equalities.ML

--- a/equalities.ML	Fri Nov 11 10:35:03 1994 +0100
+++ b/equalities.ML	Mon Nov 21 17:50:34 1994 +0100
@@ -14,168 +14,168 @@
 
 goal Set.thy "x ~: {}";
 by(fast_tac set_cs 1);
-val in_empty = result();
+qed "in_empty";
 
 goal Set.thy "x : insert(y,A) = (x=y | x:A)";
 by(fast_tac set_cs 1);
-val in_insert = result();
+qed "in_insert";
 
 (** insert **)
 
 goal Set.thy "!!a. a:A ==> insert(a,A) = A";
 by (fast_tac eq_cs 1);
-val insert_absorb = result();
+qed "insert_absorb";
 
 goal Set.thy "(insert(x,A) <= B) = (x:B & A <= B)";
 by (fast_tac set_cs 1);
-val insert_subset = result();
+qed "insert_subset";
 
 (** Image **)
 
 goal Set.thy "f``{} = {}";
 by (fast_tac eq_cs 1);
-val image_empty = result();
+qed "image_empty";
 
 goal Set.thy "f``insert(a,B) = insert(f(a), f``B)";
 by (fast_tac eq_cs 1);
-val image_insert = result();
+qed "image_insert";
 
 (** Binary Intersection **)
 
 goal Set.thy "A Int A = A";
 by (fast_tac eq_cs 1);
-val Int_absorb = result();
+qed "Int_absorb";
 
 goal Set.thy "A Int B  =  B Int A";
 by (fast_tac eq_cs 1);
-val Int_commute = result();
+qed "Int_commute";
 
 goal Set.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 Set.thy "{} Int B = {}";
 by (fast_tac eq_cs 1);
-val Int_empty_left = result();
+qed "Int_empty_left";
 
 goal Set.thy "A Int {} = {}";
 by (fast_tac eq_cs 1);
-val Int_empty_right = result();
+qed "Int_empty_right";
 
 goal Set.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 Set.thy "(A<=B) = (A Int B = A)";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Int_eq = result();
+qed "subset_Int_eq";
 
 (** Binary Union **)
 
 goal Set.thy "A Un A = A";
 by (fast_tac eq_cs 1);
-val Un_absorb = result();
+qed "Un_absorb";
 
 goal Set.thy "A Un B  =  B Un A";
 by (fast_tac eq_cs 1);
-val Un_commute = result();
+qed "Un_commute";
 
 goal Set.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 Set.thy "{} Un B = B";
 by(fast_tac eq_cs 1);
-val Un_empty_left = result();
+qed "Un_empty_left";
 
 goal Set.thy "A Un {} = A";
 by(fast_tac eq_cs 1);
-val Un_empty_right = result();
+qed "Un_empty_right";
 
 goal Set.thy "insert(a,B) Un C = insert(a,B Un C)";
 by(fast_tac eq_cs 1);
-val Un_insert_left = result();
+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);
-val Un_Int_distrib = result();
+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);
-val Un_Int_crazy = result();
+qed "Un_Int_crazy";
 
 goal Set.thy "(A<=B) = (A Un B = B)";
 by (fast_tac (eq_cs addSEs [equalityE]) 1);
-val subset_Un_eq = result();
+qed "subset_Un_eq";
 
 goal Set.thy "(A <= insert(b,C)) = (A <= C | b:A & A-{b} <= C)";
 by (fast_tac eq_cs 1);
-val subset_insert_iff = result();
+qed "subset_insert_iff";
 
 (** Simple properties of Compl -- complement of a set **)
 
 goal Set.thy "A Int Compl(A) = {}";
 by (fast_tac eq_cs 1);
-val Compl_disjoint = result();
+qed "Compl_disjoint";
 
 goal Set.thy "A Un Compl(A) = {x.True}";
 by (fast_tac eq_cs 1);
-val Compl_partition = result();
+qed "Compl_partition";
 
 goal Set.thy "Compl(Compl(A)) = A";
 by (fast_tac eq_cs 1);
-val double_complement = result();
+qed "double_complement";
 
 goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)";
 by (fast_tac eq_cs 1);
-val Compl_Un = result();
+qed "Compl_Un";
 
 goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)";
 by (fast_tac eq_cs 1);
-val Compl_Int = result();
+qed "Compl_Int";
 
 goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))";
 by (fast_tac eq_cs 1);
-val Compl_UN = result();
+qed "Compl_UN";
 
 goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))";
 by (fast_tac eq_cs 1);
-val Compl_INT = result();
+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);
-val Un_Int_assoc_eq = result();
+qed "Un_Int_assoc_eq";
 
