author | wenzelm |
Thu, 17 Aug 2000 10:34:52 +0200 | |
changeset 9622 | d9aa8ca06bc2 |
parent 9114 | de99e37effda |
child 9641 | 3b80e7cf6629 |
permissions | -rw-r--r-- |
9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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(* Title: HOL/Lambda/Type.thy |
de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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ID: $Id$ |
de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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Author: Stefan Berghofer |
de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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Copyright 2000 TU Muenchen |
de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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Simply-typed lambda terms. Subject reduction and strong normalization |
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of simply-typed lambda terms. Partly based on a paper proof by Ralph |
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Matthes. |
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9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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*) |
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Subject reduction and strong normalization of simply-typed lambda terms.
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theory Type = InductTermi: |
9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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datatype "typ" = |
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Atom nat |
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| Fun "typ" "typ" (infixr "=>" 200) |
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9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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consts |
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Subject reduction and strong normalization of simply-typed lambda terms.
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typing :: "((nat => typ) * dB * typ) set" |
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Subject reduction and strong normalization of simply-typed lambda terms.
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de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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syntax |
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"_typing" :: "[nat => typ, dB, typ] => bool" ("_ |- _ : _" [50,50,50] 50) |
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"_funs" :: "[typ list, typ] => typ" (infixl "=>>" 150) |
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9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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translations |
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Subject reduction and strong normalization of simply-typed lambda terms.
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"env |- t : T" == "(env, t, T) : typing" |
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Subject reduction and strong normalization of simply-typed lambda terms.
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"Ts =>> T" == "foldr Fun Ts T" |
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Subject reduction and strong normalization of simply-typed lambda terms.
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lemmas [intro!] = IT.BetaI IT.LambdaI IT.VarI |
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(* FIXME |
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declare IT.intros [intro!] |
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*) |
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||
9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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inductive typing |
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intros (* FIXME [intro!] *) |
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Var: "env x = T ==> env |- Var x : T" |
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Abs: "(nat_case T env) |- t : U ==> env |- (Abs t) : (T => U)" |
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App: "env |- s : T => U ==> env |- t : T ==> env |- (s $ t) : U" |
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lemmas [intro!] = App Abs Var |
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9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
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consts |
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Subject reduction and strong normalization of simply-typed lambda terms.
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"types" :: "[nat => typ, dB list, typ list] => bool" |
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Subject reduction and strong normalization of simply-typed lambda terms.
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primrec |
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Subject reduction and strong normalization of simply-typed lambda terms.
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"types e [] Ts = (Ts = [])" |
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"types e (t # ts) Ts = |
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(case Ts of |
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9114
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Subject reduction and strong normalization of simply-typed lambda terms.
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[] => False |
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Subject reduction and strong normalization of simply-typed lambda terms.
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| T # Ts => e |- t : T & types e ts Ts)" |
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Subject reduction and strong normalization of simply-typed lambda terms.
