']" using f fresh_subst_fresh by auto
then have "n\<^isub>2\<sharp>v\<^isub>2" using f fresh_preserved r by blast
then show "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>2,n\<^isub>1)" using f by (simp add: fresh_atm fresh_psubst)
qed
ultimately have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using ht' eqc by auto
then have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using f by auto
}
ultimately show "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using hS V_sum by blast
qed (force)+
theorem termination_of_evaluation:
assumes a: "[] \<turnstile> e : T"
shows "\<exists>v. e \<Down> v \<and> val v"
proof -
from a have "\<exists>v. (([]::(name \<times> trm) list)<e>) \<Down> v \<and> v \<in> V T"
by (rule termination_aux) (auto)
thus "\<exists>v. e \<Down> v \<and> val v" using V_are_values by auto
qed
end