diff -r b4f18ac786fa -r b98909faaea8 src/HOL/Lambda/Type.thy
--- a/src/HOL/Lambda/Type.thy	Mon Sep 06 13:22:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,365 +0,0 @@
-(*  Title:      HOL/Lambda/Type.thy
-    Author:     Stefan Berghofer
-    Copyright   2000 TU Muenchen
-*)
-
-header {* Simply-typed lambda terms *}
-
-theory Type imports ListApplication begin
-
-
-subsection {* Environments *}
-
-definition
-  shift :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"  ("_<_:_>" [90, 0, 0] 91) where
-  "e<i:a> = (\<lambda>j. if j < i then e j else if j = i then a else e (j - 1))"
-
-notation (xsymbols)
-  shift  ("_\<langle>_:_\<rangle>" [90, 0, 0] 91)
-
-notation (HTML output)
-  shift  ("_\<langle>_:_\<rangle>" [90, 0, 0] 91)
-
-lemma shift_eq [simp]: "i = j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = T"
-  by (simp add: shift_def)
-
-lemma shift_gt [simp]: "j < i \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e j"
-  by (simp add: shift_def)
-
-lemma shift_lt [simp]: "i < j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e (j - 1)"
-  by (simp add: shift_def)
-
-lemma shift_commute [simp]: "e\<langle>i:U\<rangle>\<langle>0:T\<rangle> = e\<langle>0:T\<rangle>\<langle>Suc i:U\<rangle>"
-  apply (rule ext)
-  apply (case_tac x)
-   apply simp
-  apply (case_tac nat)
-   apply (simp_all add: shift_def)
-  done
-
-
-subsection {* Types and typing rules *}
-
-datatype type =
-    Atom nat
-  | Fun type type    (infixr "\<Rightarrow>" 200)
-
-inductive typing :: "(nat \<Rightarrow> type) \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool"  ("_ \<turnstile> _ : _" [50, 50, 50] 50)
-  where
-    Var [intro!]: "env x = T \<Longrightarrow> env \<turnstile> Var x : T"
-  | Abs [intro!]: "env\<langle>0:T\<rangle> \<turnstile> t : U \<Longrightarrow> env \<turnstile> Abs t : (T \<Rightarrow> U)"
-  | App [intro!]: "env \<turnstile> s : T \<Rightarrow> U \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U"
-
-inductive_cases typing_elims [elim!]:
-  "e \<turnstile> Var i : T"
-  "e \<turnstile> t \<degree> u : T"
-  "e \<turnstile> Abs t : T"
-
-primrec
-  typings :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
-where
-    "typings e [] Ts = (Ts = [])"
-  | "typings e (t # ts) Ts =
-      (case Ts of
-        [] \<Rightarrow> False
-      | T # Ts \<Rightarrow> e \<turnstile> t : T \<and> typings e ts Ts)"
-
-abbreviation
-  typings_rel :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
-    ("_ ||- _ : _" [50, 50, 50] 50) where
-  "env ||- ts : Ts == typings env ts Ts"
-
-notation (latex)
-  typings_rel  ("_ \<tturnstile> _ : _" [50, 50, 50] 50)
-
-abbreviation
-  funs :: "type list \<Rightarrow> type \<Rightarrow> type"  (infixr "=>>" 200) where
-  "Ts =>> T == foldr Fun Ts T"
-
-notation (latex)
-  funs  (infixr "\<Rrightarrow>" 200)
-
-
-subsection {* Some examples *}
-
-schematic_lemma "e \<turnstile> Abs (Abs (Abs (Var 1 \<degree> (Var 2 \<degree> Var 1 \<degree> Var 0)))) : ?T"
-  by force
-
-schematic_lemma "e \<turnstile> Abs (Abs (Abs (Var 2 \<degree> Var 0 \<degree> (Var 1 \<degree> Var 0)))) : ?T"
-  by force
-
-
-subsection {* Lists of types *}
-
-lemma lists_typings:
-    "e \<tturnstile> ts : Ts \<Longrightarrow> listsp (\<lambda>t. \<exists>T. e \<turnstile> t : T) ts"
-  apply (induct ts arbitrary: Ts)
-   apply (case_tac Ts)
-     apply simp
-     apply (rule listsp.Nil)
-    apply simp
-  apply (case_tac Ts)
-   apply simp
-  apply simp
-  apply (rule listsp.Cons)
-   apply blast
-  apply blast
-  done
-
-lemma types_snoc: "e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<tturnstile> ts @ [t] : Ts @ [T]"
-  apply (induct ts arbitrary: Ts)
-  apply simp
-  apply (case_tac Ts)
-  apply simp+
-  done
-
-lemma types_snoc_eq: "e \<tturnstile> ts @ [t] : Ts @ [T] =
-  (e \<tturnstile> ts : Ts \<and> e \<turnstile> t : T)"
-  apply (induct ts arbitrary: Ts)
-  apply (case_tac Ts)
-  apply simp+
-  apply (case_tac Ts)
