--- a/equalities.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,333 +0,0 @@
-(* Title: HOL/equalities
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1994 University of Cambridge
-
-Equalities involving union, intersection, inclusion, etc.
-*)
-
-writeln"File HOL/equalities";
-
-val eq_cs = set_cs addSIs [equalityI];
-
-(** The membership relation, : **)
-
-goal Set.thy "x ~: {}";
-by(fast_tac set_cs 1);
-qed "in_empty";
-
-goal Set.thy "x : insert(y,A) = (x=y | x:A)";
-by(fast_tac set_cs 1);
-qed "in_insert";
-
-(** insert **)
-
-goal Set.thy "!!a. a:A ==> insert(a,A) = A";
-by (fast_tac eq_cs 1);
-qed "insert_absorb";
-
-goal Set.thy "(insert(x,A) <= B) = (x:B & A <= B)";
-by (fast_tac set_cs 1);
-qed "insert_subset";
-
-(** Image **)
-
-goal Set.thy "f``{} = {}";
-by (fast_tac eq_cs 1);
-qed "image_empty";
-
-goal Set.thy "f``insert(a,B) = insert(f(a), f``B)";
-by (fast_tac eq_cs 1);
-qed "image_insert";
-
-(** Binary Intersection **)
-
-goal Set.thy "A Int A = A";
-by (fast_tac eq_cs 1);
-qed "Int_absorb";
-
-goal Set.thy "A Int B = B Int A";
-by (fast_tac eq_cs 1);
-qed "Int_commute";
-
-goal Set.thy "(A Int B) Int C = A Int (B Int C)";
-by (fast_tac eq_cs 1);
-qed "Int_assoc";
-
-goal Set.thy "{} Int B = {}";
-by (fast_tac eq_cs 1);
-qed "Int_empty_left";
-
-goal Set.thy "A Int {} = {}";
-by (fast_tac eq_cs 1);
-qed "Int_empty_right";
-
-goal Set.thy "A Int (B Un C) = (A Int B) Un (A Int C)";
-by (fast_tac eq_cs 1);
-qed "Int_Un_distrib";
-
-goal Set.thy "(A<=B) = (A Int B = A)";
-by (fast_tac (eq_cs addSEs [equalityE]) 1);
-qed "subset_Int_eq";
-
-(** Binary Union **)
-
-goal Set.thy "A Un A = A";
-by (fast_tac eq_cs 1);
-qed "Un_absorb";
-
-goal Set.thy "A Un B = B Un A";
-by (fast_tac eq_cs 1);
-qed "Un_commute";
-
-goal Set.thy "(A Un B) Un C = A Un (B Un C)";
-by (fast_tac eq_cs 1);
-qed "Un_assoc";
-
-goal Set.thy "{} Un B = B";
-by(fast_tac eq_cs 1);
-qed "Un_empty_left";
-
-goal Set.thy "A Un {} = A";
-by(fast_tac eq_cs 1);
-qed "Un_empty_right";
-
-goal Set.thy "insert(a,B) Un C = insert(a,B Un C)";
-by(fast_tac eq_cs 1);
-qed "Un_insert_left";
-
-goal Set.thy "(A Int B) Un C = (A Un C) Int (B Un C)";
-by (fast_tac eq_cs 1);
-qed "Un_Int_distrib";
-
-goal Set.thy
- "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)";
-by (fast_tac eq_cs 1);
-qed "Un_Int_crazy";
-
-goal Set.thy "(A<=B) = (A Un B = B)";
-by (fast_tac (eq_cs addSEs [equalityE]) 1);
-qed "subset_Un_eq";
-
-goal Set.thy "(A <= insert(b,C)) = (A <= C | b:A & A-{b} <= C)";
-by (fast_tac eq_cs 1);
-qed "subset_insert_iff";
-
-goal Set.thy "(A Un B = {}) = (A = {} & B = {})";
-by (fast_tac (eq_cs addEs [equalityCE]) 1);
-qed "Un_empty";
-
-(** Simple properties of Compl -- complement of a set **)
-
-goal Set.thy "A Int Compl(A) = {}";
-by (fast_tac eq_cs 1);
-qed "Compl_disjoint";
-
-goal Set.thy "A Un Compl(A) = {x.True}";
-by (fast_tac eq_cs 1);
-qed "Compl_partition";
-
-goal Set.thy "Compl(Compl(A)) = A";
-by (fast_tac eq_cs 1);
-qed "double_complement";
-
-goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)";
-by (fast_tac eq_cs 1);
-qed "Compl_Un";
-
-goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)";
-by (fast_tac eq_cs 1);
-qed "Compl_Int";
-
-goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))";
-by (fast_tac eq_cs 1);
-qed "Compl_UN";
-
-goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))";
-by (fast_tac eq_cs 1);
-qed "Compl_INT";
-
-(*Halmos, Naive Set Theory, page 16.*)
-
-goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)";
-by (fast_tac (eq_cs addSEs [equalityE]) 1);
-qed "Un_Int_assoc_eq";
-
-
-(** Big Union and Intersection **)
-
-goal Set.thy "Union({}) = {}";
-by (fast_tac eq_cs 1);
-qed "Union_empty";
-
-goal Set.thy "Union(insert(a,B)) = a Un Union(B)";
-by (fast_tac eq_cs 1);
-qed "Union_insert";
-
-goal Set.thy "Union(A Un B) = Union(A) Un Union(B)";
-by (fast_tac eq_cs 1);
-qed "Union_Un_distrib";
-
-goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)";
-by (fast_tac set_cs 1);
-qed "Union_Int_subset";
-
-val prems = goal Set.thy