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(* FIXME order *) |
52 |
inductive_cases [elim!]: |
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"e |- Abs t : T" |
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"e |- t $ u : T" |
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"e |- Var i : T" |
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inductive_cases [elim!]: |
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"x # xs : lists S" |
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text {* Some tests. *} |
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lemma "\<exists>T U. e |- Abs (Abs (Abs (Var 1 $ (Var 2 $ Var 1 $ Var 0)))) : T \<and> U = T" |
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apply (intro exI conjI) |
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apply force |
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apply (rule refl) |
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done |
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lemma "\<exists>T U. e |- Abs (Abs (Abs (Var 2 $ Var 0 $ (Var 1 $ Var 0)))) : T \<and> U = T"; |
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apply (intro exI conjI) |
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apply force |
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apply (rule refl) |
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73 |
done |
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text {* n-ary function types *} |
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lemma list_app_typeD [rulify]: |
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"\<forall>t T. e |- t $$ ts : T --> (\<exists>Ts. e |- t : Ts =>> T \<and> types e ts Ts)" |
|
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apply (induct_tac ts) |
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apply simp |
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apply (intro strip) |
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apply simp |
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apply (erule_tac x = "t $ a" in allE) |
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apply (erule_tac x = T in allE) |
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apply (erule impE) |
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apply assumption |
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apply (elim exE conjE) |
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apply (ind_cases "e |- t $ u : T") |
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apply (rule_tac x = "Ta # Ts" in exI) |
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apply simp |
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done |
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lemma list_app_typeI [rulify]: |
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"\<forall>t T Ts. e |- t : Ts =>> T --> types e ts Ts --> e |- t $$ ts : T" |
|
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apply (induct_tac ts) |
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apply (intro strip) |
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apply simp |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp |
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apply simp |
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apply (erule_tac x = "t $ a" in allE) |
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apply (erule_tac x = T in allE) |
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apply (erule_tac x = lista in allE) |
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apply (erule impE) |
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apply (erule conjE) |
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apply (erule typing.App) |
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apply assumption |
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apply blast |
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done |
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lemma lists_types [rulify]: |
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"\<forall>Ts. types e ts Ts --> ts : lists {t. \<exists>T. e |- t : T}" |
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apply (induct_tac ts) |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp |
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apply (rule lists.Nil) |
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apply simp |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp |
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apply simp |
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apply (rule lists.Cons) |
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apply blast |
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apply blast |
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done |
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text {* lifting preserves termination and well-typedness *} |
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lemma lift_map [rulify, simp]: |
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"\<forall>t. lift (t $$ ts) i = lift t i $$ map (\<lambda>t. lift t i) ts" |
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apply (induct_tac ts) |
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apply simp_all |
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done |
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lemma subst_map [rulify, simp]: |
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"\<forall>t. subst (t $$ ts) u i = subst t u i $$ map (\<lambda>t. subst t u i) ts" |
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apply (induct_tac ts) |
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apply simp_all |
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done |