-  apply (case_tac "ts @ [t]")
-  apply simp+
-  done
-
-lemma rev_exhaust2 [extraction_expand]:
-  obtains (Nil) "xs = []"  |  (snoc) ys y where "xs = ys @ [y]"
-  -- {* Cannot use @{text rev_exhaust} from the @{text List}
-    theory, since it is not constructive *}
-  apply (subgoal_tac "\<forall>ys. xs = rev ys \<longrightarrow> thesis")
-  apply (erule_tac x="rev xs" in allE)
-  apply simp
-  apply (rule allI)
-  apply (rule impI)
-  apply (case_tac ys)
-  apply simp
-  apply simp
-  apply atomize
-  apply (erule allE)+
-  apply (erule mp, rule conjI)
-  apply (rule refl)+
-  done
-
-lemma types_snocE: "e \<tturnstile> ts @ [t] : Ts \<Longrightarrow>
-  (\<And>Us U. Ts = Us @ [U] \<Longrightarrow> e \<tturnstile> ts : Us \<Longrightarrow> e \<turnstile> t : U \<Longrightarrow> P) \<Longrightarrow> P"
-  apply (cases Ts rule: rev_exhaust2)
-  apply simp
-  apply (case_tac "ts @ [t]")
-  apply (simp add: types_snoc_eq)+
-  apply iprover
-  done
-
-
-subsection {* n-ary function types *}
-
-lemma list_app_typeD:
-    "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> \<exists>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<and> e \<tturnstile> ts : Ts"
-  apply (induct ts arbitrary: t T)
-   apply simp
-  apply atomize
-  apply simp
-  apply (erule_tac x = "t \<degree> a" in allE)
-  apply (erule_tac x = T in allE)
-  apply (erule impE)
-   apply assumption
-  apply (elim exE conjE)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (rule_tac x = "Ta # Ts" in exI)
-  apply simp
-  done
-
-lemma list_app_typeE:
-  "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> (\<And>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> C) \<Longrightarrow> C"
-  by (insert list_app_typeD) fast
-
-lemma list_app_typeI:
-    "e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t \<degree>\<degree> ts : T"
-  apply (induct ts arbitrary: t T Ts)
-   apply simp
-  apply atomize
-  apply (case_tac Ts)
-   apply simp
-  apply simp
-  apply (erule_tac x = "t \<degree> a" in allE)
-  apply (erule_tac x = T in allE)
-  apply (erule_tac x = list in allE)
-  apply (erule impE)
-   apply (erule conjE)
-   apply (erule typing.App)
-   apply assumption
-  apply blast
-  done
-
-text {*
-For the specific case where the head of the term is a variable,
-the following theorems allow to infer the types of the arguments
-without analyzing the typing derivation. This is crucial
-for program extraction.
-*}
-
-theorem var_app_type_eq:
-  "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow> e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> T = U"
-  apply (induct ts arbitrary: T U rule: rev_induct)
-  apply simp
-  apply (ind_cases "e \<turnstile> Var i : T" for T)
-  apply (ind_cases "e \<turnstile> Var i : T" for T)
-  apply simp
-  apply simp
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply atomize
-  apply (erule_tac x="Ta \<Rightarrow> T" in allE)
-  apply (erule_tac x="Tb \<Rightarrow> U" in allE)
-  apply (erule impE)
-  apply assumption
-  apply (erule impE)
-  apply assumption
-  apply simp
-  done
-
-lemma var_app_types: "e \<turnstile> Var i \<degree>\<degree> ts \<degree>\<degree> us : T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow>
-  e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> \<exists>Us. U = Us \<Rrightarrow> T \<and> e \<tturnstile> us : Us"
-  apply (induct us arbitrary: ts Ts U)
-  apply simp
-  apply (erule var_app_type_eq)
-  apply assumption
-  apply simp
-  apply atomize
-  apply (case_tac U)
-  apply (rule FalseE)
-  apply simp
-  apply (erule list_app_typeE)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (drule_tac T="Atom nat" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
-  apply assumption
-  apply simp
-  apply (erule_tac x="ts @ [a]" in allE)
-  apply (erule_tac x="Ts @ [type1]" in allE)
-  apply (erule_tac x="type2" in allE)
-  apply simp
-  apply (erule impE)
-  apply (rule types_snoc)
-  apply assumption
