--- a/src/ZF/equalities.ML Mon May 20 11:45:57 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,765 +0,0 @@
-(* Title: ZF/equalities
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1992 University of Cambridge
-
-Set Theory examples: Union, Intersection, Inclusion, etc.
- (Thanks also to Philippe de Groote.)
-*)
-
-(** Finite Sets **)
-
-(* cons_def refers to Upair; reversing the equality LOOPS in rewriting!*)
-Goal "{a} Un B = cons(a,B)";
-by (Blast_tac 1);
-qed "cons_eq";
-
-Goal "cons(a, cons(b, C)) = cons(b, cons(a, C))";
-by (Blast_tac 1);
-qed "cons_commute";
-
-Goal "a: B ==> cons(a,B) = B";
-by (Blast_tac 1);
-qed "cons_absorb";
-
-Goal "a: B ==> cons(a, B-{a}) = B";
-by (Blast_tac 1);
-qed "cons_Diff";
-
-Goal "[| a: C; ALL y:C. y=b |] ==> C = {b}";
-by (Blast_tac 1);
-qed "equal_singleton_lemma";
-bind_thm ("equal_singleton", ballI RSN (2,equal_singleton_lemma));
-
-
-(** Binary Intersection **)
-
-(*NOT an equality, but it seems to belong here...*)
-Goal "cons(a,B) Int C <= cons(a, B Int C)";
-by (Blast_tac 1);
-qed "Int_cons";
-
-Goal "A Int A = A";
-by (Blast_tac 1);
-qed "Int_absorb";
-Addsimps [Int_absorb];
-
-Goal "A Int (A Int B) = A Int B";
-by (Blast_tac 1);
-qed "Int_left_absorb";
-
-Goal "A Int B = B Int A";
-by (Blast_tac 1);
-qed "Int_commute";
-
-Goal "A Int (B Int C) = B Int (A Int C)";
-by (Blast_tac 1);
-qed "Int_left_commute";
-
-Goal "(A Int B) Int C = A Int (B Int C)";
-by (Blast_tac 1);
-qed "Int_assoc";
-
-(*Intersection is an AC-operator*)
-bind_thms ("Int_ac",
- [Int_assoc, Int_left_absorb, Int_commute, Int_left_commute]);
-
-Goal "A Int (B Un C) = (A Int B) Un (A Int C)";
-by (Blast_tac 1);
-qed "Int_Un_distrib";
-
-Goal "(B Un C) Int A = (B Int A) Un (C Int A)";
-by (Blast_tac 1);
-qed "Int_Un_distrib2";
-
-Goal "A<=B <-> A Int B = A";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "subset_Int_iff";
-
-Goal "A<=B <-> B Int A = A";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "subset_Int_iff2";
-
-Goal "C<=A ==> (A-B) Int C = C-B";
-by (Blast_tac 1);
-qed "Int_Diff_eq";
-
-(** Binary Union **)
-
-Goal "cons(a,B) Un C = cons(a, B Un C)";
-by (Blast_tac 1);
-qed "Un_cons";
-
-Goal "A Un A = A";
-by (Blast_tac 1);
-qed "Un_absorb";
-Addsimps [Un_absorb];
-
-Goal "A Un (A Un B) = A Un B";
-by (Blast_tac 1);
-qed "Un_left_absorb";
-
-Goal "A Un B = B Un A";
-by (Blast_tac 1);
-qed "Un_commute";
-
-Goal "A Un (B Un C) = B Un (A Un C)";
-by (Blast_tac 1);
-qed "Un_left_commute";
-
-Goal "(A Un B) Un C = A Un (B Un C)";
-by (Blast_tac 1);
