(* Title: HOL/Lex/MaxPrefix.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1998 TUM
*)
Delsplits [split_if];
Goalw [is_maxpref_def] "!(ps::'a list) res. \
\ (maxsplit P res ps qs = (xs,ys)) = \
\ (if ? us. us <= qs & P(ps@us) then xs@ys=ps@qs & is_maxpref P xs (ps@qs) \
\ else (xs,ys)=res)";
by (induct_tac "qs" 1);
by (simp_tac (simpset() addsplits [split_if]) 1);
by (Blast_tac 1);
by (Asm_simp_tac 1);
by (etac thin_rl 1);
by (Clarify_tac 1);
by (case_tac "? us. us <= list & P (ps @ a # us)" 1);
by (Asm_simp_tac 1);
by (subgoal_tac "? us. us <= a # list & P (ps @ us)" 1);
by (Asm_simp_tac 1);
by (blast_tac (claset() addIs [prefix_Cons RS iffD2]) 1);
by (subgoal_tac "~P(ps@[a])" 1);
by (Blast_tac 2);
by (Asm_simp_tac 1);
by (case_tac "? us. us <= a#list & P (ps @ us)" 1);
by (Asm_simp_tac 1);
by (Clarify_tac 1);
by (case_tac "us" 1);
by (rtac iffI 1);
by (asm_full_simp_tac (simpset() addsimps [prefix_Cons,prefix_append]) 1);
by (Blast_tac 1);
by (asm_full_simp_tac (simpset() addsimps [prefix_Cons,prefix_append]) 1);
by (Clarify_tac 1);
by (etac disjE 1);
by (fast_tac (claset() addDs [prefix_antisym]) 1);
by (Clarify_tac 1);
by (etac disjE 1);
by (Clarify_tac 1);
by (Asm_full_simp_tac 1);
by (etac disjE 1);
by (Clarify_tac 1);
by (Asm_full_simp_tac 1);
by (Blast_tac 1);
by (Asm_full_simp_tac 1);
by (subgoal_tac "~P(ps)" 1);
by (Asm_simp_tac 1);
by (fast_tac (claset() addss simpset()) 1);
qed_spec_mp "maxsplit_lemma";
Addsplits [split_if];
Goalw [is_maxpref_def]
"~(? us. us<=xs & P us) ==> is_maxpref P ps xs = (ps = [])";
by (Blast_tac 1);
qed "is_maxpref_Nil";
Addsimps [is_maxpref_Nil];
Goalw [is_maxsplitter_def]
"is_maxsplitter P (%xs. maxsplit P ([],xs) [] xs)";
by (simp_tac (simpset() addsimps [maxsplit_lemma]) 1);
by (fast_tac (claset() addss simpset()) 1);
qed "is_maxsplitter_maxsplit";
val maxsplit_eq = rewrite_rule [is_maxsplitter_def] is_maxsplitter_maxsplit;