- "(Union(C) Int A = {}) = (! B:C. B Int A = {})";
-by (fast_tac (eq_cs addSEs [equalityE]) 1);
-qed "Union_disjoint";
-
-goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)";
-by (best_tac eq_cs 1);
-qed "Inter_Un_distrib";
-
-(** Unions and Intersections of Families **)
-
-(*Basic identities*)
-
-goal Set.thy "Union(range(f)) = (UN x.f(x))";
-by (fast_tac eq_cs 1);
-qed "Union_range_eq";
-
-goal Set.thy "Inter(range(f)) = (INT x.f(x))";
-by (fast_tac eq_cs 1);
-qed "Inter_range_eq";
-
-goal Set.thy "Union(B``A) = (UN x:A. B(x))";
-by (fast_tac eq_cs 1);
-qed "Union_image_eq";
-
-goal Set.thy "Inter(B``A) = (INT x:A. B(x))";
-by (fast_tac eq_cs 1);
-qed "Inter_image_eq";
-
-goal Set.thy "!!A. a: A ==> (UN y:A. c) = c";
-by (fast_tac eq_cs 1);
-qed "UN_constant";
-
-goal Set.thy "!!A. a: A ==> (INT y:A. c) = c";
-by (fast_tac eq_cs 1);
-qed "INT_constant";
-
-goal Set.thy "(UN x.B) = B";
-by (fast_tac eq_cs 1);
-qed "UN1_constant";
-
-goal Set.thy "(INT x.B) = B";
-by (fast_tac eq_cs 1);
-qed "INT1_constant";
-
-goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})";
-by (fast_tac eq_cs 1);
-qed "UN_eq";
-
-(*Look: it has an EXISTENTIAL quantifier*)
-goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})";
-by (fast_tac eq_cs 1);
-qed "INT_eq";
-
-(*Distributive laws...*)
-
-goal Set.thy "A Int Union(B) = (UN C:B. A Int C)";
-by (fast_tac eq_cs 1);
-qed "Int_Union";
-
-(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:
- Union of a family of unions **)
-goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)";
-by (fast_tac eq_cs 1);
-qed "Un_Union_image";
-
-(*Equivalent version*)
-goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))";
-by (fast_tac eq_cs 1);
-qed "UN_Un_distrib";
-
-goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)";
-by (fast_tac eq_cs 1);
-qed "Un_Inter";
-
-goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)";
-by (best_tac eq_cs 1);
-qed "Int_Inter_image";
-
-(*Equivalent version*)
-goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))";
-by (fast_tac eq_cs 1);
-qed "INT_Int_distrib";
-
-(*Halmos, Naive Set Theory, page 35.*)
-goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
-by (fast_tac eq_cs 1);
-qed "Int_UN_distrib";
-
-goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
-by (fast_tac eq_cs 1);
-qed "Un_INT_distrib";
-
-goal Set.thy
- "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))";
-by (fast_tac eq_cs 1);
-qed "Int_UN_distrib2";
-
-goal Set.thy
- "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))";
-by (fast_tac eq_cs 1);
-qed "Un_INT_distrib2";
-
-(** Simple properties of Diff -- set difference **)
-
-goal Set.thy "A-A = {}";
-by (fast_tac eq_cs 1);
-qed "Diff_cancel";
-
-goal Set.thy "{}-A = {}";
-by (fast_tac eq_cs 1);
-qed "empty_Diff";
-
-goal Set.thy "A-{} = A";
-by (fast_tac eq_cs 1);
-qed "Diff_empty";
-
-(*NOT SUITABLE FOR REWRITING since {a} == insert(a,0)*)
-goal Set.thy "A - insert(a,B) = A - B - {a}";
-by (fast_tac eq_cs 1);
-qed "Diff_insert";
-
-(*NOT SUITABLE FOR REWRITING since {a} == insert(a,0)*)
-goal Set.thy "A - insert(a,B) = A - {a} - B";
-by (fast_tac eq_cs 1);
-qed "Diff_insert2";
-
-val prems = goal Set.thy "a:A ==> insert(a,A-{a}) = A";
-by (fast_tac (eq_cs addSIs prems) 1);
-qed "insert_Diff";
-
-goal Set.thy "A Int (B-A) = {}";
-by (fast_tac eq_cs 1);
-qed "Diff_disjoint";
-
-goal Set.thy "!!A. A<=B ==> A Un (B-A) = B";
-by (fast_tac eq_cs 1);
-qed "Diff_partition";
-
-goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)";
-by (fast_tac eq_cs 1);
-qed "double_diff";
-
-goal Set.thy "A - (B Un C) = (A-B) Int (A-C)";
-by (fast_tac eq_cs 1);
-qed "Diff_Un";
-
-goal Set.thy "A - (B Int C) = (A-B) Un (A-C)";
-by (fast_tac eq_cs 1);
-qed "Diff_Int";
-
-val set_ss = set_ss addsimps
- [in_empty,in_insert,insert_subset,
- Int_absorb,Int_empty_left,Int_empty_right,
- Un_absorb,Un_empty_left,Un_empty_right,Un_empty,
- UN1_constant,image_empty,
- Compl_disjoint,double_complement,
- Union_empty,Union_insert,empty_subsetI,subset_refl,
- Diff_cancel,empty_Diff,Diff_empty,Diff_disjoint];