--- a/src/HOL/Lambda/NormalForm.thy Tue Sep 07 11:51:53 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,247 +0,0 @@
-(* Title: HOL/Lambda/NormalForm.thy
- Author: Stefan Berghofer, TU Muenchen, 2003
-*)
-
-header {* Inductive characterization of lambda terms in normal form *}
-
-theory NormalForm
-imports ListBeta
-begin
-
-subsection {* Terms in normal form *}
-
-definition
- listall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
- "listall P xs \<equiv> (\<forall>i. i < length xs \<longrightarrow> P (xs ! i))"
-
-declare listall_def [extraction_expand_def]
-
-theorem listall_nil: "listall P []"
- by (simp add: listall_def)
-
-theorem listall_nil_eq [simp]: "listall P [] = True"
- by (iprover intro: listall_nil)
-
-theorem listall_cons: "P x \<Longrightarrow> listall P xs \<Longrightarrow> listall P (x # xs)"
- apply (simp add: listall_def)
- apply (rule allI impI)+
- apply (case_tac i)
- apply simp+
- done
-
-theorem listall_cons_eq [simp]: "listall P (x # xs) = (P x \<and> listall P xs)"
- apply (rule iffI)
- prefer 2
- apply (erule conjE)
- apply (erule listall_cons)
- apply assumption
- apply (unfold listall_def)
- apply (rule conjI)
- apply (erule_tac x=0 in allE)
- apply simp
- apply simp
- apply (rule allI)
- apply (erule_tac x="Suc i" in allE)
- apply simp
- done
-
-lemma listall_conj1: "listall (\<lambda>x. P x \<and> Q x) xs \<Longrightarrow> listall P xs"
- by (induct xs) simp_all
-
-lemma listall_conj2: "listall (\<lambda>x. P x \<and> Q x) xs \<Longrightarrow> listall Q xs"
- by (induct xs) simp_all
-
-lemma listall_app: "listall P (xs @ ys) = (listall P xs \<and> listall P ys)"
- apply (induct xs)
- apply (rule iffI, simp, simp)
- apply (rule iffI, simp, simp)
- done
-
-lemma listall_snoc [simp]: "listall P (xs @ [x]) = (listall P xs \<and> P x)"
- apply (rule iffI)
- apply (simp add: listall_app)+
- done
-
-lemma listall_cong [cong, extraction_expand]:
- "xs = ys \<Longrightarrow> listall P xs = listall P ys"
- -- {* Currently needed for strange technical reasons *}
- by (unfold listall_def) simp
-
-text {*
-@{term "listsp"} is equivalent to @{term "listall"}, but cannot be
-used for program extraction.
-*}
-
-lemma listall_listsp_eq: "listall P xs = listsp P xs"
- by (induct xs) (auto intro: listsp.intros)
-
-inductive NF :: "dB \<Rightarrow> bool"
-where
- App: "listall NF ts \<Longrightarrow> NF (Var x \<degree>\<degree> ts)"
-| Abs: "NF t \<Longrightarrow> NF (Abs t)"
-monos listall_def
-
-lemma nat_eq_dec: "\<And>n::nat. m = n \<or> m \<noteq> n"
- apply (induct m)
- apply (case_tac n)
- apply (case_tac [3] n)
- apply (simp only: nat.simps, iprover?)+
- done
-
-lemma nat_le_dec: "\<And>n::nat. m < n \<or> \<not> (m < n)"
- apply (induct m)
- apply (case_tac n)
- apply (case_tac [3] n)
- apply (simp del: simp_thms, iprover?)+
- done
-
-lemma App_NF_D: assumes NF: "NF (Var n \<degree>\<degree> ts)"
- shows "listall NF ts" using NF
- by cases simp_all
-
-
-subsection {* Properties of @{text NF} *}
-
-lemma Var_NF: "NF (Var n)"
- apply (subgoal_tac "NF (Var n \<degree>\<degree> [])")
- apply simp
- apply (rule NF.App)
- apply simp
- done
-
-lemma Abs_NF:
- assumes NF: "NF (Abs t \<degree>\<degree> ts)"
- shows "ts = []" using NF
-proof cases
- case (App us i)
- thus ?thesis by (simp add: Var_apps_neq_Abs_apps [THEN not_sym])
-next
- case (Abs u)
- thus ?thesis by simp
-qed
-
-lemma subst_terms_NF: "listall NF ts \<Longrightarrow>
- listall (\<lambda>t. \<forall>i j. NF (t[Var i/j])) ts \<Longrightarrow>
- listall NF (map (\<lambda>t. t[Var i/j]) ts)"
- by (induct ts) simp_all
-
