(* *)
(* Formalisation of some typical SOS-proofs. *)
(* *)
(* This work was inspired by 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 "HOL-Nominal.Nominal"
begin
atom_decl name
text ‹types and terms
›
nominal_datatype ty =
TVar
"nat"
| Arrow
"ty" "ty" (
‹_
→_
› [100,100] 100)
nominal_datatype trm =
Var
"name"
| Lam
"\name\trm" (
‹Lam [_]._
› [100,100] 100)
| App
"trm" "trm"
lemma fresh_ty:
fixes x::
"name"
and T::
"ty"
shows "x\T"
by (induct T rule: ty.induct)
(auto simp add: fresh_nat)
text ‹Parallel
and single substitution.
›
fun
lookup ::
"(name\trm) list \ name \ trm"
where
"lookup [] x = Var x"
|
"lookup ((y,e)#\) x = (if x=y then e else lookup \ x)"
lemma lookup_eqvt[eqvt]:
fixes pi::
"name prm"
shows "pi\(lookup \ X) = lookup (pi\\) (pi\X)"
by (induct θ) (auto simp add: eqvts)
lemma lookup_fresh:
fixes z::
"name"
assumes a:
"z\\" and b:
"z\x"
shows "z \lookup \ x"
using a b
by (induct rule: lookup.induct) (auto simp add: fresh_list_cons)
lemma lookup_fresh
':
assumes "z\\"
shows "lookup \ z = Var z"
using assms
by (induct rule: lookup.induct)
(auto simp add: fresh_list_cons fresh_prod fresh_atm)
(* parallel substitution *)
nominal_primrec
psubst ::
"(name\trm) list \ trm \ trm" (
‹_<_>
› [95,95] 105)
where
"\<(Var x)> = (lookup \ x)"
|
"\<(App e\<^sub>1 e\<^sub>2)> = App (\1>) (\2>)"
|
"x\\ \ \<(Lam [x].e)> = Lam [x].(\)"
by (finite_guess | simp add: abs_fresh | fresh_guess)+
lemma psubst_eqvt[eqvt]:
fixes pi::
"name prm"
and t::
"trm"
shows "pi\(\) = (pi\\)<(pi\t)>"
by (nominal_induct t avoiding: θ rule: trm.strong_induct)
(perm_simp add: fresh_bij lookup_eqvt)+
lemma fresh_psubst:
fixes z::
"name"
and t::
"trm"
assumes "z\t" and "z\\"
shows "z\(\)"
using assms
by (nominal_induct t avoiding: z θ t rule: trm.strong_induct)
(auto simp add: abs_fresh lookup_fresh)
lemma psubst_empty[simp]:
shows "[] = t"
by (nominal_induct t rule: trm.strong_induct)
(auto simp add: fresh_list_nil)
text ‹Single substitution
›
abbreviation
subst ::
"trm \ name \ trm \ trm" (
‹_[_::=_]
› [100,100,100] 100)
where
"t[x::=t'] \ ([(x,t')])"
lemma subst[simp]:
shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
and "(App t\<^sub>1 t\<^sub>2)[y::=t'] = App (t\<^sub>1[y::=t']) (t\<^sub>2[y::=t'])"
and "x\(y,t') \ (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"
shows "\z\s; (z=y \ z\t)\ \ z\t[y::=s]"
by (nominal_induct t avoiding: z y s rule: trm.strong_induct)
(auto simp add: abs_fresh fresh_prod fresh_atm)
lemma forget:
assumes a:
"x\e"
shows "e[x::=e'] = e"
using a
by (nominal_induct e avoiding: x e
' rule: trm.strong_induct)
(auto simp add: fresh_atm abs_fresh)
lemma psubst_subst_psubst:
assumes h:
"x\\"
shows "\[x::=e'] = ((x,e')#\)"
using h
by (nominal_induct e avoiding: θ x e
' rule: trm.strong_induct)
(auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh
')
text ‹Typing Judgements
›
inductive
valid ::
"(name\ty) list \ bool"
where
