# HG changeset patch
# User wenzelm
# Date 966501292 -7200
# Node ID d9aa8ca06bc2818c68f32b85b5f65575c489c5d1
# Parent  3047ada4bc05300730851721c47003e68a148b0c
converted to new-style theory;

diff -r 3047ada4bc05 -r d9aa8ca06bc2 src/HOL/Lambda/Type.thy
--- a/src/HOL/Lambda/Type.thy	Thu Aug 17 10:34:28 2000 +0200
+++ b/src/HOL/Lambda/Type.thy	Thu Aug 17 10:34:52 2000 +0200
@@ -3,38 +3,482 @@
     Author:     Stefan Berghofer
     Copyright   2000 TU Muenchen
 
-Simply-typed lambda terms.
+Simply-typed lambda terms.  Subject reduction and strong normalization
+of simply-typed lambda terms.  Partly based on a paper proof by Ralph
+Matthes.
 *)
 
-Type = InductTermi +
+theory Type = InductTermi:
 
-datatype typ = Atom nat
-             | Fun typ typ (infixr "=>" 200)
+datatype "typ" =
+    Atom nat
+  | Fun "typ" "typ"     (infixr "=>" 200)
 
 consts
   typing :: "((nat => typ) * dB * typ) set"
 
 syntax
-  "@type" :: "[nat => typ, dB, typ] => bool" ("_ |- _ : _" [50,50,50] 50)
-  "=>>"   :: "[typ list, typ] => typ" (infixl 150)
+  "_typing" :: "[nat => typ, dB, typ] => bool"   ("_ |- _ : _" [50,50,50] 50)
+  "_funs"   :: "[typ list, typ] => typ"         (infixl "=>>" 150)
 
 translations
   "env |- t : T" == "(env, t, T) : typing"
   "Ts =>> T" == "foldr Fun Ts T"
 
+lemmas [intro!] = IT.BetaI IT.LambdaI IT.VarI
+
+(* FIXME
+declare IT.intros [intro!]
+*)
+
 inductive typing
-intrs
-  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"
+intros (* FIXME [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"
+
+lemmas [intro!] = App Abs Var
 
 consts
   "types" :: "[nat => typ, dB list, typ list] => bool"
-
 primrec
   "types e [] Ts = (Ts = [])"
-  "types e (t # ts) Ts = (case Ts of
+  "types e (t # ts) Ts =
+    (case Ts of
       [] => False
     | T # Ts => e |- t : T & types e ts Ts)"
 
+(* FIXME order *)
+inductive_cases [elim!]:
+  "e |- Abs t : T"
+  "e |- t $ u : T"
+  "e |- Var i : T"
+
+inductive_cases [elim!]:
+  "x # xs : lists S"
+
+
+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.VarI,
+       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 (%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 simp
+    apply simp
+    apply fast
+    apply fast
+      (* FIXME 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 {* Var *}
+    apply (intro strip)
+    apply (simp (no_asm) add: subst_Var)
+    apply
+    ((rule conjI impI)+,
+      rule IT.VarI,
+      erule lists.induct,
+      simp (no_asm),
+      rule lists.Nil,
+      simp (no_asm),
+      erule IntE,
+      erule CollectE,
+      rule lists.Cons,
+      fast,
+      assumption)+
+   txt {* Lambda *}
+   apply (intro strip)
+   apply simp
+   apply (rule IT.LambdaI)
+   apply fast
+  txt {* Beta *}
+  apply (intro strip)
+  apply (simp (no_asm_use) add: subst_subst [symmetric])
+  apply (rule IT.BetaI)
+   apply auto
+  done
+
+lemma Var_IT: "Var n \<in> IT"
+  apply (subgoal_tac "Var n $$ [] \<in> IT")
+   apply simp
+  apply (rule IT.VarI)
+  apply (rule lists.Nil)
+  done
+
+lemma app_Var_IT: "t : IT ==> t $ Var i : IT"
+  apply (erule IT.induct)
+    apply (subst app_last)
+    apply (rule IT.VarI)
+    apply simp
+    apply (rule lists.Cons)
+     apply (rule Var_IT)
+    apply (rule lists.Nil)
+   apply (rule IT.BetaI [where ?ss = "[]", unfold foldl_Nil [THEN eq_reflection]])
+    apply (erule subst_Var_IT)
+   apply (rule Var_IT)
+  apply (subst app_last)
+  apply (rule IT.BetaI)
+   apply (subst app_last [symmetric])
+   apply assumption
+  apply assumption
+  done
+
+
+text {* Well-typed substitution preserves termination. *}
+
+lemma subst_type_IT [rulify]:
+  "\<forall>t. t : IT --> (\<forall>e T u i.
+    (\<lambda>j. if j < i then e j
+          else if j = i then U
+          else e (j - 1)) |- t : T -->
+    u : IT --> e |- u : U --> t[u/i] : IT)"
+  apply (rule_tac f = size and a = U in measure_induct)
+  apply (rule allI)
+  apply (rule impI)
+  apply (erule IT.induct)
+    txt {* Var *}
+    apply (intro strip)
+    apply (case_tac "n = i")
+     txt {* n=i *}
+     apply (case_tac rs)
+      apply simp
+     apply simp
+     apply (drule list_app_typeD)
+     apply (elim exE conjE)
+     apply (ind_cases "e |- t $ u : T")
+     apply (ind_cases "e |- Var i : T")
+     apply (drule_tac s = "(?T::typ) => ?U" in sym)
+     apply simp
+     apply (subgoal_tac "lift u 0 $ Var 0 : IT")
+      prefer 2
+      apply (rule app_Var_IT)
+      apply (erule lift_IT)
+     apply (subgoal_tac "(lift u 0 $ Var 0)[a[u/i]/0] : IT")
+      apply (simp (no_asm_use))
+      apply (subgoal_tac "(Var 0 $$ map (%t. lift t 0)
+        (map (%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 (%t. lift t 0) (map (%t. t[u/i]) list)" in allE)
+      apply (erule impE)
+       apply (rule IT.VarI)
+       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)
+
+(* FIXME
+       apply (tactic { * fast_tac (claset()
+  addSIs [thm "list_app_typeI", thm "lift_types", thm "subst_lemma", thm "substs_lemma"]
+  addss simpset()) 1 * }) *)
+
+     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 {* n~=i *}
+    apply (drule list_app_typeD)
+    apply (erule exE)
+    apply (erule conjE)
+    apply (drule lists_types)
+    apply (subgoal_tac "map (%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 {* Lambda *}
+   apply fastsimp
+  txt {* Beta *}
+  apply (intro strip)
+  apply (simp (no_asm))
+  apply (rule IT.BetaI)
+   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.LambdaI)
+  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.VarI], 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