(* "$Id$" *)
(* *)
(* Formalisation of some typical SOS-proofs. *)
(* *)
(* This work arose from a challenge suggested by Adam *)
(* Chlipala 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
text {* types and terms *}
nominal_datatype ty =
TVar "nat"
| Arrow "ty" "ty" ("_\<rightarrow>_" [100,100] 100)
nominal_datatype trm =
Var "name"
| Lam "\<guillemotleft>name\<guillemotright>trm" ("Lam [_]._" [100,100] 100)
| App "trm" "trm"
lemma fresh_ty:
fixes x::"name"
and T::"ty"
shows "x\<sharp>T"
by (induct T rule: ty.weak_induct)
(auto simp add: fresh_nat)
text {* Parallel and single 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[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)
lemma lookup_fresh:
fixes z::"name"
assumes a: "z\<sharp>\<theta>" and b: "z\<sharp>x"
shows "z \<sharp>lookup \<theta> x"
using a b
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)
(* 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>)"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
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)
lemma psubst_empty[simp]:
shows "[]<t> = t"
by (nominal_induct t rule: trm.induct)
(auto simp add: fresh_list_nil)
(* Single substitution *)
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'])"
by (simp_all add: fresh_list_cons fresh_list_nil)
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':
assumes "x\<sharp>e'"
shows "x\<sharp>e[x::=e']"
using assms
by (nominal_induct e avoiding: x e' rule: trm.induct)
(auto simp add: fresh_atm abs_fresh fresh_nat)
lemma forget:
fixes x::"name"
and L::"trm"
assumes "x\<sharp>e"
shows "e[x::=e'] = e"
using assms
by (nominal_induct e avoiding: x e' rule: trm.induct)
(auto simp add: fresh_atm abs_fresh)
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')
done
text {* Typing Judgements *}
inductive
valid :: "(name\<times>ty) 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_cases
valid_cons_inv_auto[elim]: "valid ((x,T)#\<Gamma>)"
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
inductive
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"
equivariance typing
nominal_inductive typing
by (simp_all add: abs_fresh fresh_ty)
lemma typing_implies_valid:
assumes a: "\<Gamma> \<turnstile> t : T"
shows "valid \<Gamma>"
using a
by (induct) (auto)
lemma t_Lam_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"
using a1 a2
by (cases rule: typing.strong_cases [where x="x"])
(auto simp add: abs_fresh fresh_ty alpha trm.inject)
abbreviation
"sub_context" :: "(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 weakening:
fixes \<Gamma>\<^isub>1 \<Gamma>\<^isub>2::"(name\<times>ty) list"
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 vc: "x\<sharp>\<Gamma>\<^isub>2" by fact
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 "valid \<Gamma>\<^isub>2" by fact
then have "valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2)" using vc by auto
moreover
have "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2" by fact
then have "(x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2" by simp
ultimately have "(x,T\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> t : T\<^isub>2" using ih by simp
with vc show "\<Gamma>\<^isub>2 \<turnstile> Lam [x].t : T\<^isub>1\<rightarrow>T\<^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 fresh_ty)
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 a: "(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>)" using a
by (cases) (auto simp add: trm.inject)
moreover
have "valid ((x,T\<^isub>1)#\<Gamma>)" using a by (simp add: typing_implies_valid)
ultimately show "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 (cases) (auto simp add: trm.inject)
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 elim: t_Lam_elim simp add: fresh_list_cons fresh_ty)
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 (simp add: 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
then show "\<Gamma> \<turnstile> (Lam [y].t)[x::=e'] : T" using vc eq by simp
next
case (App e1 e2 \<Gamma> e' x T)
have "(x,T')#\<Gamma> \<turnstile> App e1 e2 : T" by fact
then obtain Tn where a1: "(x,T')#\<Gamma> \<turnstile> e1 : Tn \<rightarrow> T"
and a2: "(x,T')#\<Gamma> \<turnstile> e2 : Tn"
by (cases) (auto simp add: trm.inject)
have a3: "\<Gamma> \<turnstile> e' : T'" by fact