v_nil[intro]:
"valid []"
| v_cons[intro]:
"\valid \;x\\\ \ valid ((x,T)#\)"
equivariance valid
inductive_cases
valid_elim[elim]:
"valid ((x,T)#\)"
lemma valid_insert:
assumes a:
"valid (\@[(x,T)]@\)"
shows "valid (\ @ \)"
using a
by (induct Δ)
(auto simp add: fresh_list_append fresh_list_cons elim!: valid_elim)
lemma fresh_set:
shows "y\xs = (\x\set xs. y\x)"
by (induct xs) (simp_all add: fresh_list_nil fresh_list_cons)
lemma context_unique:
assumes a1:
"valid \"
and a2:
"(x,T) \ set \"
and a3:
"(x,U) \ set \"
shows "T = U"
using a1 a2 a3
by (induct) (auto simp add: fresh_set fresh_prod fresh_atm)
text ‹Typing Relation
›
inductive
typing ::
"(name\ty) list\trm\ty\bool" (
‹_
⊨ _ : _
› [60,60,60] 60)
where
t_Var[intro]:
"\valid \; (x,T)\set \\ \ \ \ Var x : T"
| t_App[intro]:
"\\ \ e\<^sub>1 : T\<^sub>1\T\<^sub>2; \ \ e\<^sub>2 : T\<^sub>1\ \ \ \ App e\<^sub>1 e\<^sub>2 : T\<^sub>2"
| t_Lam[intro]:
"\x\\; (x,T\<^sub>1)#\ \ e : T\<^sub>2\ \ \ \ Lam [x].e : T\<^sub>1\T\<^sub>2"
equivariance typing
nominal_inductive typing
by (simp_all add: abs_fresh fresh_ty)
lemma typing_implies_valid:
assumes a:
"\ \ t : T"
shows "valid \"
using a
by (induct) (auto)
lemma t_App_elim:
assumes a:
"\ \ App t1 t2 : T"
obtains T
' where "\ \ t1 : T' → T
" and "Γ
⊨ t2 : T
'"
using a
by (cases) (auto simp add: trm.inject)
lemma t_Lam_elim:
assumes a:
"\ \ Lam [x].t : T" "x\\"
obtains T
🚫1
and T
🚫2
where "(x,T\<^sub>1)#\ \ t : T\<^sub>2" and "T=T\<^sub>1\T\<^sub>2"
using a
by (cases rule: typing.strong_cases [
where x=
"x"])
(auto simp add: abs_fresh fresh_ty alpha trm.inject)
abbreviation
"sub_context" ::
"(name\ty) list \ (name\ty) list \ bool" (
‹_
⊆ _
› [55,55] 55)
where
"\\<^sub>1 \ \\<^sub>2 \ \x T. (x,T)\set \\<^sub>1 \ (x,T)\set \\<^sub>2"
lemma weakening:
fixes Γ
🚫1 Γ
🚫2::
"(name\ty) list"
assumes "\\<^sub>1 \ e: T" and "valid \\<^sub>2" and "\\<^sub>1 \ \\<^sub>2"
shows "\\<^sub>2 \ e: T"
using assms
proof(nominal_induct Γ
🚫1 e T avoiding: Γ
🚫2 rule: typing.strong_induct)
case (t_Lam x Γ
🚫1 T
🚫1 t T
🚫2 Γ
🚫2)
have vc:
"x\\\<^sub>2" by fact
have ih:
"\valid ((x,T\<^sub>1)#\\<^sub>2); (x,T\<^sub>1)#\\<^sub>1 \ (x,T\<^sub>1)#\\<^sub>2\ \ (x,T\<^sub>1)#\\<^sub>2 \ t : T\<^sub>2" by fact
have "valid \\<^sub>2" by fact
then have "valid ((x,T\<^sub>1)#\\<^sub>2)" using vc
by auto
moreover
have "\\<^sub>1 \ \\<^sub>2" by fact
then have "(x,T\<^sub>1)#\\<^sub>1 \ (x,T\<^sub>1)#\\<^sub>2" by simp
ultimately have "(x,T\<^sub>1)#\\<^sub>2 \ t : T\<^sub>2" using ih
by simp
with vc
show "\\<^sub>2 \ Lam [x].t : T\<^sub>1\T\<^sub>2" by auto
qed (auto)
lemma type_substitutivity_aux:
assumes a:
"(\@[(x,T')]@\) \ e : T"
and b:
"\ \ e' : T'"
shows "(\@\) \ e[x::=e'] : T"
using a b
proof (nominal_induct Γ
≡"\@[(x,T')]@\" e T avoiding: e
' \ rule: typing.strong_induct)
case (t_Var y T e
' \)
then have a1:
"valid (\@[(x,T')]@\)"
and a2:
"(y,T) \ set (\@[(x,T')]@\)"
and a3:
"\ \ e' : T'" .
