src/ZF/equalities.ML

--- a/src/ZF/equalities.ML	Wed Dec 07 12:34:47 1994 +0100
+++ b/src/ZF/equalities.ML	Wed Dec 07 13:12:04 1994 +0100
@@ -12,23 +12,23 @@
 (* cons_def refers to Upair; reversing the equality LOOPS in rewriting!*)
 goal ZF.thy "{a} Un B = cons(a,B)";
 by (fast_tac eq_cs 1);
-val cons_eq = result();
+qed "cons_eq";
 
 goal ZF.thy "cons(a, cons(b, C)) = cons(b, cons(a, C))";
 by (fast_tac eq_cs 1);
-val cons_commute = result();
+qed "cons_commute";
 
 goal ZF.thy "!!B. a: B ==> cons(a,B) = B";
 by (fast_tac eq_cs 1);
-val cons_absorb = result();
+qed "cons_absorb";
 
 goal ZF.thy "!!B. a: B ==> cons(a, B-{a}) = B";
 by (fast_tac eq_cs 1);
-val cons_Diff = result();
+qed "cons_Diff";
 
 goal ZF.thy "!!C. [| a: C;  ALL y:C. y=b |] ==> C = {b}";
 by (fast_tac eq_cs 1);
-val equal_singleton_lemma = result();
+qed "equal_singleton_lemma";
 val equal_singleton = ballI RSN (2,equal_singleton_lemma);
 
 
@@ -36,443 +36,443 @@
 
 goal ZF.thy "0 Int A = 0";
 by (fast_tac eq_cs 1);
-val Int_0 = result();
+qed "Int_0";
 
 (*NOT an equality, but it seems to belong here...*)
 goal ZF.thy "cons(a,B) Int C <= cons(a, B Int C)";
 by (fast_tac eq_cs 1);
-val Int_cons = result();
+qed "Int_cons";
 
 goal ZF.thy "A Int A = A";
 by (fast_tac eq_cs 1);
-val Int_absorb = result();
+qed "Int_absorb";
 
 goal ZF.thy "A Int B = B Int A";
 by (fast_tac eq_cs 1);
-val Int_commute = result();
+qed "Int_commute";
 
 goal ZF.thy "(A Int B) Int C  =  A Int (B Int C)";
 by (fast_tac eq_cs 1);
-val Int_assoc = result();
+qed "Int_assoc";
 
 goal ZF.thy "(A Un B) Int C  =  (A Int C) Un (B Int C)";
 by (fast_tac eq_cs 1);
-val Int_Un_distrib = result();
+qed "Int_Un_distrib";
 
 goal ZF.thy "A<=B <-> A Int B = A";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Int_iff = result();
+qed "subset_Int_iff";
 
 goal ZF.thy "A<=B <-> B Int A = A";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Int_iff2 = result();
+qed "subset_Int_iff2";
 
 (** Binary Union **)
 
 goal ZF.thy "0 Un A = A";
 by (fast_tac eq_cs 1);
-val Un_0 = result();
+qed "Un_0";
 
 goal ZF.thy "cons(a,B) Un C = cons(a, B Un C)";
 by (fast_tac eq_cs 1);
-val Un_cons = result();
+qed "Un_cons";
 
 goal ZF.thy "A Un A = A";
 by (fast_tac eq_cs 1);
-val Un_absorb = result();
+qed "Un_absorb";
 
 goal ZF.thy "A Un B = B Un A";
 by (fast_tac eq_cs 1);
-val Un_commute = result();
+qed "Un_commute";
 
 goal ZF.thy "(A Un B) Un C  =  A Un (B Un C)";
 by (fast_tac eq_cs 1);
-val Un_assoc = result();
+qed "Un_assoc";
 
 goal ZF.thy "(A Int B) Un C  =  (A Un C) Int (B Un C)";
 by (fast_tac eq_cs 1);
-val Un_Int_distrib = result();
+qed "Un_Int_distrib";
 
 goal ZF.thy "A<=B <-> A Un B = B";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Un_iff = result();
+qed "subset_Un_iff";
 
 goal ZF.thy "A<=B <-> B Un A = B";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Un_iff2 = result();
+qed "subset_Un_iff2";
 
