(* Title: HOL/Lambda/Type.thy
ID: $Id$
Author: Stefan Berghofer
Copyright 2000 TU Muenchen
Simply-typed lambda terms. Subject reduction and strong normalization
of simply-typed lambda terms. Partly based on a paper proof by Ralph
Matthes.
*)
theory Type = InductTermi:
datatype type =
Atom nat
| Fun type type (infixr "=>" 200)
consts
typing :: "((nat => type) \<times> dB \<times> type) set"
syntax
"_typing" :: "[nat => type, dB, type] => bool" ("_ |- _ : _" [50,50,50] 50)
"_funs" :: "[type list, type] => type" (infixl "=>>" 150)
translations
"env |- t : T" == "(env, t, T) : typing"
"Ts =>> T" == "foldr Fun Ts T"
inductive typing
intros [intro!]
Var: "env x = T ==> env |- Var x : T"
Abs: "(nat_case T env) |- t : U ==> env |- (Abs t) : (T => U)"
App: "env |- s : T => U ==> env |- t : T ==> env |- (s $ t) : U"
inductive_cases [elim!]:
"e |- Var i : T"
"e |- t $ u : T"
"e |- Abs t : T"
consts
"types" :: "[nat => type, dB list, type list] => bool"
primrec
"types e [] Ts = (Ts = [])"
"types e (t # ts) Ts =
(case Ts of
[] => False
| T # Ts => e |- t : T \<and> types e ts Ts)"
inductive_cases [elim!]:
"x # xs : lists S"
declare IT.intros [intro!]
text {* Some tests. *}
lemma "\<exists>T U. e |- Abs (Abs (Abs (Var 1 $ (Var 2 $ Var 1 $ Var 0)))) : T \<and> U = T"
apply (intro exI conjI)
apply force
apply (rule refl)
done
lemma "\<exists>T U. e |- Abs (Abs (Abs (Var 2 $ Var 0 $ (Var 1 $ Var 0)))) : T \<and> U = T";
apply (intro exI conjI)
apply force
apply (rule refl)
done
text {* n-ary function types *}
lemma list_app_typeD [rulify]:
"\<forall>t T. e |- t $$ ts : T --> (\<exists>Ts. e |- t : Ts =>> T \<and> types e ts Ts)"
apply (induct_tac ts)
apply simp
apply (intro strip)
apply simp
apply (erule_tac x = "t $ a" in allE)
apply (erule_tac x = T in allE)
apply (erule impE)
apply assumption
apply (elim exE conjE)
apply (ind_cases "e |- t $ u : T")
apply (rule_tac x = "Ta # Ts" in exI)
apply simp
done
lemma list_app_typeI [rulify]:
"\<forall>t T Ts. e |- t : Ts =>> T --> types e ts Ts --> e |- t $$ ts : T"
apply (induct_tac ts)
apply (intro strip)
apply simp
apply (intro strip)
apply (case_tac Ts)
apply simp
apply simp
apply (erule_tac x = "t $ a" in allE)
apply (erule_tac x = T in allE)
apply (erule_tac x = lista in allE)
apply (erule impE)
apply (erule conjE)
apply (erule typing.App)
apply assumption
apply blast
done
lemma lists_types [rulify]:
"\<forall>Ts. types e ts Ts --> ts : lists {t. \<exists>T. e |- t : T}"
apply (induct_tac ts)
apply (intro strip)
apply (case_tac Ts)
apply simp
apply (rule lists.Nil)
apply simp
apply (intro strip)
apply (case_tac Ts)
apply simp
apply simp
apply (rule lists.Cons)
apply blast
apply blast
done
text {* lifting preserves termination and well-typedness *}
lemma lift_map [rulify, simp]:
"\<forall>t. lift (t $$ ts) i = lift t i $$ map (\<lambda>t. lift t i) ts"
apply (induct_tac ts)
apply simp_all
done
lemma subst_map [rulify, simp]:
"\<forall>t. subst (t $$ ts) u i = subst t u i $$ map (\<lambda>t. subst t u i) ts"
apply (induct_tac ts)
apply simp_all
done
lemma lift_IT [rulify, intro!]:
"t : IT ==> \<forall>i. lift t i : IT"
apply (erule IT.induct)
apply (rule allI)
apply (simp (no_asm))
apply (rule conjI)
apply
(rule impI,
rule IT.Var,
erule lists.induct,
simp (no_asm),
rule lists.Nil,
simp (no_asm),
erule IntE,
rule lists.Cons,
blast,
assumption)+
apply auto
done
lemma lifts_IT [rulify]:
"ts : lists IT --> map (\<lambda>t. lift t 0) ts : lists IT"
apply (induct_tac ts)
apply auto
done
lemma shift_env [simp]:
"nat_case T
(\<lambda>j. if j < i then e j else if j = i then Ua else e (j - 1)) =
(\<lambda>j. if j < Suc i then nat_case T e j else if j = Suc i then Ua
else nat_case T e (j - 1))"
apply (rule ext)
apply (case_tac j)
apply simp
apply (case_tac nat)
apply simp_all
done
lemma lift_type' [rulify]:
"e |- t : T ==> \<forall>i U.
