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src/HOL/equalities.ML
changeset 923 ff1574a81019
child 1179 7678408f9751 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/equalities.ML	Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,333 @@
+(*  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];
isabelle