 (** Simple properties of Diff -- set difference **)
 
 goal ZF.thy "A-A = 0";
 by (fast_tac eq_cs 1);
-val Diff_cancel = result();
+qed "Diff_cancel";
 
 goal ZF.thy "0-A = 0";
 by (fast_tac eq_cs 1);
-val empty_Diff = result();
+qed "empty_Diff";
 
 goal ZF.thy "A-0 = A";
 by (fast_tac eq_cs 1);
-val Diff_0 = result();
+qed "Diff_0";
 
 (*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
 goal ZF.thy "A - cons(a,B) = A - B - {a}";
 by (fast_tac eq_cs 1);
-val Diff_cons = result();
+qed "Diff_cons";
 
 (*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
 goal ZF.thy "A - cons(a,B) = A - {a} - B";
 by (fast_tac eq_cs 1);
-val Diff_cons2 = result();
+qed "Diff_cons2";
 
 goal ZF.thy "A Int (B-A) = 0";
 by (fast_tac eq_cs 1);
-val Diff_disjoint = result();
+qed "Diff_disjoint";
 
 goal ZF.thy "!!A B. A<=B ==> A Un (B-A) = B";
 by (fast_tac eq_cs 1);
-val Diff_partition = result();
+qed "Diff_partition";
 
 goal ZF.thy "!!A B. [| A<=B; B<=C |] ==> B-(C-A) = A";
 by (fast_tac eq_cs 1);
-val double_complement = result();
+qed "double_complement";
 
 goal ZF.thy "(A Un B) - (B-A) = A";
 by (fast_tac eq_cs 1);
-val double_complement_Un = result();
+qed "double_complement_Un";
 
 goal ZF.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);
-val Un_Int_crazy = result();
+qed "Un_Int_crazy";
 
 goal ZF.thy "A - (B Un C) = (A-B) Int (A-C)";
 by (fast_tac eq_cs 1);
-val Diff_Un = result();
+qed "Diff_Un";
 
 goal ZF.thy "A - (B Int C) = (A-B) Un (A-C)";
 by (fast_tac eq_cs 1);
-val Diff_Int = result();
+qed "Diff_Int";
 
 (*Halmos, Naive Set Theory, page 16.*)
 goal ZF.thy "(A Int B) Un C = A Int (B Un C)  <->  C<=A";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val Un_Int_assoc_iff = result();
+qed "Un_Int_assoc_iff";
 
 
 (** Big Union and Intersection **)
 
 goal ZF.thy "Union(0) = 0";
 by (fast_tac eq_cs 1);
-val Union_0 = result();
+qed "Union_0";
 
 goal ZF.thy "Union(cons(a,B)) = a Un Union(B)";
 by (fast_tac eq_cs 1);
-val Union_cons = result();
+qed "Union_cons";
 
 goal ZF.thy "Union(A Un B) = Union(A) Un Union(B)";
 by (fast_tac eq_cs 1);
-val Union_Un_distrib = result();
+qed "Union_Un_distrib";
 
 goal ZF.thy "Union(A Int B) <= Union(A) Int Union(B)";
 by (fast_tac ZF_cs 1);
-val Union_Int_subset = result();
+qed "Union_Int_subset";
 
 goal ZF.thy "Union(C) Int A = 0 <-> (ALL B:C. B Int A = 0)";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val Union_disjoint = result();
+qed "Union_disjoint";
 
 (* A good challenge: Inter is ill-behaved on the empty set *)
 goal ZF.thy "!!A B. [| a:A;  b:B |] ==> Inter(A Un B) = Inter(A) Int Inter(B)";
 by (fast_tac eq_cs 1);
-val Inter_Un_distrib = result();
+qed "Inter_Un_distrib";
 
 goal ZF.thy "Union({b}) = b";
 by (fast_tac eq_cs 1);
-val Union_singleton = result();
+qed "Union_singleton";
 
 goal ZF.thy "Inter({b}) = b";
 by (fast_tac eq_cs 1);
-val Inter_singleton = result();
+qed "Inter_singleton";
 
 (** Unions and Intersections of Families **)
 
 goal ZF.thy "Union(A) = (UN x:A. x)";
 by (fast_tac eq_cs 1);
-val Union_eq_UN = result();
+qed "Union_eq_UN";
 
 goalw ZF.thy [Inter_def] "Inter(A) = (INT x:A. x)";
 by (fast_tac eq_cs 1);
-val Inter_eq_INT = result();
+qed "Inter_eq_INT";
 
 goal ZF.thy "(UN i:0. A(i)) = 0";
 by (fast_tac eq_cs 1);
-val UN_0 = result();
+qed "UN_0";
 