(\<lambda>j. if j < i then e j
else if j = i then U
else e (j - 1)) |- lift t i : T"
apply (erule typing.induct)
apply auto
done
lemma lift_type [intro!]:
"e |- t : T ==> nat_case U e |- lift t 0 : T"
apply (subgoal_tac
"nat_case U e =
(\<lambda>j. if j < 0 then e j
else if j = 0 then U else e (j - 1))")
apply (erule ssubst)
apply (erule lift_type')
apply (rule ext)
apply (case_tac j)
apply simp_all
done
lemma lift_types [rulify]:
"\<forall>Ts. types e ts Ts -->
types (\<lambda>j. if j < i then e j
else if j = i then U
else e (j - 1)) (map (\<lambda>t. lift t i) ts) Ts"
apply (induct_tac ts)
apply simp
apply (intro strip)
apply (case_tac Ts)
apply simp_all
apply (rule lift_type')
apply (erule conjunct1)
done
text {* substitution lemma *}
lemma subst_lemma [rulify]:
"e |- t : T ==> \<forall>e' i U u.
e = (\<lambda>j. if j < i then e' j
else if j = i then U
else e' (j-1)) -->
e' |- u : U --> e' |- t[u/i] : T"
apply (erule typing.induct)
apply (intro strip)
apply (case_tac "x = i")
apply simp
apply (frule linorder_neq_iff [THEN iffD1])
apply (erule disjE)
apply simp
apply (rule typing.Var)
apply assumption
apply (frule order_less_not_sym)
apply (simp only: subst_gt split: split_if add: if_False)
apply (rule typing.Var)
apply assumption
apply fastsimp
apply fastsimp
done
lemma substs_lemma [rulify]:
"e |- u : T ==>
\<forall>Ts. types (\<lambda>j. if j < i then e j
else if j = i then T else e (j - 1)) ts Ts -->
types e (map (\<lambda>t. t[u/i]) ts) Ts"
apply (induct_tac ts)
apply (intro strip)
apply (case_tac Ts)
apply simp
apply simp
apply (intro strip)
apply (case_tac Ts)
apply simp
apply simp
apply (erule conjE)
apply (erule subst_lemma)
apply (rule refl)
apply assumption
done
text {* subject reduction *}
lemma subject_reduction [rulify]:
"e |- t : T ==> \<forall>t'. t -> t' --> e |- t' : T"
apply (erule typing.induct)
apply blast
apply blast
apply (intro strip)
apply (ind_cases "s $ t -> t'")
apply hypsubst
apply (ind_cases "env |- Abs t : T => U")
apply (rule subst_lemma)
apply assumption
prefer 2
apply assumption
apply (rule ext)
apply (case_tac j)
apply auto
done
text {* additional lemmas *}
lemma app_last: "(t $$ ts) $ u = t $$ (ts @ [u])"
apply simp
done
lemma subst_Var_IT [rulify]: "r : IT ==> \<forall>i j. r[Var i/j] : IT"
apply (erule IT.induct)
txt {* @{term Var} *}
apply (intro strip)
apply (simp (no_asm) add: subst_Var)
apply
((rule conjI impI)+,
rule IT.Var,
erule lists.induct,
simp (no_asm),
rule lists.Nil,
simp (no_asm),
erule IntE,
erule CollectE,
rule lists.Cons,
fast,
assumption)+
txt {* @{term Lambda} *}
apply (intro strip)
apply simp
apply (rule IT.Lambda)
apply fast
txt {* @{term Beta} *}
apply (intro strip)
apply (simp (no_asm_use) add: subst_subst [symmetric])
apply (rule IT.Beta)
apply auto
done
lemma Var_IT: "Var n \<in> IT"
apply (subgoal_tac "Var n $$ [] \<in> IT")
apply simp
apply (rule IT.Var)
apply (rule lists.Nil)
done
lemma app_Var_IT: "t : IT ==> t $ Var i : IT"
apply (erule IT.induct)
apply (subst app_last)
apply (rule IT.Var)
apply simp
apply (rule lists.Cons)
apply (rule Var_IT)
apply (rule lists.Nil)
apply (rule IT.Beta [where ?ss = "[]", unfold foldl_Nil [THEN eq_reflection]])
apply (erule subst_Var_IT)
apply (rule Var_IT)
apply (subst app_last)
apply (rule IT.Beta)
apply (subst app_last [symmetric])
apply assumption
apply assumption
done
text {* Well-typed substitution preserves termination. *}
lemma subst_type_IT [rulify]:
"\<forall>t. t : IT --> (\<forall>e T u i.
