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(* Title: HOL/UNITY/UNITYMisc.ML
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ID: $Id$
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Author: Sidi O Ehmety, Computer Laboratory
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Copyright 2001 University of Cambridge
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Some miscellaneous and add-hoc set theory concepts.
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*)
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Goalw [measure_def, less_than_def]
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"less_than(A) = {<x,y>:A*A. x<y}";
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by Auto_tac;
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qed "less_than_equals";
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Goalw [less_than_def] "wf(less_than(A))";
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by (rtac wf_measure 1);
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qed "wf_less_than";
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Goalw [less_than_def, measure_def]
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"less_than(A)<= A*A";
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by Auto_tac;
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qed "less_than_subset";
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Goalw [less_than_def, measure_def]
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"<x,y>:less_than(A) <-> (x:A & y:A & x<y)";
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by Auto_tac;
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qed "less_than_iff";
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Goalw [lessThan_def]
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"i:lessThan(k,A) <-> i:A & i<k";
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by Auto_tac;
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qed "lessThan_iff";
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Goalw [greaterThan_def]
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"i:greaterThan(k,A) <-> i:A & k<i";
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by Auto_tac;
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qed "greaterThan_iff";
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(** Needed for WF reasoning in WFair.ML **)
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Goal "k:A ==>less_than(A)``{k} = greaterThan(k, A)";
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by (rtac equalityI 1);
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by (auto_tac (claset(), simpset() addsimps
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[less_than_iff,greaterThan_iff]));
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qed "Image_less_than";
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Goal "k:A ==> less_than(A)-`` {k} = lessThan(k, A)";
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by (rtac equalityI 1);
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by (auto_tac (claset(), simpset() addsimps
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[less_than_iff,lessThan_iff]));
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qed "Image_inverse_less_than";
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Addsimps [Image_less_than, Image_inverse_less_than];
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(** Ad-hoc set-theory rules **)
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Goal "Union(B) Int A = (UN b:B. b Int A)";
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by Auto_tac;
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qed "Int_Union_Union";
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Goal "A Int Union(B) = (UN b:B. A Int b)";
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by Auto_tac;
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qed "Int_Union_Union2";
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Goal "A Un B - (A - B) = B";
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by (Blast_tac 1);
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qed "Un_Diff_Diff";
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AddIffs [Un_Diff_Diff];
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(** Needed in State theory for the current definition of variables
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where they are indexed by lists **)
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Goal "i:list(nat) ==> i:univ(0)";
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by (dres_inst_tac [("B", "0")] list_into_univ 1);
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by (blast_tac (claset() addIs [nat_into_univ]) 1);
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by (assume_tac 1);
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qed "list_nat_into_univ";
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(** To be moved to Update theory **)
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Goalw [update_def]
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"[| f:Pi(A,B); x:A; y:B(x) |] ==> f(x:=y):Pi(A, B)";
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by (asm_simp_tac (simpset() addsimps [domain_of_fun, cons_absorb,
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apply_funtype, lam_type]) 1);
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qed "update_type2";
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(** Simplication rules for Collect; To be moved elsewhere **)
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Goal "{x:A. P(x)} Int A = {x:A. P(x)}";
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by Auto_tac;
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qed "Collect_Int2";
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Goal "A Int {x:A. P(x)} = {x:A. P(x)}";
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by Auto_tac;
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qed "Collect_Int3";
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AddIffs [Collect_Int2, Collect_Int3];
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Goal "{x:A. P(x) | Q(x)} = Collect(A, P) Un Collect(A, Q)";
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by Auto_tac;
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qed "Collect_disj_eq";
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Goal "{x:A. P(x) & Q(x)} = Collect(A, P) Int Collect(A, Q)";
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by Auto_tac;
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qed "Collect_conj_eq";
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