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(* Title: HOL/LessThan/LessThan
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1998 University of Cambridge
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lessThan, greaterThan, atLeast, atMost
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*)
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(*** lessThan ***)
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Goalw [lessThan_def] "(i: lessThan k) = (i<k)";
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by (Blast_tac 1);
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qed "lessThan_iff";
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AddIffs [lessThan_iff];
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Goalw [lessThan_def] "lessThan 0 = {}";
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by (Simp_tac 1);
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qed "lessThan_0";
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Addsimps [lessThan_0];
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Goalw [lessThan_def] "lessThan (Suc k) = insert k (lessThan k)";
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by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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by (Blast_tac 1);
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qed "lessThan_Suc";
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Goalw [lessThan_def, atMost_def] "lessThan (Suc k) = atMost k";
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by (simp_tac (simpset() addsimps [less_Suc_eq_le]) 1);
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qed "lessThan_Suc_atMost";
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Goal "finite (lessThan k)";
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by (induct_tac "k" 1);
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by (simp_tac (simpset() addsimps [lessThan_Suc]) 2);
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by Auto_tac;
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qed "finite_lessThan";
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Goal "(UN m. lessThan m) = UNIV";
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by (Blast_tac 1);
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qed "UN_lessThan_UNIV";
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Goalw [lessThan_def, atLeast_def, le_def]
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"-lessThan k = atLeast k";
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by (Blast_tac 1);
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qed "Compl_lessThan";
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Goal "{k} - lessThan k = {k}";
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by (Blast_tac 1);
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qed "single_Diff_lessThan";
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Addsimps [single_Diff_lessThan];
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(*** greaterThan ***)
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Goalw [greaterThan_def] "(i: greaterThan k) = (k<i)";
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by (Blast_tac 1);
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qed "greaterThan_iff";
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AddIffs [greaterThan_iff];
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Goalw [greaterThan_def] "greaterThan 0 = range Suc";
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by (blast_tac (claset() addIs [Suc_pred RS sym]) 1);
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qed "greaterThan_0";
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Addsimps [greaterThan_0];
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Goalw [greaterThan_def] "greaterThan (Suc k) = greaterThan k - {Suc k}";
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by (auto_tac (claset() addEs [linorder_neqE], simpset()));
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qed "greaterThan_Suc";
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Goal "(INT m. greaterThan m) = {}";
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by (Blast_tac 1);
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qed "INT_greaterThan_UNIV";
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Goalw [greaterThan_def, atMost_def, le_def]
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"-greaterThan k = atMost k";
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by (Blast_tac 1);
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qed "Compl_greaterThan";
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Goalw [greaterThan_def, atMost_def, le_def]
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"-atMost k = greaterThan k";
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by (Blast_tac 1);
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qed "Compl_atMost";
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Goal "less_than ^^ {k} = greaterThan k";
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by (Blast_tac 1);
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qed "Image_less_than";
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Addsimps [Compl_greaterThan, Compl_atMost, Image_less_than];
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(*** atLeast ***)
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Goalw [atLeast_def] "(i: atLeast k) = (k<=i)";
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by (Blast_tac 1);
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qed "atLeast_iff";
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AddIffs [atLeast_iff];
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Goalw [atLeast_def, UNIV_def] "atLeast 0 = UNIV";
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by (Simp_tac 1);
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qed "atLeast_0";
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Addsimps [atLeast_0];
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Goalw [atLeast_def] "atLeast (Suc k) = atLeast k - {k}";
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by (simp_tac (simpset() addsimps [Suc_le_eq]) 1);
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by (simp_tac (simpset() addsimps [order_le_less]) 1);
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by (Blast_tac 1);
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qed "atLeast_Suc";
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Goal "(UN m. atLeast m) = UNIV";
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by (Blast_tac 1);
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qed "UN_atLeast_UNIV";
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Goalw [lessThan_def, atLeast_def, le_def]
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"-atLeast k = lessThan k";
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by (Blast_tac 1);
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qed "Compl_atLeast";
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Goal "less_than^-1 ^^ {k} = lessThan k";
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by (Blast_tac 1);
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qed "Image_inverse_less_than";
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Addsimps [Compl_lessThan, Compl_atLeast, Image_inverse_less_than];
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(*** atMost ***)
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Goalw [atMost_def] "(i: atMost k) = (i<=k)";
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by (Blast_tac 1);
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qed "atMost_iff";
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AddIffs [atMost_iff];
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Goalw [atMost_def] "atMost 0 = {0}";
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by (Simp_tac 1);
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qed "atMost_0";
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Addsimps [atMost_0];
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Goalw [atMost_def] "atMost (Suc k) = insert (Suc k) (atMost k)";
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by (simp_tac (simpset() addsimps [less_Suc_eq, order_le_less]) 1);
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by (Blast_tac 1);
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qed "atMost_Suc";
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Goal "finite (atMost k)";
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by (induct_tac "k" 1);
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by (simp_tac (simpset() addsimps [atMost_Suc]) 2);
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by Auto_tac;
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qed "finite_atMost";
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Goal "(UN m. atMost m) = UNIV";
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by (Blast_tac 1);
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qed "UN_atMost_UNIV";
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Goalw [atMost_def, le_def]
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"-atMost k = greaterThan k";
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by (Blast_tac 1);
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qed "Compl_atMost";
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Addsimps [Compl_atMost];
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(*** Combined properties ***)
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Goal "atMost n Int atLeast n = {n}";
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by (blast_tac (claset() addIs [order_antisym]) 1);
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qed "atMost_Int_atLeast";
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(*** Finally, a few set-theoretic laws...
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CAN BOOLEAN SIMPLIFICATION BE AUTOMATED? ***)
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context Set.thy;
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(** Rewrite rules to eliminate A. Conditions can be satisfied by letting B
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be any set including A Int C and contained in A Un C, such as B=A or B=C.
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**)
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Goal "[| A Int C <= B; B <= A Un C |] \
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\ ==> (A Int B) Un (-A Int C) = B Un C";
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by (Blast_tac 1);
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qed "set_cancel1";
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Goal "[| A Int C <= B; B <= A Un C |] \
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\ ==> (A Un B) Int (-A Un C) = B Int C";
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by (Blast_tac 1);
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qed "set_cancel2";
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(*The base B=A*)
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Goal "A Un (-A Int C) = A Un C";
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by (Blast_tac 1);
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qed "set_cancel3";
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Goal "A Int (-A Un C) = A Int C";
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by (Blast_tac 1);
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qed "set_cancel4";
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(*The base B=C*)
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Goal "(A Int C) Un (-A Int C) = C";
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by (Blast_tac 1);
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qed "set_cancel5";
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Goal "(A Un C) Int (-A Un C) = C";
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by (Blast_tac 1);
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qed "set_cancel6";
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Addsimps [set_cancel1, set_cancel2, set_cancel3,
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set_cancel4, set_cancel5, set_cancel6];
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(** More ad-hoc rules **)
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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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Addsimps [Un_Diff_Diff];
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Goal "A Int (B - C) Un C = A Int B Un C";
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by (Blast_tac 1);
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qed "Int_Diff_Un";
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Goal "Union(B) Int A = (UN C:B. C Int A)";
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by (Blast_tac 1);
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qed "Int_Union2";
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Goal "Union(B) Int A = Union((%C. C Int A)``B)";
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by (Blast_tac 1);
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qed "Int_Union_Union";
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