 (*Halmos, Naive Set Theory, page 35.*)
 goal ZF.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
 by (fast_tac eq_cs 1);
-val Int_UN_distrib = result();
+qed "Int_UN_distrib";
 
 goal ZF.thy "!!A B. i:I ==> B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
 by (fast_tac eq_cs 1);
-val Un_INT_distrib = result();
+qed "Un_INT_distrib";
 
 goal ZF.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);
-val Int_UN_distrib2 = result();
+qed "Int_UN_distrib2";
 
 goal ZF.thy
     "!!I J. [| 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 (fast_tac eq_cs 1);
-val Un_INT_distrib2 = result();
+qed "Un_INT_distrib2";
 
 goal ZF.thy "!!A. a: A ==> (UN y:A. c) = c";
 by (fast_tac eq_cs 1);
-val UN_constant = result();
+qed "UN_constant";
 
 goal ZF.thy "!!A. a: A ==> (INT y:A. c) = c";
 by (fast_tac eq_cs 1);
-val INT_constant = result();
+qed "INT_constant";
 
 (** Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: 
     Union of a family of unions **)
 
 goal ZF.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);
-val UN_Un_distrib = result();
+qed "UN_Un_distrib";
 
 goal ZF.thy
     "!!A B. i:I ==> \
 \           (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);
-val INT_Int_distrib = result();
+qed "INT_Int_distrib";
 
 (** Devlin, page 12, exercise 5: Complements **)
 
 goal ZF.thy "!!A B. i:I ==> B - (UN i:I. A(i)) = (INT i:I. B - A(i))";
 by (fast_tac eq_cs 1);
-val Diff_UN = result();
+qed "Diff_UN";
 
 goal ZF.thy "!!A B. i:I ==> B - (INT i:I. A(i)) = (UN i:I. B - A(i))";
 by (fast_tac eq_cs 1);
-val Diff_INT = result();
+qed "Diff_INT";
 
 (** Unions and Intersections with General Sum **)
 
 goal ZF.thy "Sigma(cons(a,B), C) = ({a}*C(a)) Un Sigma(B,C)";
 by (fast_tac eq_cs 1);
-val Sigma_cons = result();
+qed "Sigma_cons";
 
 goal ZF.thy "(SUM x:(UN y:A. C(y)). B(x)) = (UN y:A. SUM x:C(y). B(x))";
 by (fast_tac eq_cs 1);
-val SUM_UN_distrib1 = result();
+qed "SUM_UN_distrib1";
 
 goal ZF.thy "(SUM i:I. UN j:J. C(i,j)) = (UN j:J. SUM i:I. C(i,j))";
 by (fast_tac eq_cs 1);
-val SUM_UN_distrib2 = result();
+qed "SUM_UN_distrib2";
 
 goal ZF.thy "(SUM i:I Un J. C(i)) = (SUM i:I. C(i)) Un (SUM j:J. C(j))";
 by (fast_tac eq_cs 1);
-val SUM_Un_distrib1 = result();
+qed "SUM_Un_distrib1";
 
 goal ZF.thy "(SUM i:I. A(i) Un B(i)) = (SUM i:I. A(i)) Un (SUM i:I. B(i))";
 by (fast_tac eq_cs 1);
-val SUM_Un_distrib2 = result();
+qed "SUM_Un_distrib2";
 
 (*First-order version of the above, for rewriting*)
 goal ZF.thy "I * (A Un B) = I*A Un I*B";
 by (resolve_tac [SUM_Un_distrib2] 1);
-val prod_Un_distrib2 = result();
+qed "prod_Un_distrib2";
 
 goal ZF.thy "(SUM i:I Int J. C(i)) = (SUM i:I. C(i)) Int (SUM j:J. C(j))";
 by (fast_tac eq_cs 1);
-val SUM_Int_distrib1 = result();
+qed "SUM_Int_distrib1";
 
 goal ZF.thy
     "(SUM i:I. A(i) Int B(i)) = (SUM i:I. A(i)) Int (SUM i:I. B(i))";
 by (fast_tac eq_cs 1);
-val SUM_Int_distrib2 = result();
+qed "SUM_Int_distrib2";
 