from a1
have a4:
"valid (\@\)" by (rule valid_insert)
{
assume eq:
"x=y"
from a1 a2
have "T=T'" using eq
by (auto intro: context_unique)
with a3
have "\@\ \ Var y[x::=e'] : T" using eq a4
by (auto intro: weakening)
}
moreover
{
assume ineq:
"x\y"
from a2
have "(y,T) \ set (\@\)" using ineq
by simp
then have "\@\ \ Var y[x::=e'] : T" using ineq a4
by auto
}
ultimately show "\@\ \ Var y[x::=e'] : T" by blast
qed (force simp add: fresh_list_append fresh_list_cons)+
corollary type_substitutivity:
assumes a:
"(x,T')#\ \ e : T"
and b:
"\ \ e' : T'"
shows "\ \ e[x::=e'] : T"
using a b type_substitutivity_aux[
where Δ=
"[]"]
by (auto)
text ‹Values
›
inductive
val ::
"trm\bool"
where
v_Lam[intro]:
"val (Lam [x].e)"
equivariance val
lemma not_val_App[simp]:
shows
"\ val (App e\<^sub>1 e\<^sub>2)"
"\ val (Var x)"
by (auto elim: val.cases)
text ‹Big-Step Evaluation
›
inductive
big ::
"trm\trm\bool" (
‹_
⇓ _
› [80,80] 80)
where
b_Lam[intro]:
"Lam [x].e \ Lam [x].e"
| b_App[intro]:
"\x\(e\<^sub>1,e\<^sub>2,e'); e\<^sub>1\Lam [x].e; e\<^sub>2\e\<^sub>2'; e[x::=e\<^sub>2']\e'\ \ App e\<^sub>1 e\<^sub>2 \ e'"
equivariance big
nominal_inductive big
by (simp_all add: abs_fresh)
lemma big_preserves_fresh:
fixes x::
"name"
assumes a:
"e \ e'" "x\e"
shows "x\e'"
using a
by (induct) (auto simp add: abs_fresh fresh_subst)
lemma b_App_elim:
assumes a:
"App e\<^sub>1 e\<^sub>2 \ e'" "x\(e\<^sub>1,e\<^sub>2,e')"
obtains f
🚫1
and f
🚫2
where "e\<^sub>1 \ Lam [x]. f\<^sub>1" "e\<^sub>2 \ f\<^sub>2" "f\<^sub>1[x::=f\<^sub>2] \ e'"
using a
by (cases rule: big.strong_cases[
where x=
"x" and xa=
"x"])
(auto simp add: trm.inject)
lemma subject_reduction:
assumes a:
"e \ e'" and b:
"\ \ e : T"
shows "\ \ e' : T"
using a b
proof (nominal_induct avoiding: Γ arbitrary: T rule: big.strong_induct)
case (b_App x e
🚫1 e
🚫2 e
' e e\<^sub>2' Γ T)
have vc:
"x\\" by fact
have "\ \ App e\<^sub>1 e\<^sub>2 : T" by fact
then obtain T
' where a1: "\ \ e\<^sub>1 : T'→T
" and a2: "Γ
⊨ e
🚫2 : T
'"
by (cases) (auto simp add: trm.inject)
have ih1:
"\ \ e\<^sub>1 : T' \ T \ \ \ Lam [x].e : T' \ T" by fact
have ih2:
"\ \ e\<^sub>2 : T' \ \ \ e\<^sub>2' : T'" by fact
have ih3:
"\ \ e[x::=e\<^sub>2'] : T \ \ \ e' : T" by fact
have "\ \ Lam [x].e : T'\T" using ih1 a1
by simp
then have "((x,T')#\) \ e : T" using vc
by (auto elim: t_Lam_elim simp add: ty.inject)
moreover
have "\ \ e\<^sub>2': T'" using ih2 a2
by simp
ultimately have "\ \ e[x::=e\<^sub>2'] : T" by (simp add: type_substitutivity)
thus "\ \ e' : T" using ih3
by simp
qed (blast)
lemma subject_reduction2:
assumes a:
"e \ e'" and b:
"\ \ e : T"
shows "\ \ e' : T"
using a b
by (nominal_induct avoiding: Γ T rule: big.strong_induct)
(force elim: t_App_elim t_Lam_elim simp add: ty.inject type_substitutivity)+
lemma unicity_of_evaluation:
assumes a:
"e \ e\<^sub>1"
and b:
"e \ e\<^sub>2"