-qed "Un_assoc";
-
-(*Union is an AC-operator*)
-bind_thms ("Un_ac", [Un_assoc, Un_left_absorb, Un_commute, Un_left_commute]);
-
-Goal "(A Int B) Un C = (A Un C) Int (B Un C)";
-by (Blast_tac 1);
-qed "Un_Int_distrib";
-
-Goal "A<=B <-> A Un B = B";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "subset_Un_iff";
-
-Goal "A<=B <-> B Un A = B";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "subset_Un_iff2";
-
-Goal "(A Un B = 0) <-> (A = 0 & B = 0)";
-by (Blast_tac 1);
-qed "Un_empty";
-AddIffs[Un_empty];
-
-Goal "A Un B = Union({A, B})";
-by (Blast_tac 1);
-qed "Un_eq_Union";
-
-(** Simple properties of Diff -- set difference **)
-
-Goal "A - A = 0";
-by (Blast_tac 1);
-qed "Diff_cancel";
-
-Goal "A Int B = 0 ==> A - B = A";
-by (Blast_tac 1);
-qed "Diff_triv";
-
-Goal "0 - A = 0";
-by (Blast_tac 1);
-qed "empty_Diff";
-Addsimps[empty_Diff];
-
-Goal "A - 0 = A";
-by (Blast_tac 1);
-qed "Diff_0";
-Addsimps[Diff_0];
-
-Goal "A - B = 0 <-> A <= B";
-by (blast_tac (claset() addEs [equalityE]) 1);
-qed "Diff_eq_0_iff";
-
-(*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
-Goal "A - cons(a,B) = A - B - {a}";
-by (Blast_tac 1);
-qed "Diff_cons";
-
-(*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
-Goal "A - cons(a,B) = A - {a} - B";
-by (Blast_tac 1);
-qed "Diff_cons2";
-
-Goal "A Int (B-A) = 0";
-by (Blast_tac 1);
-qed "Diff_disjoint";
-
-Goal "A<=B ==> A Un (B-A) = B";
-by (Blast_tac 1);
-qed "Diff_partition";
-
-Goal "A <= B Un (A - B)";
-by (Blast_tac 1);
-qed "subset_Un_Diff";
-
-Goal "[| A<=B; B<=C |] ==> B-(C-A) = A";
-by (Blast_tac 1);
-qed "double_complement";
-
-Goal "(A Un B) - (B-A) = A";
-by (Blast_tac 1);
-qed "double_complement_Un";
-
-Goal
- "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)";
-by (Blast_tac 1);
-qed "Un_Int_crazy";
-
-Goal "A - (B Un C) = (A-B) Int (A-C)";
-by (Blast_tac 1);
-qed "Diff_Un";
-
-Goal "A - (B Int C) = (A-B) Un (A-C)";
-by (Blast_tac 1);
-qed "Diff_Int";
-
-Goal "(A Un B) - C = (A - C) Un (B - C)";
-by (Blast_tac 1);
-qed "Un_Diff";
-
-Goal "(A Int B) - C = A Int (B - C)";
-by (Blast_tac 1);
-qed "Int_Diff";
-
-Goal "C Int (A-B) = (C Int A) - (C Int B)";
-by (Blast_tac 1);
-qed "Diff_Int_distrib";
-
-Goal "(A-B) Int C = (A Int C) - (B Int C)";
-by (Blast_tac 1);
-qed "Diff_Int_distrib2";
-
-(*Halmos, Naive Set Theory, page 16.*)
-Goal "(A Int B) Un C = A Int (B Un C) <-> C<=A";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "Un_Int_assoc_iff";
-
-
-(** Big Union and Intersection **)
-
