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(* Title: HOL/Lex/MaxPrefix.thy
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1998 TUM
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*)
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14428
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theory MaxPrefix = List_Prefix:
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constdefs
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is_maxpref :: "('a list => bool) => 'a list => 'a list => bool"
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"is_maxpref P xs ys ==
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xs <= ys & (xs=[] | P xs) & (!zs. zs <= ys & P zs --> zs <= xs)"
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types 'a splitter = "'a list => 'a list * 'a list"
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constdefs
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is_maxsplitter :: "('a list => bool) => 'a splitter => bool"
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"is_maxsplitter P f ==
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(!xs ps qs. f xs = (ps,qs) = (xs=ps@qs & is_maxpref P ps xs))"
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consts
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4910
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maxsplit :: "('a list => bool) => 'a list * 'a list => 'a list => 'a splitter"
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primrec
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"maxsplit P res ps [] = (if P ps then (ps,[]) else res)"
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"maxsplit P res ps (q#qs) = maxsplit P (if P ps then (ps,q#qs) else res)
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(ps@[q]) qs"
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declare split_if[split del]
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lemma maxsplit_lemma: "!!(ps::'a list) res.
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(maxsplit P res ps qs = (xs,ys)) =
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(if EX us. us <= qs & P(ps@us) then xs@ys=ps@qs & is_maxpref P xs (ps@qs)
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else (xs,ys)=res)"
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apply(unfold is_maxpref_def)
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apply (induct "qs")
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apply (simp split: split_if)
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apply blast
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apply simp
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apply (erule thin_rl)
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apply clarify
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apply (case_tac "EX us. us <= list & P (ps @ a # us)")
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apply (subgoal_tac "EX us. us <= a # list & P (ps @ us)")
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apply simp
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apply (blast intro: prefix_Cons[THEN iffD2])
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apply (subgoal_tac "~P(ps@[a])")
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prefer 2 apply blast
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apply (simp (no_asm_simp))
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apply (case_tac "EX us. us <= a#list & P (ps @ us)")
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apply simp
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apply clarify
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apply (case_tac "us")
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apply (rule iffI)
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apply (simp add: prefix_Cons prefix_append)
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apply blast
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apply (simp add: prefix_Cons prefix_append)
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apply clarify
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apply (erule disjE)
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apply (fast dest: order_antisym)
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apply clarify
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apply (erule disjE)
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apply clarify
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apply simp
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apply (erule disjE)
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apply clarify
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apply simp
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apply blast
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apply simp
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apply (subgoal_tac "~P(ps)")
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apply (simp (no_asm_simp))
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apply fastsimp
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done
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declare split_if[split add]
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lemma is_maxpref_Nil[simp]:
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"~(? us. us<=xs & P us) ==> is_maxpref P ps xs = (ps = [])"
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apply(unfold is_maxpref_def)
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apply blast
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done
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lemma is_maxsplitter_maxsplit:
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"is_maxsplitter P (%xs. maxsplit P ([],xs) [] xs)"
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apply(unfold is_maxsplitter_def)
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apply (simp add: maxsplit_lemma)
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apply (fastsimp)
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done
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lemmas maxsplit_eq = is_maxsplitter_maxsplit[simplified is_maxsplitter_def]
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end
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