shows "e\<^sub>1 = e\<^sub>2"
using a b
proof (nominal_induct e e
🚫1 avoiding: e
🚫2 rule: big.strong_induct)
case (b_Lam x e t
🚫2)
have "Lam [x].e \ t\<^sub>2" by fact
thus "Lam [x].e = t\<^sub>2" by cases (simp_all add: trm.inject)
next
case (b_App x e
🚫1 e
🚫2 e
' e\<^sub>1' e
🚫2
' t\<^sub>2)
have ih1:
"\t. e\<^sub>1 \ t \ Lam [x].e\<^sub>1' = t" by fact
have ih2:
"\t. e\<^sub>2 \ t \ e\<^sub>2' = t" by fact
have ih3:
"\t. e\<^sub>1'[x::=e\<^sub>2'] \ t \ e' = t" by fact
have app:
"App e\<^sub>1 e\<^sub>2 \ t\<^sub>2" by fact
have vc:
"x\e\<^sub>1" "x\e\<^sub>2" "x\t\<^sub>2" by fact+
then have "x\App e\<^sub>1 e\<^sub>2" by auto
from app vc
obtain f
🚫1 f
🚫2
where x1:
"e\<^sub>1 \ Lam [x]. f\<^sub>1" and x2:
"e\<^sub>2 \ f\<^sub>2" and x3:
"f\<^sub>1[x::=f\<^sub>2] \ t\<^sub>2"
by (auto elim!: b_App_elim)
then have "Lam [x]. f\<^sub>1 = Lam [x]. e\<^sub>1'" using ih1
by simp
then
have "f\<^sub>1 = e\<^sub>1'" by (auto simp add: trm.inject alpha)
moreover
have "f\<^sub>2 = e\<^sub>2'" using x2 ih2
by simp
ultimately have "e\<^sub>1'[x::=e\<^sub>2'] \ t\<^sub>2" using x3
by simp
thus "e' = t\<^sub>2" using ih3
by simp
qed
lemma reduces_evaluates_to_values:
assumes h:
"t \ t'"
shows "val t'"
using h
by (induct) (auto)
(* Valuation *)
nominal_primrec
V ::
"ty \ trm set"
where
"V (TVar x) = {e. val e}"
|
"V (T\<^sub>1 \ T\<^sub>2) = {Lam [x].e | x e. \ v \ (V T\<^sub>1). \ v'. e[x::=v] \ v' \ v' \ V T\<^sub>2}"
by (rule TrueI)+
lemma V_eqvt:
fixes pi::
"name prm"
assumes "x \ V T"
shows "(pi\x) \ V T"
using assms
proof (nominal_induct T arbitrary: pi x rule: ty.strong_induct)
case (TVar nat)
then show ?
case
by (simp add: val.eqvt)
next
case (Arrow T
🚫1 T
🚫2 pi x)
obtain a e
where ae:
"x = Lam [a]. e" "\v\V T\<^sub>1. \v'. e[a::=v] \ v' \ v' \ V T\<^sub>2"
using Arrow.prems
by auto
have "\v'. (pi \ e)[(pi \ a)::=v] \ v' \ v' \ V T\<^sub>2" if v:
"v \ V T\<^sub>1" for v
proof -
have "rev pi \ v \ V T\<^sub>1"
by (simp add: Arrow.hyps(1) v)
then obtain v
' where "e[a::=(rev pi \ v)] \ v'" "v
' \ V T\<^sub>2"
using ae(2)
by blast
then have "(pi \ e)[(pi \ a)::=v] \ pi \ v'"
by (metis (no_types, lifting) big.eqvt cons_eqvt nil_eqvt perm_pi_simp(1) perm_prod.simp
s psubst_eqvt)
then show ?thesis
using Arrow.hyps ‹v' \ V T\<^sub>2\ by blast
qed
with ae show ?case by force
qed
lemma V_arrow_elim_weak:
assumes h:"u \ V (T\<^sub>1 \ T\<^sub>2)"
obtains a t where "u = Lam [a].t" and "\ v \ (V T\<^sub>1). \ v'. t[a::=v] \ v' \ v' \ V T\<^sub>2"
using h by (auto)
lemma V_arrow_elim_strong:
fixes c::"'a::fs_name"
assumes h: "u \ V (T\<^sub>1 \ T\<^sub>2)"
obtains a t where "a\c" "u = Lam [a].t" "\v \ (V T\<^sub>1). \ v'. t[a::=v] \ v' \ v' \ V T\<^sub>2"
proof -
obtain a t where "u = Lam [a].t"
and at: "\v. v \ (V T\<^sub>1) \ \ v'. t[a::=v] \ v' \ v' \ V T\<^sub>2"
using V_arrow_elim_weak [OF assms] by metis