 
 (** Big Union and Intersection **)
 
 goal Set.thy "Union({}) = {}";
 by (fast_tac eq_cs 1);
-val Union_empty = result();
+qed "Union_empty";
 
 goal Set.thy "Union(insert(a,B)) = a Un Union(B)";
 by (fast_tac eq_cs 1);
-val Union_insert = result();
+qed "Union_insert";
 
 goal Set.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 Set.thy "Union(A Int B) <= Union(A) Int Union(B)";
 by (fast_tac set_cs 1);
-val Union_Int_subset = result();
+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);
-val Union_disjoint = result();
+qed "Union_disjoint";
 
 goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)";
 by (best_tac eq_cs 1);
-val Inter_Un_distrib = result();
+qed "Inter_Un_distrib";
 
 (** Unions and Intersections of Families **)
 
@@ -183,141 +183,141 @@
 
 goal Set.thy "Union(range(f)) = (UN x.f(x))";
 by (fast_tac eq_cs 1);
-val Union_range_eq = result();
+qed "Union_range_eq";
 
 goal Set.thy "Inter(range(f)) = (INT x.f(x))";
 by (fast_tac eq_cs 1);
-val Inter_range_eq = result();
+qed "Inter_range_eq";
 
 goal Set.thy "Union(B``A) = (UN x:A. B(x))";
 by (fast_tac eq_cs 1);
-val Union_image_eq = result();
+qed "Union_image_eq";
 
 goal Set.thy "Inter(B``A) = (INT x:A. B(x))";
 by (fast_tac eq_cs 1);
-val Inter_image_eq = result();
+qed "Inter_image_eq";
 
 goal Set.thy "!!A. a: A ==> (UN y:A. c) = c";
 by (fast_tac eq_cs 1);
-val UN_constant = result();
+qed "UN_constant";
 
 goal Set.thy "!!A. a: A ==> (INT y:A. c) = c";
 by (fast_tac eq_cs 1);
-val INT_constant = result();
+qed "INT_constant";
 
 goal Set.thy "(UN x.B) = B";
 by (fast_tac eq_cs 1);
-val UN1_constant = result();
+qed "UN1_constant";
 
 goal Set.thy "(INT x.B) = B";
 by (fast_tac eq_cs 1);
-val INT1_constant = result();
+qed "INT1_constant";
 
 goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})";
 by (fast_tac eq_cs 1);
-val UN_eq = result();
+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);
-val INT_eq = result();
+qed "INT_eq";
 
 (*Distributive laws...*)
 
 goal Set.thy "A Int Union(B) = (UN C:B. A Int C)";
 by (fast_tac eq_cs 1);
-val Int_Union = result();
+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);
-val Un_Union_image = result();
+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);
-val UN_Un_distrib = result();
+qed "UN_Un_distrib";
 
 goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)";
 by (fast_tac eq_cs 1);
-val Un_Inter = result();
+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);
-val Int_Inter_image = result();
+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);
-val INT_Int_distrib = result();
+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);
-val Int_UN_distrib = result();
+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);
-val Un_INT_distrib = result();
+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);
-val Int_UN_distrib2 = result();
+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);
-val Un_INT_distrib2 = result();
+qed "Un_INT_distrib2";
 
 (** Simple properties of Diff -- set difference **)
 
 goal Set.thy "A-A = {}";
 by (fast_tac eq_cs 1);
-val Diff_cancel = result();
+qed "Diff_cancel";
 
 goal Set.thy "{}-A = {}";
 by (fast_tac eq_cs 1);
-val empty_Diff = result();
+qed "empty_Diff";
 
 goal Set.thy "A-{} = A";
 by (fast_tac eq_cs 1);
-val Diff_empty = result();
+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);
-val Diff_insert = result();
+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);
-val Diff_insert2 = result();
+qed "Diff_insert2";
 
 val prems = goal Set.thy "a:A ==> insert(a,A-{a}) = A";
 by (fast_tac (eq_cs addSIs prems) 1);
-val insert_Diff = result();
+qed "insert_Diff";
 
 goal Set.thy "A Int (B-A) = {}";
 by (fast_tac eq_cs 1);
-val Diff_disjoint = result();
+qed "Diff_disjoint";
 
 goal Set.thy "!!A. A<=B ==> A Un (B-A) = B";
 by (fast_tac eq_cs 1);
-val Diff_partition = result();
+qed "Diff_partition";
 
 goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)";
 by (fast_tac eq_cs 1);
-val double_diff = result();
+qed "double_diff";
 
 goal Set.thy "A - (B Un C) = (A-B) Int (A-C)";
 by (fast_tac eq_cs 1);
-val Diff_Un = result();
+qed "Diff_Un";
 
 goal Set.thy "A - (B Int C) = (A-B) Un (A-C)";
 by (fast_tac eq_cs 1);
-val Diff_Int = result();
+qed "Diff_Int";
 
 val set_ss = set_ss addsimps
   [in_empty,in_insert,insert_subset,