-Goal "Union(cons(a,B)) = a Un Union(B)";
-by (Blast_tac 1);
-qed "Union_cons";
-Addsimps [Union_cons];
-
-Goal "Union(A Un B) = Union(A) Un Union(B)";
-by (Blast_tac 1);
-qed "Union_Un_distrib";
-
-Goal "Union(A Int B) <= Union(A) Int Union(B)";
-by (Blast_tac 1);
-qed "Union_Int_subset";
-
-Goal "Union(C) Int A = 0 <-> (ALL B:C. B Int A = 0)";
-by (blast_tac (claset() addSEs [equalityE]) 1);
-qed "Union_disjoint";
-
-Goal "Union(A) = 0 <-> (ALL B:A. B=0)";
-by (Blast_tac 1);
-qed "Union_empty_iff";
-
-Goalw [Inter_def] "Inter(0) = 0";
-by (Blast_tac 1);
-qed "Inter_0";
-
-Goal "[| z:A; z:B |] ==> Inter(A) Un Inter(B) <= Inter(A Int B)";
-by (Blast_tac 1);
-qed "Inter_Un_subset";
-
-(* A good challenge: Inter is ill-behaved on the empty set *)
-Goal "[| a:A; b:B |] ==> Inter(A Un B) = Inter(A) Int Inter(B)";
-by (Blast_tac 1);
-qed "Inter_Un_distrib";
-
-Goal "Union({b}) = b";
-by (Blast_tac 1);
-qed "Union_singleton";
-
-Goal "Inter({b}) = b";
-by (Blast_tac 1);
-qed "Inter_singleton";
-
-Goal "Inter(cons(a,B)) = (if B=0 then a else a Int Inter(B))";
-by (Simp_tac 1);
-by (Blast_tac 1);
-qed "Inter_cons";
-Addsimps [Inter_cons];
-
-(** Unions and Intersections of Families **)
-
-Goal "Union(A) = (UN x:A. x)";
-by (Blast_tac 1);
-qed "Union_eq_UN";
-
-Goalw [Inter_def] "Inter(A) = (INT x:A. x)";
-by (Blast_tac 1);
-qed "Inter_eq_INT";
-
-Goal "(UN i:0. A(i)) = 0";
-by (Blast_tac 1);
-qed "UN_0";
-Addsimps [UN_0];
-
-Goal "(UN x:A. {x}) = A";
-by (Blast_tac 1);
-qed "UN_singleton";
-
-Goal "(UN i: A Un B. C(i)) = (UN i: A. C(i)) Un (UN i:B. C(i))";
-by (Blast_tac 1);
-qed "UN_Un";
-
-Goal "(INT i:I Un J. A(i)) = (if I=0 then INT j:J. A(j) \
-\ else if J=0 then INT i:I. A(i) \
-\ else ((INT i:I. A(i)) Int (INT j:J. A(j))))";
-by Auto_tac;
-by (blast_tac (claset() addSIs [equalityI]) 1);
-qed "INT_Un";
-
-Goal "(UN x : (UN y:A. B(y)). C(x)) = (UN y:A. UN x: B(y). C(x))";
-by (Blast_tac 1);
-qed "UN_UN_flatten";
-
-(*Halmos, Naive Set Theory, page 35.*)
-Goal "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
-by (Blast_tac 1);
-qed "Int_UN_distrib";
-
-Goal "i:I ==> B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
-by (Blast_tac 1);
-qed "Un_INT_distrib";
-
-Goal "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))";
-by (Blast_tac 1);
-qed "Int_UN_distrib2";
-
-Goal "[| i:I; j:J |] ==> \
-\ (INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))";
-by (Blast_tac 1);
-qed "Un_INT_distrib2";
-
-Goal "a: A ==> (UN y:A. c) = c";
-by (Blast_tac 1);
-qed "UN_constant";
-