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lemma lift_IT [rulify, intro!]: |
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"t : IT ==> \<forall>i. lift t i : IT" |
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apply (erule IT.induct) |
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apply (rule allI) |
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apply (simp (no_asm)) |
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apply (rule conjI) |
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apply |
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(rule impI, |
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rule IT.VarI, |
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erule lists.induct, |
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simp (no_asm), |
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rule lists.Nil, |
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simp (no_asm), |
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erule IntE, |
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rule lists.Cons, |
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blast, |
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assumption)+ |
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apply auto |
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done |
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lemma lifts_IT [rulify]: |
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"ts : lists IT --> map (\<lambda>t. lift t 0) ts : lists IT" |
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apply (induct_tac ts) |
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apply auto |
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done |
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lemma shift_env [simp]: |
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"nat_case T |
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(\<lambda>j. if j < i then e j else if j = i then Ua else e (j - 1)) = |
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(\<lambda>j. if j < Suc i then nat_case T e j else if j = Suc i then Ua |
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else nat_case T e (j - 1))" |
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apply (rule ext) |
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apply (case_tac j) |
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apply simp |
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apply (case_tac nat) |
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apply simp_all |
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done |
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lemma lift_type' [rulify]: |
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"e |- t : T ==> \<forall>i U. |
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(\<lambda>j. if j < i then e j |
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else if j = i then U |
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else e (j - 1)) |- lift t i : T" |
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apply (erule typing.induct) |
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apply auto |
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done |
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lemma lift_type [intro!]: |
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"e |- t : T ==> nat_case U e |- lift t 0 : T" |
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apply (subgoal_tac |
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"nat_case U e = |
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(\<lambda>j. if j < 0 then e j |
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else if j = 0 then U else e (j - 1))") |
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apply (erule ssubst) |
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apply (erule lift_type') |
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apply (rule ext) |
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apply (case_tac j) |
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apply simp_all |
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done |
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lemma lift_types [rulify]: |
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"\<forall>Ts. types e ts Ts --> |
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types (\<lambda>j. if j < i then e j |
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else if j = i then U |
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else e (j - 1)) (map (\<lambda>t. lift t i) ts) Ts" |
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apply (induct_tac ts) |
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apply simp |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp_all |
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apply (rule lift_type') |
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apply (erule conjunct1) |
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done |
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text {* substitution lemma *} |
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lemma subst_lemma [rulify]: |
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"e |- t : T ==> \<forall>e' i U u. |
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e = (\<lambda>j. if j < i then e' j |
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else if j = i then U |
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else e' (j-1)) --> |
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e' |- u : U --> e' |- t[u/i] : T" |
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apply (erule typing.induct) |
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apply (intro strip) |