have ih1: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> e1 : Tn\<rightarrow>T; \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> e1[x::=e'] : Tn\<rightarrow>T" by fact
have ih2: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> e2 : Tn; \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> e2[x::=e'] : Tn" by fact
then show ?case using a1 a2 a3 ih1 ih2 by auto
qed
text {* Values *}
inductive
val :: "trm\<Rightarrow>bool"
where
v_Lam[intro]: "val (Lam [x].e)"
equivariance val
lemma not_val_App[simp]:
shows
"\<not> val (App e\<^isub>1 e\<^isub>2)"
"\<not> val (Var x)"
by (auto elim: val.cases)
text {* Big-Step Evaluation *}
inductive
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'"
equivariance big
nominal_inductive big
by (simp_all add: abs_fresh)
lemma big_preserves_fresh:
fixes x::"name"
assumes a: "e \<Down> e'" "x\<sharp>e"
shows "x\<sharp>e'"
using a by (induct) (auto simp add: abs_fresh fresh_subst)
lemma b_App_elim:
assumes a: "App e\<^isub>1 e\<^isub>2 \<Down> e'" "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 a
by (cases rule: big.strong_cases[where x="x" and xa="x"])
(auto simp add: trm.inject)
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 (cases) (auto simp add: trm.inject)
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 elim: t_Lam_elim 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
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" "x\<sharp>t\<^isub>2" by fact
then have "x\<sharp>App e\<^isub>1 e\<^isub>2" by auto
from app 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"
by (cases rule: big.strong_cases[where x="x" and xa="x"])
(auto simp add: trm.inject)
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 "e' = t\<^isub>2" using ih3 by simp
qed
lemma reduces_evaluates_to_values:
assumes h:"t \<Down> t'"
shows "val t'"
using h by (induct) (auto)
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)
thus ?thesis using prems by auto
qed
(* Valuation *)
consts
V :: "ty \<Rightarrow> trm set"
nominal_primrec
"V (TVar x) = {e. val e}"
"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}"
by (rule TrueI)+
(* can go with strong inversion rules *)
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(simp add: eqvts)
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:
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:
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 Vs_are_values:
assumes a: "e \<in> V T"
shows "val e"
using a by (nominal_induct T arbitrary: e rule: ty.induct) (auto)
lemma values_reduce_to_themselves:
assumes a: "val v"
shows "v \<Down> v"
using a by (induct) (auto)
lemma Vs_reduce_to_themselves:
assumes a: "v \<in> V T"
shows "v \<Down> v"
using a by (simp add: values_reduce_to_themselves Vs_are_values)
text {* '\<theta> maps x to e' asserts that \<theta> substitutes x with e *}
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:
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
then obtain T' where "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> e\<^isub>2 : T'"
by (cases) (auto simp add: trm.inject)
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 elim: V_arrow_elim_strong)
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: big_preserves_fresh)
then have "x\<sharp>e'[x::=v\<^isub>2]" by (simp add: fresh_subst')
then have fr\<^isub>3: "x\<sharp>v\<^isub>3" using "(v)" by (simp add: big_preserves_fresh)
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 (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" using fs
by (cases rule: typing.strong_cases[where x="x"])
(auto simp add: trm.inject alpha abs_fresh fresh_ty)
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 (Var x \<Gamma> \<theta> T)
have "\<Gamma> \<turnstile> (Var x) : T" by fact
then have "(x,T)\<in>set \<Gamma>" by (cases) (auto simp add: trm.inject)
with prems have "\<theta><Var x> \<Down> \<theta><Var x> \<and> \<theta><Var x>\<in> V T"
by (auto intro!: Vs_reduce_to_themselves)
then show "\<exists>v. \<theta><Var x> \<Down> v \<and> v \<in> V T" by auto
qed
theorem termination_of_evaluation:
assumes a: "[] \<turnstile> e : T"
shows "\<exists>v. e \<Down> v \<and> val v"
proof -
from a have "\<exists>v. []<e> \<Down> v \<and> v \<in> V T" by (rule termination_aux) (auto)
thus "\<exists>v. e \<Down> v \<and> val v" using Vs_are_values by auto
qed
end