-lemma subst_Var_NF: "NF t \<Longrightarrow> NF (t[Var i/j])"
- apply (induct arbitrary: i j set: NF)
- apply simp
- apply (frule listall_conj1)
- apply (drule listall_conj2)
- apply (drule_tac i=i and j=j in subst_terms_NF)
- apply assumption
- apply (rule_tac m=x and n=j in nat_eq_dec [THEN disjE, standard])
- apply simp
- apply (erule NF.App)
- apply (rule_tac m=j and n=x in nat_le_dec [THEN disjE, standard])
- apply simp
- apply (iprover intro: NF.App)
- apply simp
- apply (iprover intro: NF.App)
- apply simp
- apply (iprover intro: NF.Abs)
- done
-
-lemma app_Var_NF: "NF t \<Longrightarrow> \<exists>t'. t \<degree> Var i \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
- apply (induct set: NF)
- apply (simplesubst app_last) --{*Using @{text subst} makes extraction fail*}
- apply (rule exI)
- apply (rule conjI)
- apply (rule rtranclp.rtrancl_refl)
- apply (rule NF.App)
- apply (drule listall_conj1)
- apply (simp add: listall_app)
- apply (rule Var_NF)
- apply (rule exI)
- apply (rule conjI)
- apply (rule rtranclp.rtrancl_into_rtrancl)
- apply (rule rtranclp.rtrancl_refl)
- apply (rule beta)
- apply (erule subst_Var_NF)
- done
-
-lemma lift_terms_NF: "listall NF ts \<Longrightarrow>
- listall (\<lambda>t. \<forall>i. NF (lift t i)) ts \<Longrightarrow>
- listall NF (map (\<lambda>t. lift t i) ts)"
- by (induct ts) simp_all
-
-lemma lift_NF: "NF t \<Longrightarrow> NF (lift t i)"
- apply (induct arbitrary: i set: NF)
- apply (frule listall_conj1)
- apply (drule listall_conj2)
- apply (drule_tac i=i in lift_terms_NF)
- apply assumption
- apply (rule_tac m=x and n=i in nat_le_dec [THEN disjE, standard])
- apply simp
- apply (rule NF.App)
- apply assumption
- apply simp
- apply (rule NF.App)
- apply assumption
- apply simp
- apply (rule NF.Abs)
- apply simp
- done
-
-text {*
-@{term NF} characterizes exactly the terms that are in normal form.
-*}
-
-lemma NF_eq: "NF t = (\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t')"
-proof
- assume "NF t"
- then have "\<And>t'. \<not> t \<rightarrow>\<^sub>\<beta> t'"
- proof induct
- case (App ts t)
- show ?case
- proof
- assume "Var t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta> t'"
- then obtain rs where "ts => rs"
- by (iprover dest: head_Var_reduction)
- with App show False
- by (induct rs arbitrary: ts) auto
- qed
- next
- case (Abs t)
- show ?case
- proof
- assume "Abs t \<rightarrow>\<^sub>\<beta> t'"
- then show False using Abs by cases simp_all
- qed
- qed
- then show "\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t'" ..
-next
- assume H: "\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t'"
- then show "NF t"
- proof (induct t rule: Apps_dB_induct)
- case (1 n ts)
- then have "\<forall>ts'. \<not> ts => ts'"
- by (iprover intro: apps_preserves_betas)
- with 1(1) have "listall NF ts"
- by (induct ts) auto
- then show ?case by (rule NF.App)
- next
- case (2 u ts)
- show ?case
- proof (cases ts)
- case Nil
- from 2 have "\<forall>u'. \<not> u \<rightarrow>\<^sub>\<beta> u'"
- by (auto intro: apps_preserves_beta)
- then have "NF u" by (rule 2)
- then have "NF (Abs u)" by (rule NF.Abs)
- with Nil show ?thesis by simp
- next
- case (Cons r rs)
- have "Abs u \<degree> r \<rightarrow>\<^sub>\<beta> u[r/0]" ..
- then have "Abs u \<degree> r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> u[r/0] \<degree>\<degree> rs"
- by (rule apps_preserves_beta)
- with Cons have "Abs u \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta> u[r/0] \<degree>\<degree> rs"
- by simp
- with 2 show ?thesis by iprover
- qed
- qed
-qed
-
-end