-Goal "a: A ==> (INT y:A. c) = c";
-by (Blast_tac 1);
-qed "INT_constant";
-
-Goal "(UN y: RepFun(A,f). B(y)) = (UN x:A. B(f(x)))";
-by (Blast_tac 1);
-qed "UN_RepFun";
-Addsimps [UN_RepFun];
-
-Goal "(INT x:RepFun(A,f). B(x)) = (INT a:A. B(f(a)))";
-by (auto_tac (claset(), simpset() addsimps [Inter_def]));
-qed "INT_RepFun";
-Addsimps [INT_RepFun];
-
-Goal "0 ~: A ==> (INT x: Union(A). B(x)) = (INT y:A. INT x:y. B(x))";
-by (asm_full_simp_tac (simpset() addsimps [Inter_def]) 1);
-by (subgoal_tac "ALL x:A. x~=0" 1);
-by (Blast_tac 2);
-by (rtac equalityI 1);
-by (Clarify_tac 1);
-by (blast_tac (claset() addIs []) 1);
-by (blast_tac (claset() addSDs [bspec]) 1);
-qed "INT_Union_eq";
-
-Goal "(ALL x:A. B(x) ~= 0) \
-\ ==> (INT z: (UN x:A. B(x)). C(z)) = (INT x:A. INT z: B(x). C(z))";
-by (stac INT_Union_eq 1);
-by (Blast_tac 1);
-by (simp_tac (simpset() addsimps [Inter_def]) 1);
-qed "INT_UN_eq";
-
-
-(** Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:
- Union of a family of unions **)
-
-Goal "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))";
-by (Blast_tac 1);
-qed "UN_Un_distrib";
-
-Goal "i:I ==> (INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))";
-by (Blast_tac 1);
-qed "INT_Int_distrib";
-
-Goal "(UN z:I Int J. A(z)) <= (UN z:I. A(z)) Int (UN z:J. A(z))";
-by (Blast_tac 1);
-qed "UN_Int_subset";
-
-(** Devlin, page 12, exercise 5: Complements **)
-
-Goal "i:I ==> B - (UN i:I. A(i)) = (INT i:I. B - A(i))";
-by (Blast_tac 1);
-qed "Diff_UN";
-
-Goal "i:I ==> B - (INT i:I. A(i)) = (UN i:I. B - A(i))";
-by (Blast_tac 1);
-qed "Diff_INT";
-
-(** Unions and Intersections with General Sum **)
-
-(*Not suitable for rewriting: LOOPS!*)
-Goal "Sigma(cons(a,B), C) = ({a}*C(a)) Un Sigma(B,C)";
-by (Blast_tac 1);
-qed "Sigma_cons1";
-
-(*Not suitable for rewriting: LOOPS!*)
-Goal "A * cons(b,B) = A*{b} Un A*B";
-by (Blast_tac 1);
-qed "Sigma_cons2";
-
-Goal "Sigma(succ(A), B) = ({A}*B(A)) Un Sigma(A,B)";
-by (Blast_tac 1);
-qed "Sigma_succ1";
-
-Goal "A * succ(B) = A*{B} Un A*B";
-by (Blast_tac 1);
-qed "Sigma_succ2";
-
-Goal "(SUM x:(UN y:A. C(y)). B(x)) = (UN y:A. SUM x:C(y). B(x))";
-by (Blast_tac 1);
-qed "SUM_UN_distrib1";
-
-Goal "(SUM i:I. UN j:J. C(i,j)) = (UN j:J. SUM i:I. C(i,j))";
-by (Blast_tac 1);
-qed "SUM_UN_distrib2";
-
-Goal "(SUM i:I Un J. C(i)) = (SUM i:I. C(i)) Un (SUM j:J. C(j))";
-by (Blast_tac 1);
-qed "SUM_Un_distrib1";
-
-Goal "(SUM i:I. A(i) Un B(i)) = (SUM i:I. A(i)) Un (SUM i:I. B(i))";
-by (Blast_tac 1);
-qed "SUM_Un_distrib2";
-