(\<lambda>j. if j < i then e j
else if j = i then U
else e (j - 1)) |- t : T -->
u : IT --> e |- u : U --> t[u/i] : IT)"
apply (rule_tac f = size and a = U in measure_induct)
apply (rule allI)
apply (rule impI)
apply (erule IT.induct)
txt {* @{term Var} *}
apply (intro strip)
apply (case_tac "n = i")
txt {* @{term "n = i"} *}
apply (case_tac rs)
apply simp
apply simp
apply (drule list_app_typeD)
apply (elim exE conjE)
apply (ind_cases "e |- t $ u : T")
apply (ind_cases "e |- Var i : T")
apply (drule_tac s = "(?T::type) => ?U" in sym)
apply simp
apply (subgoal_tac "lift u 0 $ Var 0 : IT")
prefer 2
apply (rule app_Var_IT)
apply (erule lift_IT)
apply (subgoal_tac "(lift u 0 $ Var 0)[a[u/i]/0] : IT")
apply (simp (no_asm_use))
apply (subgoal_tac "(Var 0 $$ map (\<lambda>t. lift t 0)
(map (\<lambda>t. t[u/i]) list))[(u $ a[u/i])/0] : IT")
apply (simp (no_asm_use) del: map_compose add: map_compose [symmetric] o_def)
apply (erule_tac x = "Ts =>> T" in allE)
apply (erule impE)
apply simp
apply (erule_tac x = "Var 0 $$
map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) list)" in allE)
apply (erule impE)
apply (rule IT.Var)
apply (rule lifts_IT)
apply (drule lists_types)
apply
(ind_cases "x # xs : lists (Collect P)",
erule lists_IntI [THEN lists.induct],
assumption)
apply fastsimp
apply fastsimp
apply (erule_tac x = e in allE)
apply (erule_tac x = T in allE)
apply (erule_tac x = "u $ a[u/i]" in allE)
apply (erule_tac x = 0 in allE)
apply (fastsimp intro!: list_app_typeI lift_types subst_lemma substs_lemma)
apply (erule_tac x = Ta in allE)
apply (erule impE)
apply simp
apply (erule_tac x = "lift u 0 $ Var 0" in allE)
apply (erule impE)
apply assumption
apply (erule_tac x = e in allE)
apply (erule_tac x = "Ts =>> T" in allE)
apply (erule_tac x = "a[u/i]" in allE)
apply (erule_tac x = 0 in allE)
apply (erule impE)
apply (rule typing.App)
apply (erule lift_type')
apply (rule typing.Var)
apply simp
apply (fast intro!: subst_lemma)
txt {* @{term "n ~= i"} *}
apply (drule list_app_typeD)
apply (erule exE)
apply (erule conjE)
apply (drule lists_types)
apply (subgoal_tac "map (\<lambda>x. x[u/i]) rs : lists IT")
apply (simp add: subst_Var)
apply fast
apply (erule lists_IntI [THEN lists.induct])
apply assumption
apply fastsimp
apply fastsimp
txt {* @{term Lambda} *}
apply fastsimp
txt {* @{term Beta} *}
apply (intro strip)
apply (simp (no_asm))
apply (rule IT.Beta)
apply (simp (no_asm) del: subst_map add: subst_subst subst_map [symmetric])
apply (drule subject_reduction)
apply (rule apps_preserves_beta)
apply (rule beta.beta)
apply fast
apply (drule list_app_typeD)
apply fast
done
text {* main theorem: well-typed terms are strongly normalizing *}
lemma type_implies_IT: "e |- t : T ==> t : IT"
apply (erule typing.induct)
apply (rule Var_IT)
apply (erule IT.Lambda)
apply (subgoal_tac "(Var 0 $ lift t 0)[s/0] : IT")
apply simp
apply (rule subst_type_IT)
apply (rule lists.Nil [THEN 2 lists.Cons [THEN IT.Var], unfold foldl_Nil [THEN eq_reflection]
foldl_Cons [THEN eq_reflection]])
apply (erule lift_IT)
apply (rule typing.App)
apply (rule typing.Var)
apply simp
apply (erule lift_type')
apply assumption
apply assumption
done
theorem type_implies_termi: "e |- t : T ==> t : termi beta"
apply (rule IT_implies_termi)
apply (erule type_implies_IT)
done
end