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apply (case_tac "x = i") |
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apply simp |
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apply (frule linorder_neq_iff [THEN iffD1]) |
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apply (erule disjE) |
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apply simp |
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apply (rule typing.Var) |
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apply assumption |
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apply (frule order_less_not_sym) |
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apply (simp only: subst_gt split: split_if add: if_False) |
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apply (rule typing.Var) |
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apply assumption |
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apply fastsimp |
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apply fastsimp |
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done |
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lemma substs_lemma [rulify]: |
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"e |- u : T ==> |
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\<forall>Ts. types (\<lambda>j. if j < i then e j |
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else if j = i then T else e (j - 1)) ts Ts --> |
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types e (map (%t. t[u/i]) ts) Ts" |
|
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apply (induct_tac ts) |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp |
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apply simp |
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apply (intro strip) |
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apply (case_tac Ts) |
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apply simp |
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apply simp |
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apply (erule conjE) |
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apply (erule subst_lemma) |
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apply (rule refl) |
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apply assumption |
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done |
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text {* subject reduction *} |
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lemma subject_reduction [rulify]: |
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"e |- t : T ==> \<forall>t'. t -> t' --> e |- t' : T" |
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apply (erule typing.induct) |
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apply blast |
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apply blast |
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apply (intro strip) |
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apply (ind_cases "s $ t -> t'") |
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apply hypsubst |
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apply (ind_cases "env |- Abs t : T => U") |
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apply (rule subst_lemma) |
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apply assumption |
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prefer 2 |
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apply assumption |
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apply (rule ext) |
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apply (case_tac j) |
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apply simp |
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apply simp |
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apply fast |
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apply fast |
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(* FIXME apply auto *) |
|
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done |
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text {* additional lemmas *} |
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lemma app_last: "(t $$ ts) $ u = t $$ (ts @ [u])" |
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apply simp |
|
297 |
done |
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298 |
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lemma subst_Var_IT [rulify]: "r : IT ==> \<forall>i j. r[Var i/j] : IT" |
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apply (erule IT.induct) |
|
302 |
txt {* Var *} |
|
303 |
apply (intro strip) |
|
304 |
apply (simp (no_asm) add: subst_Var) |
|
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apply |
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306 |
((rule conjI impI)+, |
|
307 |
rule IT.VarI, |
|
308 |
erule lists.induct, |
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simp (no_asm), |
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rule lists.Nil, |
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simp (no_asm), |
|
312 |
erule IntE, |
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313 |
erule CollectE, |
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rule lists.Cons, |
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315 |
fast, |
|
316 |
assumption)+ |
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txt {* Lambda *} |
|
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apply (intro strip) |
|
319 |
apply simp |
|
320 |
apply (rule IT.LambdaI) |
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321 |
apply fast |
|
322 |
txt {* Beta *} |
|
323 |
apply (intro strip) |
|
324 |
apply (simp (no_asm_use) add: subst_subst [symmetric]) |
|
325 |
apply (rule IT.BetaI) |
|
326 |
apply auto |
|
327 |
done |
|
328 |
||
329 |
lemma Var_IT: "Var n \<in> IT" |
