author | clasohm |
Sun, 24 Apr 1994 11:27:38 +0200 | |
changeset 70 | 9459592608e2 |
parent 57 | 194d088c1511 |
child 76 | fb4fe9f8c3cd |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: HOL/equalities |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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Equalities involving union, intersection, inclusion, etc. |
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*) |
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writeln"File HOL/equalities"; |
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val eq_cs = set_cs addSIs [equalityI]; |
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(** : **) |
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goal Set.thy "x ~: {}"; |
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by(fast_tac set_cs 1); |
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val in_empty = result(); |
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goal Set.thy "x : insert(y,A) = (x=y | x:A)"; |
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by(fast_tac set_cs 1); |
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val in_insert = result(); |
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(** insert **) |
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goal Set.thy "!!a. a:A ==> insert(a,A) = A"; |
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by (fast_tac eq_cs 1); |
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val insert_absorb = result(); |
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goal Set.thy "(insert(x,A) <= B) = (x:B & A <= B)"; |
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by (fast_tac set_cs 1); |
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val insert_subset = result(); |
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(** Image **) |
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goal Set.thy "f``{} = {}"; |
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by (fast_tac eq_cs 1); |
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val image_empty = result(); |
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goal Set.thy "f``insert(a,B) = insert(f(a), f``B)"; |
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by (fast_tac eq_cs 1); |
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val image_insert = result(); |
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(** Binary Intersection **) |
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goal Set.thy "A Int A = A"; |
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by (fast_tac eq_cs 1); |
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val Int_absorb = result(); |
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goal Set.thy "A Int B = B Int A"; |
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by (fast_tac eq_cs 1); |
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val Int_commute = result(); |
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goal Set.thy "(A Int B) Int C = A Int (B Int C)"; |
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by (fast_tac eq_cs 1); |
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val Int_assoc = result(); |
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goal Set.thy "{} Int B = {}"; |
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by (fast_tac eq_cs 1); |
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val Int_empty_left = result(); |
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goal Set.thy "A Int {} = {}"; |
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by (fast_tac eq_cs 1); |
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val Int_empty_right = result(); |
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goal Set.thy "(A Un B) Int C = (A Int C) Un (B Int C)"; |
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by (fast_tac eq_cs 1); |
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val Int_Un_distrib = result(); |
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goal Set.thy "(A<=B) = (A Int B = A)"; |
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by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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val subset_Int_eq = result(); |
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(** Binary Union **) |
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goal Set.thy "A Un A = A"; |
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by (fast_tac eq_cs 1); |
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val Un_absorb = result(); |
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goal Set.thy "A Un B = B Un A"; |
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by (fast_tac eq_cs 1); |
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val Un_commute = result(); |
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goal Set.thy "(A Un B) Un C = A Un (B Un C)"; |
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by (fast_tac eq_cs 1); |
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val Un_assoc = result(); |
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goal Set.thy "{} Un B = B"; |
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by(fast_tac eq_cs 1); |
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val Un_empty_left = result(); |
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goal Set.thy "A Un {} = A"; |
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by(fast_tac eq_cs 1); |
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val Un_empty_right = result(); |
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goal Set.thy "insert(a,B) Un C = insert(a,B Un C)"; |
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by(fast_tac eq_cs 1); |
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val Un_insert_left = result(); |
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goal Set.thy "(A Int B) Un C = (A Un C) Int (B Un C)"; |
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by (fast_tac eq_cs 1); |
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val Un_Int_distrib = result(); |
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goal Set.thy |
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"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"; |
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by (fast_tac eq_cs 1); |
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val Un_Int_crazy = result(); |
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goal Set.thy "(A<=B) = (A Un B = B)"; |
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by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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val subset_Un_eq = result(); |
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goal Set.thy "(A <= insert(b,C)) = (A <= C | b:A & A-{b} <= C)"; |
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by (fast_tac eq_cs 1); |
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val subset_insert_iff = result(); |
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(** Simple properties of Compl -- complement of a set **) |
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goal Set.thy "A Int Compl(A) = {}"; |
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by (fast_tac eq_cs 1); |
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val Compl_disjoint = result(); |
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goal Set.thy "A Un Compl(A) = {x.True}"; |
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by (fast_tac eq_cs 1); |
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val Compl_partition = result(); |
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goal Set.thy "Compl(Compl(A)) = A"; |
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by (fast_tac eq_cs 1); |
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val double_complement = result(); |
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goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)"; |
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by (fast_tac eq_cs 1); |
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val Compl_Un = result(); |
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goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)"; |
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by (fast_tac eq_cs 1); |
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val Compl_Int = result(); |
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goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"; |
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by (fast_tac eq_cs 1); |
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val Compl_UN = result(); |
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goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"; |
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by (fast_tac eq_cs 1); |
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val Compl_INT = result(); |
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(*Halmos, Naive Set Theory, page 16.*) |
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goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)"; |
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by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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val Un_Int_assoc_eq = result(); |
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(** Big Union and Intersection **) |