-(*First-order version of the above, for rewriting*)
-Goal "I * (A Un B) = I*A Un I*B";
-by (rtac SUM_Un_distrib2 1);
-qed "prod_Un_distrib2";
-
-Goal "(SUM i:I Int J. C(i)) = (SUM i:I. C(i)) Int (SUM j:J. C(j))";
-by (Blast_tac 1);
-qed "SUM_Int_distrib1";
-
-Goal "(SUM i:I. A(i) Int B(i)) = (SUM i:I. A(i)) Int (SUM i:I. B(i))";
-by (Blast_tac 1);
-qed "SUM_Int_distrib2";
-
-(*First-order version of the above, for rewriting*)
-Goal "I * (A Int B) = I*A Int I*B";
-by (rtac SUM_Int_distrib2 1);
-qed "prod_Int_distrib2";
-
-(*Cf Aczel, Non-Well-Founded Sets, page 115*)
-Goal "(SUM i:I. A(i)) = (UN i:I. {i} * A(i))";
-by (Blast_tac 1);
-qed "SUM_eq_UN";
-
-(** Domain **)
-
-Goal "b:B ==> domain(A*B) = A";
-by (Blast_tac 1);
-qed "domain_of_prod";
-
-Goal "domain(0) = 0";
-by (Blast_tac 1);
-qed "domain_0";
-Addsimps [domain_0];
-
-Goal "domain(cons(<a,b>,r)) = cons(a, domain(r))";
-by (Blast_tac 1);
-qed "domain_cons";
-Addsimps [domain_cons];
-
-Goal "domain(A Un B) = domain(A) Un domain(B)";
-by (Blast_tac 1);
-qed "domain_Un_eq";
-Addsimps [domain_Un_eq];
-
-Goal "domain(A Int B) <= domain(A) Int domain(B)";
-by (Blast_tac 1);
-qed "domain_Int_subset";
-
-Goal "domain(A) - domain(B) <= domain(A - B)";
-by (Blast_tac 1);
-qed "domain_Diff_subset";
-
-Goal "domain(converse(r)) = range(r)";
-by (Blast_tac 1);
-qed "domain_converse";
-Addsimps [domain_converse];
-
-Goal "domain(UN x:A. B(x)) = (UN x:A. domain(B(x)))";
-by (Blast_tac 1);
-qed "domain_UN";
-
-Goal "domain(Union(A)) = (UN x:A. domain(x))";
-by (Blast_tac 1);
-qed "domain_Union";
-
-
-(** Range **)
-
-Goal "a:A ==> range(A*B) = B";
-by (Blast_tac 1);
-qed "range_of_prod";
-
-Goal "range(0) = 0";
-by (Blast_tac 1);
-qed "range_0";
-
-Goal "range(cons(<a,b>,r)) = cons(b, range(r))";
-by (Blast_tac 1);
-qed "range_cons";
-
-Goal "range(A Un B) = range(A) Un range(B)";
-by (Blast_tac 1);
-qed "range_Un_eq";
-
-Goal "range(A Int B) <= range(A) Int range(B)";
-by (Blast_tac 1);
-qed "range_Int_subset";
-
-Goal "range(A) - range(B) <= range(A - B)";
-by (Blast_tac 1);
-qed "range_Diff_subset";
-
-Goal "range(converse(r)) = domain(r)";
-by (Blast_tac 1);
-qed "range_converse";
-
-Addsimps [range_0, range_cons, range_Un_eq, range_converse];
-
-
-(** Field **)
-
-Goal "field(A*A) = A";
-by (Blast_tac 1);
-qed "field_of_prod";
-
-Goal "field(0) = 0";
-by (Blast_tac 1);
-qed "field_0";
-
-Goal "field(cons(<a,b>,r)) = cons(a, cons(b, field(r)))";
-by (rtac equalityI 1);
-by (ALLGOALS Blast_tac) ;
-qed "field_cons";
-
-Goal "field(A Un B) = field(A) Un field(B)";