obtain a'::name where a': "a'\(a,t,c)"
by (meson exists_fresh fs_name_class.axioms)
then have "u = Lam [a'].([(a, a')] \ t)"
unfolding ‹u = Lam [a].t›
by (smt (verit) alpha fresh_atm fresh_prod perm_swap trm.inject(2))
moreover
have "\ v'. ([(a, a')] \ t)[a'::=v] \ v' \ v' \ V T\<^sub>2" if "v \ (V T\<^sub>1)" for v
proof -
obtain v' where v': "t[a::=([(a, a')] \ v)] \ v' \ v' \ V T\<^sub>2"
using V_eqvt ‹v ∈ V T🚫1› at by blast
then have "([(a, a')] \ t[a::=([(a, a')] \ v)]) \ [(a, a')] \ v'"
by (simp add: big.eqvt)
then show ?thesis
by (smt (verit) V_eqvt cons_eqvt nil_eqvt perm_prod.simps perm_swap(1) psubst_eqvt swap_simps(1) v')
qed
ultimately show thesis
by (metis fresh_prod that a')
qed
lemma Vs_are_values:
assumes a: "e \ V T"
shows "val e"
using a by (nominal_induct T arbitrary: e rule: ty.strong_induct) (auto)
lemma values_reduce_to_themselves:
assumes a: "val v"
shows "v \ v"
using a by (induct) (auto)
lemma Vs_reduce_to_themselves:
assumes a: "v \ V T"
shows "v \ v"
using a by (simp add: values_reduce_to_themselves Vs_are_values)
text ‹'\ maps x to e' asserts that θ substitutes x with e›
abbreviation
mapsto :: "(name\trm) list \ name \ trm \ bool" (‹_ maps _ to _› [55,55,55] 55)
where
"\ maps x to e \ (lookup \ x) = e"
abbreviation
v_closes :: "(name\trm) list \ (name\ty) list \ bool" (‹_ Vcloses _› [55,55] 55)
where
"\ Vcloses \ \ \x T. (x,T) \ set \ \ (\v. \ maps x to v \ v \ V T)"
lemma case_distinction_on_context:
fixes Γ::"(name\ty) list"
assumes asm1: "valid ((m,t)#\)"
and asm2: "(n,U) \ set ((m,T)#\)"
shows "(n,U) = (m,T) \ ((n,U) \ set \ \ n \ m)"
proof -
from asm2 have "(n,U) \ set [(m,T)] \ (n,U) \ set \" by auto
moreover
{ assume eq: "m=n"
assume "(n,U) \ set \"
then have "\ n\\"
by (induct Γ) (auto simp add: fresh_list_cons fresh_prod fresh_atm)
moreover have "m\\" using asm1 by auto
ultimately have False using eq by auto
}
ultimately show ?thesis by auto
qed
lemma monotonicity:
fixes m::"name"
fixes θ::"(name \ trm) list"
assumes h1: "\ Vcloses \"
and h2: "e \ V T"
and h3: "valid ((x,T)#\)"
shows "(x,e)#\ Vcloses (x,T)#\"
proof(intro strip)
fix x' T'
assume "(x',T') \ set ((x,T)#\)"
then have "((x',T')=(x,T)) \ ((x',T')\set \ \ x'\x)" using h3
by (rule_tac case_distinction_on_context)
moreover
{ (* first case *)
assume "(x',T') = (x,T)"
then have "\e'. ((x,e)#\) maps x to e' \ e' \ V T'" using h2 by auto
}
moreover
{ (* second case *)
assume "(x',T') \ set \" and neq:"x' \ x"
then have "\e'. \ maps x' to e' \ e' \ V T'" using h1 by auto
then have "\e'. ((x,e)#\) maps x' to e' \ e' \ V T'" using neq by auto
}
ultimately show "\e'. ((x,e)#\) maps x' to e' \ e' \ V T'" by blast
qed
lemma termination_aux:
assumes h1: "\ \ e : T"
and h2: "\ Vcloses \"
shows "\v. \ \ v \ v \ V T"
using h2 h1
proof(nominal_induct e avoiding: Γ θ arbitrary: T rule: trm.strong_induct)
