author | berghofe |
Thu, 23 May 1996 15:17:23 +0200 | |
changeset 1763 | fb07e359b59f |
parent 1754 | 852093aeb0ab |
child 1786 | 8a31d85d27b8 |
permissions | -rw-r--r-- |
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(* 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 1994 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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AddSIs [equalityI]; |
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val eq_cs = set_cs addSIs [equalityI]; |
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section "{}"; |
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goal Set.thy "{x.False} = {}"; |
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by (Fast_tac 1); |
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qed "Collect_False_empty"; |
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Addsimps [Collect_False_empty]; |
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goal Set.thy "(A <= {}) = (A = {})"; |
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by (Fast_tac 1); |
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qed "subset_empty"; |
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Addsimps [subset_empty]; |
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section ":"; |
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goal Set.thy "x ~: {}"; |
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by (Fast_tac 1); |
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qed "in_empty"; |
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Addsimps[in_empty]; |
923 | 33 |
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goal Set.thy "x : insert y A = (x=y | x:A)"; |
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by (Fast_tac 1); |
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qed "in_insert"; |
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Addsimps[in_insert]; |
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section "insert"; |
923 | 40 |
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(*NOT SUITABLE FOR REWRITING since {a} == insert a {}*) |
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goal Set.thy "insert a A = {a} Un A"; |
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by (Fast_tac 1); |
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qed "insert_is_Un"; |
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goal Set.thy "insert a A ~= {}"; |
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by (fast_tac (!claset addEs [equalityCE]) 1); |
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qed"insert_not_empty"; |
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Addsimps[insert_not_empty]; |
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bind_thm("empty_not_insert",insert_not_empty RS not_sym); |
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Addsimps[empty_not_insert]; |
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goal Set.thy "!!a. a:A ==> insert a A = A"; |
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by (Fast_tac 1); |
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qed "insert_absorb"; |
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goal Set.thy "insert x (insert x A) = insert x A"; |
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by (Fast_tac 1); |
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qed "insert_absorb2"; |
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Addsimps [insert_absorb2]; |
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goal Set.thy "(insert x A <= B) = (x:B & A <= B)"; |
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by (Fast_tac 1); |
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qed "insert_subset"; |
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Addsimps[insert_subset]; |
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(* use new B rather than (A-{a}) to avoid infinite unfolding *) |
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goal Set.thy "!!a. a:A ==> ? B. A = insert a B & a ~: B"; |
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by (res_inst_tac [("x","A-{a}")] exI 1); |
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by (Fast_tac 1); |
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qed "mk_disjoint_insert"; |
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section "``"; |
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goal Set.thy "f``{} = {}"; |
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by (Fast_tac 1); |
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qed "image_empty"; |
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Addsimps[image_empty]; |
923 | 80 |
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goal Set.thy "f``insert a B = insert (f a) (f``B)"; |
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by (Fast_tac 1); |
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qed "image_insert"; |
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Addsimps[image_insert]; |
923 | 85 |
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1660 | 86 |
qed_goal "ball_image" Set.thy "(!y:F``S. P y) = (!x:S. P (F x))" |
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(fn _ => [Fast_tac 1]); |
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goalw Set.thy [image_def] |
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"(%x. if P x then f x else g x) `` S \ |
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\ = (f `` ({x.x:S & P x})) Un (g `` ({x.x:S & ~(P x)}))"; |
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by(split_tac [expand_if] 1); |
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by(Fast_tac 1); |
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qed "if_image_distrib"; |
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Addsimps[if_image_distrib]; |
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section "range"; |
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qed_goal "ball_range" Set.thy "(!y:range f. P y) = (!x. P (f x))" |
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(fn _ => [Fast_tac 1]); |
1660 | 102 |
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qed_goalw "image_range" Set.thy [image_def, range_def] |
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"f``range g = range (%x. f (g x))" (fn _ => [ |
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rtac Collect_cong 1, |
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Fast_tac 1]); |
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section "Int"; |
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goal Set.thy "A Int A = A"; |
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by (Fast_tac 1); |
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qed "Int_absorb"; |
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Addsimps[Int_absorb]; |
923 | 114 |
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goal Set.thy "A Int B = B Int A"; |
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by (Fast_tac 1); |
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qed "Int_commute"; |
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goal Set.thy "(A Int B) Int C = A Int (B Int C)"; |
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by (Fast_tac 1); |
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qed "Int_assoc"; |
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goal Set.thy "{} Int B = {}"; |
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by (Fast_tac 1); |
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qed "Int_empty_left"; |
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Addsimps[Int_empty_left]; |
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goal Set.thy "A Int {} = {}"; |
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by (Fast_tac 1); |
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qed "Int_empty_right"; |
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Addsimps[Int_empty_right]; |
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goal Set.thy "UNIV Int B = B"; |
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by (Fast_tac 1); |
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qed "Int_UNIV_left"; |
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Addsimps[Int_UNIV_left]; |
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goal Set.thy "A Int UNIV = A"; |
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by (Fast_tac 1); |
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qed "Int_UNIV_right"; |
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Addsimps[Int_UNIV_right]; |
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goal Set.thy "A Int (B Un C) = (A Int B) Un (A Int C)"; |
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by (Fast_tac 1); |