-by (Blast_tac 1);
-qed "field_Un_eq";
-
-Goal "field(A Int B) <= field(A) Int field(B)";
-by (Blast_tac 1);
-qed "field_Int_subset";
-
-Goal "field(A) - field(B) <= field(A - B)";
-by (Blast_tac 1);
-qed "field_Diff_subset";
-
-Goal "field(converse(r)) = field(r)";
-by (Blast_tac 1);
-qed "field_converse";
-
-Addsimps [field_0, field_cons, field_Un_eq, field_converse];
-
-
-(** Image **)
-
-Goal "r``0 = 0";
-by (Blast_tac 1);
-qed "image_0";
-
-Goal "r``(A Un B) = (r``A) Un (r``B)";
-by (Blast_tac 1);
-qed "image_Un";
-
-Goal "r``(A Int B) <= (r``A) Int (r``B)";
-by (Blast_tac 1);
-qed "image_Int_subset";
-
-Goal "(r Int A*A)``B <= (r``B) Int A";
-by (Blast_tac 1);
-qed "image_Int_square_subset";
-
-Goal "B<=A ==> (r Int A*A)``B = (r``B) Int A";
-by (Blast_tac 1);
-qed "image_Int_square";
-
-Addsimps [image_0, image_Un];
-
-(*Image laws for special relations*)
-Goal "0``A = 0";
-by (Blast_tac 1);
-qed "image_0_left";
-Addsimps [image_0_left];
-
-Goal "(r Un s)``A = (r``A) Un (s``A)";
-by (Blast_tac 1);
-qed "image_Un_left";
-
-Goal "(r Int s)``A <= (r``A) Int (s``A)";
-by (Blast_tac 1);
-qed "image_Int_subset_left";
-
-
-(** Inverse Image **)
-
-Goal "r-``0 = 0";
-by (Blast_tac 1);
-qed "vimage_0";
-Addsimps [vimage_0];
-
-Goal "r-``(A Un B) = (r-``A) Un (r-``B)";
-by (Blast_tac 1);
-qed "vimage_Un";
-Addsimps [vimage_Un];
-
-Goal "r-``(A Int B) <= (r-``A) Int (r-``B)";
-by (Blast_tac 1);
-qed "vimage_Int_subset";
-
-(*NOT suitable for rewriting*)
-Goal "f -``B = (UN y:B. f-``{y})";
-by (Blast_tac 1);
-qed "vimage_eq_UN";
-
-Goalw [function_def] "function(f) ==> f-``(A Int B) = (f-``A) Int (f-``B)";
-by (Blast_tac 1);
-qed "function_vimage_Int";
-
-Goalw [function_def] "function(f) ==> f-``(A-B) = (f-``A) - (f-``B)";
-by (Blast_tac 1);
-qed "function_vimage_Diff";
-
-Goalw [function_def] "function(f) ==> f `` (f-`` A) <= A";
-by (Blast_tac 1);
-qed "function_image_vimage";
-
-Goal "(r Int A*A)-``B <= (r-``B) Int A";
-by (Blast_tac 1);
-qed "vimage_Int_square_subset";
-
-Goal "B<=A ==> (r Int A*A)-``B = (r-``B) Int A";
-by (Blast_tac 1);
-qed "vimage_Int_square";
-
-
-
-(*Invese image laws for special relations*)
-Goal "0-``A = 0";
-by (Blast_tac 1);
-qed "vimage_0_left";
-Addsimps [vimage_0_left];
-
-Goal "(r Un s)-``A = (r-``A) Un (s-``A)";
-by (Blast_tac 1);
-qed "vimage_Un_left";
-
-Goal "(r Int s)-``A <= (r-``A) Int (s-``A)";
-by (Blast_tac 1);
-qed "vimage_Int_subset_left";
-
-
-(** Converse **)
-
-Goal "converse(A Un B) = converse(A) Un converse(B)";
-by (Blast_tac 1);