case (App e🚫1 e🚫2 Γ θ T)
have ih🚫1: "\\ \ T. \\ Vcloses \; \ \ e\<^sub>1 : T\ \ \v. \1> \ v \ v \ V T" by fact
have ih🚫2: "\\ \ T. \\ Vcloses \; \ \ e\<^sub>2 : T\ \ \v. \2> \ v \ v \ V T" by fact
have as🚫1: "\ Vcloses \" by fact
have as🚫2: "\ \ App e\<^sub>1 e\<^sub>2 : T" by fact
then obtain T' where "\ \ e\<^sub>1 : T' → T" and "Γ ⊨ e🚫2 : T'" by (auto elim: t_App_elim)
then obtain v🚫1 v🚫2 where "(i)": "\1> \ v\<^sub>1" "v\<^sub>1 \ V (T' \ T)"
and "(ii)": "\2> \ v\<^sub>2" "v\<^sub>2 \ V T'" using ih🚫1 ih🚫2 as🚫1 by blast
from "(i)" obtain x e'
where "v\<^sub>1 = Lam [x].e'"
and "(iii)": "(\v \ (V T').\ v'. e'[x::=v] \ v' \ v' \ V T)"
and "(iv)": "\1> \ Lam [x].e'"
and fr: "x\(\,e\<^sub>1,e\<^sub>2)" by (blast elim: V_arrow_elim_strong)
from fr have fr🚫1: "x\\1>" and fr🚫2: "x\\2>" by (simp_all add: fresh_psubst)
from "(ii)" "(iii)" obtain v🚫3 where "(v)": "e'[x::=v\<^sub>2] \ v\<^sub>3" "v\<^sub>3 \ V T" by auto
from fr🚫2 "(ii)" have "x\v\<^sub>2" by (simp add: big_preserves_fresh)
then have "x\e'[x::=v\<^sub>2]" by (simp add: fresh_subst)
then have fr🚫3: "x\v\<^sub>3" using "(v)" by (simp add: big_preserves_fresh)
from fr🚫1 fr🚫2 fr🚫3 have "x\(\1>,\2>,v\<^sub>3)" by simp
with "(iv)" "(ii)" "(v)" have "App (\1>) (\2>) \ v\<^sub>3" by auto
then show "\v. \1 e\<^sub>2> \ v \ v \ V T" using "(v)" by auto
next
case (Lam x e Γ θ T)
have ih:"\\ \ T. \\ Vcloses \; \ \ e : T\ \ \v. \ \ v \ v \ V T" by fact
have as🚫1: "\ Vcloses \" by fact
have as🚫2: "\ \ Lam [x].e : T" by fact
have fs: "x\\" "x\\" by fact+
from as🚫2 fs obtain T🚫1 T🚫2
where "(i)": "(x,T\<^sub>1)#\ \ e:T\<^sub>2" and "(ii)": "T = T\<^sub>1 \ T\<^sub>2" using fs
by (auto elim: t_Lam_elim)
from "(i)" have "(iii)": "valid ((x,T\<^sub>1)#\)" by (simp add: typing_implies_valid)
have "\v \ (V T\<^sub>1). \v'. (\)[x::=v] \ v' \ v' \ V T\<^sub>2"
proof
fix v
assume "v \ (V T\<^sub>1)"
with "(iii)" as🚫1 have "(x,v)#\ Vcloses (x,T\<^sub>1)#\" using monotonicity by auto
with ih "(i)" obtain v' where "((x,v)#\) \ v' ∧ v' \ V T\<^sub>2" by blast
then have "\[x::=v] \ v' \ v' \ V T\<^sub>2" using fs by (simp add: psubst_subst_psubst)
then show "\v'. \[x::=v] \ v' \ v' \ V T\<^sub>2" by auto
qed
then have "Lam[x].\ \ V (T\<^sub>1 \ T\<^sub>2)" by auto
then have "\ \ Lam [x].\ \ Lam [x].\ \ V (T\<^sub>1\T\<^sub>2)" using fs by auto
thus "\v. \ \ v \ v \ V T" using "(ii)" by auto
next
case (Var x Γ θ T)
have "\ \ (Var x) : T" by fact
then have "(x,T)\set \" by (cases) (auto simp add: trm.inject)
with Var have "\ \ \ \ \\ V T"
by (auto intro!: Vs_reduce_to_themselves)
then show "\v. \ \ v \ v \ V T" by auto
qed
theorem termination_of_evaluation:
assumes a: "[] \ e : T"
shows "\v. e \ v \ val v"
proof -
from a have "\v. [] \ v \ v \ V T" by (rule termination_aux) (auto)
thus "\v. e \ v \ val v" using Vs_are_values by auto
qed
end