|
330 |
apply (subgoal_tac "Var n $$ [] \<in> IT") |
|
331 |
apply simp |
|
332 |
apply (rule IT.VarI) |
|
333 |
apply (rule lists.Nil) |
|
334 |
done |
|
335 |
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336 |
lemma app_Var_IT: "t : IT ==> t $ Var i : IT" |
|
337 |
apply (erule IT.induct) |
|
338 |
apply (subst app_last) |
|
339 |
apply (rule IT.VarI) |
|
340 |
apply simp |
|
341 |
apply (rule lists.Cons) |
|
342 |
apply (rule Var_IT) |
|
343 |
apply (rule lists.Nil) |
|
344 |
apply (rule IT.BetaI [where ?ss = "[]", unfold foldl_Nil [THEN eq_reflection]]) |
|
345 |
apply (erule subst_Var_IT) |
|
346 |
apply (rule Var_IT) |
|
347 |
apply (subst app_last) |
|
348 |
apply (rule IT.BetaI) |
|
349 |
apply (subst app_last [symmetric]) |
|
350 |
apply assumption |
|
351 |
apply assumption |
|
352 |
done |
|
353 |
||
354 |
||
355 |
text {* Well-typed substitution preserves termination. *} |
|
356 |
||
357 |
lemma subst_type_IT [rulify]: |
|
358 |
"\<forall>t. t : IT --> (\<forall>e T u i. |
|
359 |
(\<lambda>j. if j < i then e j |
|
360 |
else if j = i then U |
|
361 |
else e (j - 1)) |- t : T --> |
|
362 |
u : IT --> e |- u : U --> t[u/i] : IT)" |
|
363 |
apply (rule_tac f = size and a = U in measure_induct) |
|
364 |
apply (rule allI) |
|
365 |
apply (rule impI) |
|
366 |
apply (erule IT.induct) |
|
367 |
txt {* Var *} |
|
368 |
apply (intro strip) |
|
369 |
apply (case_tac "n = i") |
|
370 |
txt {* n=i *} |
|
371 |
apply (case_tac rs) |
|
372 |
apply simp |
|
373 |
apply simp |
|
374 |
apply (drule list_app_typeD) |
|
375 |
apply (elim exE conjE) |
|
376 |
apply (ind_cases "e |- t $ u : T") |
|
377 |
apply (ind_cases "e |- Var i : T") |
|
378 |
apply (drule_tac s = "(?T::typ) => ?U" in sym) |
|
379 |
apply simp |
|
380 |
apply (subgoal_tac "lift u 0 $ Var 0 : IT") |
|
381 |
prefer 2 |
|
382 |
apply (rule app_Var_IT) |
|
383 |
apply (erule lift_IT) |
|
384 |
apply (subgoal_tac "(lift u 0 $ Var 0)[a[u/i]/0] : IT") |
|
385 |
apply (simp (no_asm_use)) |
|
386 |
apply (subgoal_tac "(Var 0 $$ map (%t. lift t 0) |
|
387 |
(map (%t. t[u/i]) list))[(u $ a[u/i])/0] : IT") |
|
388 |
apply (simp (no_asm_use) del: map_compose add: map_compose [symmetric] o_def) |
|
389 |
apply (erule_tac x = "Ts =>> T" in allE) |
|
390 |
apply (erule impE) |
|
391 |
apply simp |
|
392 |
apply (erule_tac x = "Var 0 $$ |
|
393 |
map (%t. lift t 0) (map (%t. t[u/i]) list)" in allE) |
|
394 |
apply (erule impE) |
|
395 |
apply (rule IT.VarI) |
|
396 |
apply (rule lifts_IT) |
|
397 |
apply (drule lists_types) |
|
398 |
apply |
|
399 |
(ind_cases "x # xs : lists (Collect P)", |
|
400 |
erule lists_IntI [THEN lists.induct], |
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401 |
assumption) |
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402 |
apply fastsimp |
|
403 |
apply fastsimp |
|
404 |
apply (erule_tac x = e in allE) |
|
405 |
apply (erule_tac x = T in allE) |
|
406 |
apply (erule_tac x = "u $ a[u/i]" in allE) |
|
407 |
apply (erule_tac x = 0 in allE) |
|
408 |
apply (fastsimp intro!: list_app_typeI lift_types subst_lemma substs_lemma) |
|
409 |
||
410 |
(* FIXME |
|
411 |
apply (tactic { * fast_tac (claset() |
|
412 |
addSIs [thm "list_app_typeI", thm "lift_types", thm "subst_lemma", thm "substs_lemma"] |
|
413 |
addss simpset()) 1 * }) *) |
|
414 |
||
415 |
apply (erule_tac x = Ta in allE) |
|
416 |
apply (erule impE) |
|
417 |
apply simp |
|
418 |
apply (erule_tac x = "lift u 0 $ Var 0" in allE) |
|
419 |
apply (erule impE) |
|
420 |
apply assumption |
|
421 |
apply (erule_tac x = e in allE) |
|
422 |
apply (erule_tac x = "Ts =>> T" in allE) |
|
423 |
apply (erule_tac x = "a[u/i]" in allE) |
|
424 |
apply (erule_tac x = 0 in allE) |
|
425 |
apply (erule impE) |
|
426 |
apply (rule typing.App) |
|
427 |
apply (erule lift_type') |
|
428 |
apply (rule typing.Var) |
|
429 |
apply simp |
|
430 |
apply (fast intro!: subst_lemma) |
|
431 |
txt {* n~=i *} |
|
432 |
apply (drule list_app_typeD) |
|
433 |
apply (erule exE) |
|
434 |
apply (erule conjE) |
|
435 |
apply (drule lists_types) |
|
436 |
apply (subgoal_tac "map (%x. x[u/i]) rs : lists IT") |
|
437 |
apply (simp add: subst_Var) |
|
438 |
apply fast |
|
439 |
apply (erule lists_IntI [THEN lists.induct]) |
|
440 |
apply assumption |
|
441 |
apply fastsimp |
|
442 |
apply fastsimp |
|
443 |
txt {* Lambda *} |
|
444 |
apply fastsimp |
|
445 |
txt {* Beta *} |
|
446 |
apply (intro strip) |
|
447 |
apply (simp (no_asm)) |
|
448 |
apply (rule IT.BetaI) |
|
449 |
apply (simp (no_asm) del: subst_map add: subst_subst subst_map [symmetric]) |
|
450 |
apply (drule subject_reduction) |
|
451 |
apply (rule apps_preserves_beta) |
|
452 |
apply (rule beta.beta) |
|
453 |
apply fast |
|
454 |
apply (drule list_app_typeD) |
|
455 |
apply fast |
|
456 |
done |
|
457 |
||
458 |
||
459 |
text {* main theorem: well-typed terms are strongly normalizing *} |
|
460 |
||
461 |
lemma type_implies_IT: "e |- t : T ==> t : IT" |
|
462 |
apply (erule typing.induct) |
|
463 |
apply (rule Var_IT) |
|
464 |
apply (erule IT.LambdaI) |
|
465 |
apply (subgoal_tac "(Var 0 $ lift t 0)[s/0] : IT") |
|
466 |
apply simp |
|
467 |
apply (rule subst_type_IT) |
|
468 |
apply (rule lists.Nil [THEN 2 lists.Cons [THEN IT.VarI], unfold foldl_Nil [THEN eq_reflection] |
|
469 |
foldl_Cons [THEN eq_reflection]]) |
|
470 |
apply (erule lift_IT) |
|
471 |
apply (rule typing.App) |
|
472 |
apply (rule typing.Var) |
|
473 |
apply simp |
|
474 |
apply (erule lift_type') |
|
475 |
apply assumption |
|
476 |
apply assumption |
|
477 |
done |
|
478 |
||
479 |
theorem type_implies_termi: "e |- t : T ==> t : termi beta" |
|
480 |
apply (rule IT_implies_termi) |
|
481 |
apply (erule type_implies_IT) |
|
482 |
done |
|
483 |
||
9114
de99e37effda
Subject reduction and strong normalization of simply-typed lambda terms.
berghofe
parents:
diff
changeset
|
484 |
end |