-qed "converse_Un";
-
-Goal "converse(A Int B) = converse(A) Int converse(B)";
-by (Blast_tac 1);
-qed "converse_Int";
-
-Goal "converse(A - B) = converse(A) - converse(B)";
-by (Blast_tac 1);
-qed "converse_Diff";
-
-Goal "converse(UN x:A. B(x)) = (UN x:A. converse(B(x)))";
-by (Blast_tac 1);
-qed "converse_UN";
-
-(*Unfolding Inter avoids using excluded middle on A=0*)
-Goalw [Inter_def] "converse(INT x:A. B(x)) = (INT x:A. converse(B(x)))";
-by (Blast_tac 1);
-qed "converse_INT";
-
-Addsimps [converse_Un, converse_Int, converse_Diff, converse_UN, converse_INT];
-
-(** Pow **)
-
-Goal "Pow(0) = {0}";
-by (Blast_tac 1);
-qed "Pow_0";
-
-Goal "Pow (cons(a,A)) = Pow(A) Un {cons(a,X) . X: Pow(A)}";
-by (rtac equalityI 1);
-by Safe_tac;
-by (etac swap 1);
-by (res_inst_tac [("a", "x-{a}")] RepFun_eqI 1);
-by (ALLGOALS Blast_tac);
-qed "Pow_insert";
-
-Goal "Pow(A) Un Pow(B) <= Pow(A Un B)";
-by (Blast_tac 1);
-qed "Un_Pow_subset";
-
-Goal "(UN x:A. Pow(B(x))) <= Pow(UN x:A. B(x))";
-by (Blast_tac 1);
-qed "UN_Pow_subset";
-
-Goal "A <= Pow(Union(A))";
-by (Blast_tac 1);
-qed "subset_Pow_Union";
-
-Goal "Union(Pow(A)) = A";
-by (Blast_tac 1);
-qed "Union_Pow_eq";
-
-Goal "Pow(A Int B) = Pow(A) Int Pow(B)";
-by (Blast_tac 1);
-qed "Pow_Int_eq";
-
-Goal "x:A ==> Pow(INT x:A. B(x)) = (INT x:A. Pow(B(x)))";
-by (Blast_tac 1);
-qed "Pow_INT_eq";
-
-Addsimps [Pow_0, Union_Pow_eq, Pow_Int_eq];
-
-(** RepFun **)
-
-Goal "{f(x).x:A}=0 <-> A=0";
-by (Blast_tac 1);
-qed "RepFun_eq_0_iff";
-Addsimps [RepFun_eq_0_iff];
-
-Goal "{c. x:A} = (if A=0 then 0 else {c})";
-by Auto_tac;
-by (Blast_tac 1);
-qed "RepFun_constant";
-Addsimps [RepFun_constant];
-
-(** Collect **)
-
-Goal "Collect(A Un B, P) = Collect(A,P) Un Collect(B,P)";
-by (Blast_tac 1);
-qed "Collect_Un";
-
-Goal "Collect(A Int B, P) = Collect(A,P) Int Collect(B,P)";
-by (Blast_tac 1);
-qed "Collect_Int";
-
-Goal "Collect(A - B, P) = Collect(A,P) - Collect(B,P)";
-by (Blast_tac 1);
-qed "Collect_Diff";
-
-Goal "{x:cons(a,B). P(x)} = \
-\ (if P(a) then cons(a, {x:B. P(x)}) else {x:B. P(x)})";
-by (simp_tac (simpset() addsplits [split_if]) 1);
-by (Blast_tac 1);
-qed "Collect_cons";
-
-Goal "A Int Collect(A,P) = Collect(A,P)";
-by (Blast_tac 1);
-qed "Int_Collect_self_eq";
-
-Goal "Collect(Collect(A,P), Q) = Collect(A, %x. P(x) & Q(x))";
-by (Blast_tac 1);
-qed "Collect_Collect_eq";
-Addsimps [Collect_Collect_eq];
-
-Goal "Collect(A,P) Int Collect(A,Q) = Collect(A, %x. P(x) & Q(x))";
-by (Blast_tac 1);
-qed "Collect_Int_Collect_eq";