 (*First-order version of the above, for rewriting*)
 goal ZF.thy "I * (A Int B) = I*A Int I*B";
 by (resolve_tac [SUM_Int_distrib2] 1);
-val prod_Int_distrib2 = result();
+qed "prod_Int_distrib2";
 
 (*Cf Aczel, Non-Well-Founded Sets, page 115*)
 goal ZF.thy "(SUM i:I. A(i)) = (UN i:I. {i} * A(i))";
 by (fast_tac eq_cs 1);
-val SUM_eq_UN = result();
+qed "SUM_eq_UN";
 
 (** Domain **)
 
-val domain_of_prod = prove_goal ZF.thy "!!A B. b:B ==> domain(A*B) = A"
+qed_goal "domain_of_prod" ZF.thy "!!A B. b:B ==> domain(A*B) = A"
  (fn _ => [ fast_tac eq_cs 1 ]);
 
-val domain_0 = prove_goal ZF.thy "domain(0) = 0"
+qed_goal "domain_0" ZF.thy "domain(0) = 0"
  (fn _ => [ fast_tac eq_cs 1 ]);
 
-val domain_cons = prove_goal ZF.thy
+qed_goal "domain_cons" ZF.thy
     "domain(cons(<a,b>,r)) = cons(a, domain(r))"
  (fn _ => [ fast_tac eq_cs 1 ]);
 
 goal ZF.thy "domain(A Un B) = domain(A) Un domain(B)";
 by (fast_tac eq_cs 1);
-val domain_Un_eq = result();
+qed "domain_Un_eq";
 
 goal ZF.thy "domain(A Int B) <= domain(A) Int domain(B)";
 by (fast_tac eq_cs 1);
-val domain_Int_subset = result();
+qed "domain_Int_subset";
 
 goal ZF.thy "domain(A) - domain(B) <= domain(A - B)";
 by (fast_tac eq_cs 1);
-val domain_diff_subset = result();
+qed "domain_diff_subset";
 
 goal ZF.thy "domain(converse(r)) = range(r)";
 by (fast_tac eq_cs 1);
-val domain_converse = result();
+qed "domain_converse";
 
 
 
 (** Range **)
 
-val range_of_prod = prove_goal ZF.thy
+qed_goal "range_of_prod" ZF.thy
     "!!a A B. a:A ==> range(A*B) = B"
  (fn _ => [ fast_tac eq_cs 1 ]);
 
-val range_0 = prove_goal ZF.thy "range(0) = 0"
+qed_goal "range_0" ZF.thy "range(0) = 0"
  (fn _ => [ fast_tac eq_cs 1 ]); 
 
-val range_cons = prove_goal ZF.thy
+qed_goal "range_cons" ZF.thy
     "range(cons(<a,b>,r)) = cons(b, range(r))"
  (fn _ => [ fast_tac eq_cs 1 ]);
 
 goal ZF.thy "range(A Un B) = range(A) Un range(B)";
 by (fast_tac eq_cs 1);
-val range_Un_eq = result();
+qed "range_Un_eq";
 
 goal ZF.thy "range(A Int B) <= range(A) Int range(B)";
 by (fast_tac ZF_cs 1);
-val range_Int_subset = result();
+qed "range_Int_subset";
 
 goal ZF.thy "range(A) - range(B) <= range(A - B)";
 by (fast_tac eq_cs 1);
-val range_diff_subset = result();
+qed "range_diff_subset";
 
 goal ZF.thy "range(converse(r)) = domain(r)";
 by (fast_tac eq_cs 1);
-val range_converse = result();
+qed "range_converse";
 
 (** Field **)
 
-val field_of_prod = prove_goal ZF.thy "field(A*A) = A"
+qed_goal "field_of_prod" ZF.thy "field(A*A) = A"
  (fn _ => [ fast_tac eq_cs 1 ]); 
 
-val field_0 = prove_goal ZF.thy "field(0) = 0"
+qed_goal "field_0" ZF.thy "field(0) = 0"
  (fn _ => [ fast_tac eq_cs 1 ]); 
 