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goal Set.thy "Union({}) = {}"; |
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by (fast_tac eq_cs 1); |
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val Union_empty = result(); |
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goal Set.thy "Union(insert(a,B)) = a Un Union(B)"; |
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by (fast_tac eq_cs 1); |
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val Union_insert = result(); |
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goal Set.thy "Union(A Un B) = Union(A) Un Union(B)"; |
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by (fast_tac eq_cs 1); |
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val Union_Un_distrib = result(); |
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goal Set.thy "Union(A Un B) = Union(A) Un Union(B)"; |
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by (fast_tac eq_cs 1); |
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val Union_Un_distrib = result(); |
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val prems = goal Set.thy |
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"(Union(C) Int A = {}) = (! B:C. B Int A = {})"; |
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by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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val Union_disjoint = result(); |
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goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)"; |
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by (best_tac eq_cs 1); |
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val Inter_Un_distrib = result(); |
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(** Unions and Intersections of Families **) |
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goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
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by (fast_tac eq_cs 1); |
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val UN_eq = result(); |
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(*Look: it has an EXISTENTIAL quantifier*) |
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goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
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by (fast_tac eq_cs 1); |
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val INT_eq = result(); |
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goal Set.thy "A Int Union(B) = (UN C:B. A Int C)"; |
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by (fast_tac eq_cs 1); |
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2e9a86203d59
HOL/equalities/Int_Union_image, Un_Inter_image: renamed Int_Union, Un_Inter
lcp
parents:
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diff
changeset
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val Int_Union = result(); |
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(* Devlin, page 12: Union of a family of unions **) |
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goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
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by (fast_tac eq_cs 1); |
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val Un_Union_image = result(); |
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goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)"; |
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by (fast_tac eq_cs 1); |
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2e9a86203d59
HOL/equalities/Int_Union_image, Un_Inter_image: renamed Int_Union, Un_Inter
lcp
parents:
27
diff
changeset
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val Un_Inter = result(); |
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goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
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by (best_tac eq_cs 1); |
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val Int_Inter_image = result(); |
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(** Other identities about Unions and Intersections **) |
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goal Set.thy "Union(range(f)) = (UN x.f(x))"; |
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by (fast_tac eq_cs 1); |
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val Union_range_eq = result(); |
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goal Set.thy "Inter(range(f)) = (INT x.f(x))"; |
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by (fast_tac eq_cs 1); |
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val Inter_range_eq = result(); |
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goal Set.thy "Union(B``A) = (UN x:A. B(x))"; |
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by (fast_tac eq_cs 1); |
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val Union_image_eq = result(); |
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goal Set.thy "Inter(B``A) = (INT x:A. B(x))"; |
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by (fast_tac eq_cs 1); |
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val Inter_image_eq = result(); |
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goal Set.thy "(UN x.B) = B"; |
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by (fast_tac eq_cs 1); |
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val constant_UN = result(); |
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(** Simple properties of Diff -- set difference **) |
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goal Set.thy "A-A = {}"; |
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by (fast_tac eq_cs 1); |
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val Diff_cancel = result(); |
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goal Set.thy "{}-A = {}"; |
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by (fast_tac eq_cs 1); |
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val empty_Diff = result(); |
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goal Set.thy "A-{} = A"; |
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by (fast_tac eq_cs 1); |
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val Diff_empty = result(); |
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(*NOT SUITABLE FOR REWRITING since {a} == insert(a,0)*) |
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goal Set.thy "A - insert(a,B) = A - B - {a}"; |
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by (fast_tac eq_cs 1); |
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val Diff_insert = result(); |
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(*NOT SUITABLE FOR REWRITING since {a} == insert(a,0)*) |
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goal Set.thy "A - insert(a,B) = A - {a} - B"; |
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by (fast_tac eq_cs 1); |
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val Diff_insert2 = result(); |
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val prems = goal Set.thy "a:A ==> insert(a,A-{a}) = A"; |
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by (fast_tac (eq_cs addSIs prems) 1); |
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val insert_Diff = result(); |
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goal Set.thy "A Int (B-A) = {}"; |
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by (fast_tac eq_cs 1); |
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val Diff_disjoint = result(); |
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goal Set.thy "!!A. A<=B ==> A Un (B-A) = B"; |
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by (fast_tac eq_cs 1); |
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val Diff_partition = result(); |
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(* |
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goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C-A)) = A"; |
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by (cut_facts_tac prems 1); |
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by (fast_tac eq_cs 1); |
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val double_complement = result(); |
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*) |
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goal Set.thy "A - (B Un C) = (A-B) Int (A-C)"; |
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by (fast_tac eq_cs 1); |
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val Diff_Un = result(); |
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goal Set.thy "A - (B Int C) = (A-B) Un (A-C)"; |
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by (fast_tac eq_cs 1); |
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val Diff_Int = result(); |
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val set_ss = set_ss addsimps |
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[in_empty,in_insert,insert_subset, |
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Int_absorb,Int_empty_left,Int_empty_right, |
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Un_absorb,Un_empty_left,Un_empty_right, |
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constant_UN,image_empty, |
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Compl_disjoint,double_complement, |
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Union_empty,Union_insert,empty_subsetI,subset_refl, |
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Diff_cancel,empty_Diff,Diff_empty,Diff_disjoint]; |