-  apply (erule list_app_typeE)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (drule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
-  apply assumption
-  apply simp
-  apply (erule impE)
-  apply (rule typing.App)
-  apply assumption
-  apply (erule list_app_typeE)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (frule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
-  apply assumption
-  apply simp
-  apply (erule exE)
-  apply (rule_tac x="type1 # Us" in exI)
-  apply simp
-  apply (erule list_app_typeE)
-  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
-  apply (frule_tac T="type1 \<Rightarrow> Us \<Rrightarrow> T" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
-  apply assumption
-  apply simp
-  done
-
-lemma var_app_typesE: "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow>
-  (\<And>Ts. e \<turnstile> Var i : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> P) \<Longrightarrow> P"
-  apply (drule var_app_types [of _ _ "[]", simplified])
-  apply (iprover intro: typing.Var)+
-  done
-
-lemma abs_typeE: "e \<turnstile> Abs t : T \<Longrightarrow> (\<And>U V. e\<langle>0:U\<rangle> \<turnstile> t : V \<Longrightarrow> P) \<Longrightarrow> P"
-  apply (cases T)
-  apply (rule FalseE)
-  apply (erule typing.cases)
-  apply simp_all
-  apply atomize
-  apply (erule_tac x="type1" in allE)
-  apply (erule_tac x="type2" in allE)
-  apply (erule mp)
-  apply (erule typing.cases)
-  apply simp_all
-  done
-
-
-subsection {* Lifting preserves well-typedness *}
-
-lemma lift_type [intro!]: "e \<turnstile> t : T \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> lift t i : T"
-  by (induct arbitrary: i U set: typing) auto
-
-lemma lift_types:
-  "e \<tturnstile> ts : Ts \<Longrightarrow> e\<langle>i:U\<rangle> \<tturnstile> (map (\<lambda>t. lift t i) ts) : Ts"
-  apply (induct ts arbitrary: Ts)
-   apply simp
-  apply (case_tac Ts)
-   apply auto
-  done
-
-
-subsection {* Substitution lemmas *}
-
-lemma subst_lemma:
-    "e \<turnstile> t : T \<Longrightarrow> e' \<turnstile> u : U \<Longrightarrow> e = e'\<langle>i:U\<rangle> \<Longrightarrow> e' \<turnstile> t[u/i] : T"
-  apply (induct arbitrary: e' i U u set: typing)
-    apply (rule_tac x = x and y = i in linorder_cases)
-      apply auto
-  apply blast
-  done
-
-lemma substs_lemma:
-  "e \<turnstile> u : T \<Longrightarrow> e\<langle>i:T\<rangle> \<tturnstile> ts : Ts \<Longrightarrow>
-     e \<tturnstile> (map (\<lambda>t. t[u/i]) ts) : Ts"
-  apply (induct ts arbitrary: Ts)
-   apply (case_tac Ts)
-    apply simp
-   apply simp
-  apply atomize
-  apply (case_tac Ts)
-   apply simp
-  apply simp
-  apply (erule conjE)
-  apply (erule (1) subst_lemma)
-  apply (rule refl)
-  done
-
-
-subsection {* Subject reduction *}
-
-lemma subject_reduction: "e \<turnstile> t : T \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> t' \<Longrightarrow> e \<turnstile> t' : T"
-  apply (induct arbitrary: t' set: typing)
-    apply blast
-   apply blast
-  apply atomize
-  apply (ind_cases "s \<degree> t \<rightarrow>\<^sub>\<beta> t'" for s t t')
-    apply hypsubst
-    apply (ind_cases "env \<turnstile> Abs t : T \<Rightarrow> U" for env t T U)
-    apply (rule subst_lemma)
-      apply assumption
-     apply assumption
-    apply (rule ext)
-    apply (case_tac x)
-     apply auto
-  done
-
-theorem subject_reduction': "t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<turnstile> t' : T"
-  by (induct set: rtranclp) (iprover intro: subject_reduction)+
-
-
-subsection {* Alternative induction rule for types *}
-
-lemma type_induct [induct type]:
-  assumes
-  "(\<And>T. (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T1) \<Longrightarrow>
-    (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T2) \<Longrightarrow> P T)"
-  shows "P T"
-proof (induct T)
-  case Atom
-  show ?case by (rule assms) simp_all
-next
-  case Fun
-  show ?case by (rule assms) (insert Fun, simp_all)
-qed
-
-end