-val field_cons = prove_goal ZF.thy
+qed_goal "field_cons" ZF.thy
     "field(cons(<a,b>,r)) = cons(a, cons(b, field(r)))"
  (fn _ => [ rtac equalityI 1, ALLGOALS (fast_tac ZF_cs) ]);
 
 goal ZF.thy "field(A Un B) = field(A) Un field(B)";
 by (fast_tac eq_cs 1);
-val field_Un_eq = result();
+qed "field_Un_eq";
 
 goal ZF.thy "field(A Int B) <= field(A) Int field(B)";
 by (fast_tac eq_cs 1);
-val field_Int_subset = result();
+qed "field_Int_subset";
 
 goal ZF.thy "field(A) - field(B) <= field(A - B)";
 by (fast_tac eq_cs 1);
-val field_diff_subset = result();
+qed "field_diff_subset";
 
 
 (** Image **)
 
 goal ZF.thy "r``0 = 0";
 by (fast_tac eq_cs 1);
-val image_0 = result();
+qed "image_0";
 
 goal ZF.thy "r``(A Un B) = (r``A) Un (r``B)";
 by (fast_tac eq_cs 1);
-val image_Un = result();
+qed "image_Un";
 
 goal ZF.thy "r``(A Int B) <= (r``A) Int (r``B)";
 by (fast_tac ZF_cs 1);
-val image_Int_subset = result();
+qed "image_Int_subset";
 
 goal ZF.thy "(r Int A*A)``B <= (r``B) Int A";
 by (fast_tac ZF_cs 1);
-val image_Int_square_subset = result();
+qed "image_Int_square_subset";
 
 goal ZF.thy "!!r. B<=A ==> (r Int A*A)``B = (r``B) Int A";
 by (fast_tac eq_cs 1);
-val image_Int_square = result();
+qed "image_Int_square";
 
 
 (** Inverse Image **)
 
 goal ZF.thy "r-``0 = 0";
 by (fast_tac eq_cs 1);
-val vimage_0 = result();
+qed "vimage_0";
 
 goal ZF.thy "r-``(A Un B) = (r-``A) Un (r-``B)";
 by (fast_tac eq_cs 1);
-val vimage_Un = result();
+qed "vimage_Un";
 
 goal ZF.thy "r-``(A Int B) <= (r-``A) Int (r-``B)";
 by (fast_tac ZF_cs 1);
-val vimage_Int_subset = result();
+qed "vimage_Int_subset";
 
 goal ZF.thy "(r Int A*A)-``B <= (r-``B) Int A";
 by (fast_tac ZF_cs 1);
-val vimage_Int_square_subset = result();
+qed "vimage_Int_square_subset";
 
 goal ZF.thy "!!r. B<=A ==> (r Int A*A)-``B = (r-``B) Int A";
 by (fast_tac eq_cs 1);
-val vimage_Int_square = result();
+qed "vimage_Int_square";
 
 
 (** Converse **)
 
 goal ZF.thy "converse(A Un B) = converse(A) Un converse(B)";
 by (fast_tac eq_cs 1);
-val converse_Un = result();
+qed "converse_Un";
 
 goal ZF.thy "converse(A Int B) = converse(A) Int converse(B)";
 by (fast_tac eq_cs 1);
-val converse_Int = result();
+qed "converse_Int";
 
 goal ZF.thy "converse(A) - converse(B) = converse(A - B)";
 by (fast_tac eq_cs 1);
-val converse_diff = result();
+qed "converse_diff";
 
 (** Pow **)
 
 goal ZF.thy "Pow(A) Un Pow(B) <= Pow(A Un B)";
 by (fast_tac upair_cs 1);
-val Un_Pow_subset = result();
+qed "Un_Pow_subset";
 
 goal ZF.thy "(UN x:A. Pow(B(x))) <= Pow(UN x:A. B(x))";
 by (fast_tac upair_cs 1);
-val UN_Pow_subset = result();
+qed "UN_Pow_subset";
 
 goal ZF.thy "A <= Pow(Union(A))";
 by (fast_tac upair_cs 1);
-val subset_Pow_Union = result();
+qed "subset_Pow_Union";
 
 goal ZF.thy "Union(Pow(A)) = A";
 by (fast_tac eq_cs 1);
-val Union_Pow_eq = result();
+qed "Union_Pow_eq";
 
 goal ZF.thy "Pow(A) Int Pow(B) = Pow(A Int B)";
 by (fast_tac eq_cs 1);
-val Int_Pow_eq = result();
+qed "Int_Pow_eq";
 
 goal ZF.thy "!!x A. x:A ==> (INT x:A. Pow(B(x))) = Pow(INT x:A. B(x))";
 by (fast_tac eq_cs 1);
-val INT_Pow_subset = result();
+qed "INT_Pow_subset";