src/HOL/Map.ML
author oheimb
Tue, 02 Jun 1998 15:08:42 +0200
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child 5069 3ea049f7979d
permissions -rw-r--r--
added option_map_o_empty added option_map_o_empty and option_map_o_update to simpset()
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(*  Title:      HOL/Map.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1997 TU Muenchen
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Map lemmas
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*)
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goalw thy [empty_def] "empty k = None";
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by (Simp_tac 1);
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qed "empty_def2";
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Addsimps [empty_def2];
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goalw thy [update_def] "(m[a|->b])x = (if x=a then Some b else m x)";
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by (Simp_tac 1);
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qed "update_def2";
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Addsimps [update_def2];
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qed_goal "update_same" thy "(t[k|->x]) k = Some x" 
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	(K [Simp_tac 1]);
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qed_goal "update_other" thy "!!X. l~=k ==> (t[k|->x]) l = t l"
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	(K [Asm_simp_tac 1]);
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qed_goal "update_triv" thy "!!X. t k = Some x ==> t[k|->x] = t"
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	(K [rtac ext 1, Asm_simp_tac 1]);
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(*Addsimps [update_same, update_other, update_triv];*)
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section "option_map";
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qed_goal "option_map_o_empty" thy 
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         "option_map f o empty = empty" (K [rtac ext 1, Simp_tac 1]);
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qed_goal "option_map_o_update" thy 
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	 "option_map f o m[a|->b] = (option_map f o m)[a|->f b]" 
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	(K [rtac ext 1, Simp_tac 1]);
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Addsimps[option_map_o_empty, option_map_o_update];
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section "++";
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goalw thy [override_def] "m ++ empty = m";
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by (Simp_tac 1);
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qed "override_empty";
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Addsimps [override_empty];
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goalw thy [override_def] "empty ++ m = m";
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by (Simp_tac 1);
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by (rtac ext 1);
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by (split_tac [split_option_case] 1);
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by (Simp_tac 1);
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qed "empty_override";
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Addsimps [empty_override];
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goalw thy [override_def]
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 "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)";
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by (simp_tac (simpset() addsplits [split_option_case]) 1);
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qed_spec_mp "override_Some_iff";
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bind_thm ("override_SomeD", standard(override_Some_iff RS iffD1));
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goalw thy [override_def]
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 "((m ++ n) k = None) = (n k = None & m k = None)";
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by (simp_tac (simpset() addsplits [split_option_case]) 1);
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qed "override_None";
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AddIffs [override_None];
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goalw thy [override_def] "map_of(xs@ys) = map_of ys ++ map_of xs";
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by (induct_tac "xs" 1);
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by (Simp_tac 1);
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by (rtac ext 1);
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by (asm_simp_tac (simpset() addsplits [split_option_case]) 1);
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qed "map_of_append";
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Addsimps [map_of_append];
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goal thy "map_of xs k = Some y --> (k,y):set xs";
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by (list.induct_tac "xs" 1);
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by  (Simp_tac 1);
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by (split_all_tac 1);
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by (Asm_simp_tac 1);
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qed_spec_mp "map_of_SomeD";
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section "dom";
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goalw thy [dom_def] "dom empty = {}";
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by (Simp_tac 1);
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qed "dom_empty";
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Addsimps [dom_empty];
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goalw thy [dom_def] "dom(m[a|->b]) = insert a (dom m)";
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by (Simp_tac 1);
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by (Blast_tac 1);
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qed "dom_update";
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Addsimps [dom_update];
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qed_goalw "finite_dom_map_of" Map.thy [dom_def] "finite (dom (map_of l))" (K[
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	list.induct_tac "l" 1,
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	 ALLGOALS Simp_tac,
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	stac (insert_Collect RS sym) 1,
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	Asm_simp_tac 1]);
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goalw thy [dom_def] "dom(m++n) = dom n Un dom m";
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by (Blast_tac 1);
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qed "dom_override";
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Addsimps [dom_override];
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section "ran";
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goalw thy [ran_def] "ran empty = {}";
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by (Simp_tac 1);
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qed "ran_empty";
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Addsimps [ran_empty];
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goalw thy [ran_def] "!!X. m a = None ==> ran(m[a|->b]) = insert b (ran m)";
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by Auto_tac;
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by (subgoal_tac "~(aa = a)" 1);
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by Auto_tac;
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qed "ran_update";
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Addsimps [ran_update];