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theory ListBeta(* Title: HOL/Lambda/ListBeta.thy ID: $Id$ Author: Tobias Nipkow Copyright 1998 TU Muenchen *) header {* Lifting beta-reduction to lists *} theory ListBeta imports ListApplication ListOrder begin text {* Lifting beta-reduction to lists of terms, reducing exactly one element. *} abbreviation list_beta :: "dB list => dB list => bool" (infixl "=>" 50) where "rs => ss == step1 beta rs ss" lemma head_Var_reduction: "Var n °° rs ->β v ==> ∃ss. rs => ss ∧ v = Var n °° ss" apply (induct u == "Var n °° rs" v arbitrary: rs set: beta) apply simp apply (rule_tac xs = rs in rev_exhaust) apply simp apply (atomize, force intro: append_step1I) apply (rule_tac xs = rs in rev_exhaust) apply simp apply (auto 0 3 intro: disjI2 [THEN append_step1I]) done lemma apps_betasE [elim!]: assumes major: "r °° rs ->β s" and cases: "!!r'. [| r ->β r'; s = r' °° rs |] ==> R" "!!rs'. [| rs => rs'; s = r °° rs' |] ==> R" "!!t u us. [| r = Abs t; rs = u # us; s = t[u/0] °° us |] ==> R" shows R proof - from major have "(∃r'. r ->β r' ∧ s = r' °° rs) ∨ (∃rs'. rs => rs' ∧ s = r °° rs') ∨ (∃t u us. r = Abs t ∧ rs = u # us ∧ s = t[u/0] °° us)" apply (induct u == "r °° rs" s arbitrary: r rs set: beta) apply (case_tac r) apply simp apply (simp add: App_eq_foldl_conv) apply (split split_if_asm) apply simp apply blast apply simp apply (simp add: App_eq_foldl_conv) apply (split split_if_asm) apply simp apply simp apply (drule App_eq_foldl_conv [THEN iffD1]) apply (split split_if_asm) apply simp apply blast apply (force intro!: disjI1 [THEN append_step1I]) apply (drule App_eq_foldl_conv [THEN iffD1]) apply (split split_if_asm) apply simp apply blast apply (clarify, auto 0 3 intro!: exI intro: append_step1I) done with cases show ?thesis by blast qed lemma apps_preserves_beta [simp]: "r ->β s ==> r °° ss ->β s °° ss" by (induct ss rule: rev_induct) auto lemma apps_preserves_beta2 [simp]: "r ->> s ==> r °° ss ->> s °° ss" apply (induct set: rtranclp) apply blast apply (blast intro: apps_preserves_beta rtranclp.rtrancl_into_rtrancl) done lemma apps_preserves_betas [simp]: "rs => ss ==> r °° rs ->β r °° ss" apply (induct rs arbitrary: ss rule: rev_induct) apply simp apply simp apply (rule_tac xs = ss in rev_exhaust) apply simp apply simp apply (drule Snoc_step1_SnocD) apply blast done end