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qed "Int_Un_distrib"; |
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goal Set.thy "(B Un C) Int A = (B Int A) Un (C Int A)"; |
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by (Fast_tac 1); |
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qed "Int_Un_distrib2"; |
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goal Set.thy "(A<=B) = (A Int B = A)"; |
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by (fast_tac (!claset addSEs [equalityE]) 1); |
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qed "subset_Int_eq"; |
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goal Set.thy "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
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by (fast_tac (!claset addEs [equalityCE]) 1); |
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qed "Int_UNIV"; |
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Addsimps[Int_UNIV]; |
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section "Un"; |
923 | 161 |
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goal Set.thy "A Un A = A"; |
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by (Fast_tac 1); |
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qed "Un_absorb"; |
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Addsimps[Un_absorb]; |
923 | 166 |
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goal Set.thy "A Un B = B Un A"; |
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by (Fast_tac 1); |
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qed "Un_commute"; |
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goal Set.thy "(A Un B) Un C = A Un (B Un C)"; |
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by (Fast_tac 1); |
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qed "Un_assoc"; |
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goal Set.thy "{} Un B = B"; |
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by (Fast_tac 1); |
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qed "Un_empty_left"; |
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Addsimps[Un_empty_left]; |
923 | 179 |
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goal Set.thy "A Un {} = A"; |
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by (Fast_tac 1); |
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qed "Un_empty_right"; |
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Addsimps[Un_empty_right]; |
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goal Set.thy "UNIV Un B = UNIV"; |
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by (Fast_tac 1); |
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qed "Un_UNIV_left"; |
188 |
Addsimps[Un_UNIV_left]; |
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goal Set.thy "A Un UNIV = UNIV"; |
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by (Fast_tac 1); |
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qed "Un_UNIV_right"; |
193 |
Addsimps[Un_UNIV_right]; |
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923 | 194 |
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goal Set.thy "insert a B Un C = insert a (B Un C)"; |
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by (Fast_tac 1); |
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qed "Un_insert_left"; |
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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 1); |
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qed "Un_Int_distrib"; |
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203 |
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 1); |
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qed "Un_Int_crazy"; |
207 |
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208 |
goal Set.thy "(A<=B) = (A Un B = B)"; |
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by (fast_tac (!claset addSEs [equalityE]) 1); |
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qed "subset_Un_eq"; |
211 |
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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 1); |
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qed "subset_insert_iff"; |
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216 |
goal Set.thy "(A Un B = {}) = (A = {} & B = {})"; |
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217 |
by (fast_tac (!claset addEs [equalityCE]) 1); |
923 | 218 |
qed "Un_empty"; |
1531 | 219 |
Addsimps[Un_empty]; |
923 | 220 |
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1548 | 221 |
section "Compl"; |
923 | 222 |
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223 |
goal Set.thy "A Int Compl(A) = {}"; |
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by (Fast_tac 1); |
923 | 225 |
qed "Compl_disjoint"; |
1531 | 226 |
Addsimps[Compl_disjoint]; |
923 | 227 |
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1531 | 228 |
goal Set.thy "A Un Compl(A) = UNIV"; |
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229 |
by (Fast_tac 1); |
923 | 230 |
qed "Compl_partition"; |
231 |
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232 |
goal Set.thy "Compl(Compl(A)) = A"; |
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233 |
by (Fast_tac 1); |
923 | 234 |
qed "double_complement"; |
1531 | 235 |
Addsimps[double_complement]; |
923 | 236 |
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237 |
goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)"; |
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238 |
by (Fast_tac 1); |
923 | 239 |
qed "Compl_Un"; |
240 |
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241 |
goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)"; |
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242 |
by (Fast_tac 1); |
923 | 243 |
qed "Compl_Int"; |
244 |
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goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"; |
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246 |
by (Fast_tac 1); |
923 | 247 |
qed "Compl_UN"; |
248 |
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249 |
goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"; |
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250 |
by (Fast_tac 1); |
923 | 251 |
qed "Compl_INT"; |
252 |
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253 |
(*Halmos, Naive Set Theory, page 16.*) |
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254 |
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255 |
goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)"; |
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256 |
by (fast_tac (!claset addSEs [equalityE]) 1); |
923 | 257 |
qed "Un_Int_assoc_eq"; |
258 |
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259 |
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1548 | 260 |
section "Union"; |
923 | 261 |
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262 |
goal Set.thy "Union({}) = {}"; |
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263 |
by (Fast_tac 1); |
923 | 264 |
qed "Union_empty"; |
1531 | 265 |
Addsimps[Union_empty]; |
266 |
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267 |
goal Set.thy "Union(UNIV) = UNIV"; |
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268 |
by (Fast_tac 1); |
1531 | 269 |
qed "Union_UNIV"; |
270 |
Addsimps[Union_UNIV]; |
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923 | 271 |
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272 |
goal Set.thy "Union(insert a B) = a Un Union(B)"; |
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273 |
by (Fast_tac 1); |
923 | 274 |
qed "Union_insert"; |
1531 | 275 |
Addsimps[Union_insert]; |
923 | 276 |
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277 |
goal Set.thy "Union(A Un B) = Union(A) Un Union(B)"; |
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|
278 |
by (Fast_tac 1); |
923 | 279 |
qed "Union_Un_distrib"; |
1531 | 280 |
Addsimps[Union_Un_distrib]; |
923 | 281 |
|
282 |
goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)"; |
|
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|
283 |
by (Fast_tac 1); |
923 | 284 |
qed "Union_Int_subset"; |
285 |
||
286 |
val prems = goal Set.thy |
|
287 |
"(Union(C) Int A = {}) = (! B:C. B Int A = {})"; |
|
1754
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changeset
|
288 |
by (fast_tac (!claset addSEs [equalityE]) 1); |
923 | 289 |
qed "Union_disjoint"; |
290 |
||
1548 | 291 |
section "Inter"; |
292 |
||
1531 | 293 |
goal Set.thy "Inter({}) = UNIV"; |
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changeset
|
294 |
by (Fast_tac 1); |
1531 | 295 |
qed "Inter_empty"; |
296 |
Addsimps[Inter_empty]; |
|
297 |
||
298 |
goal Set.thy "Inter(UNIV) = {}"; |
|
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|
299 |
by (Fast_tac 1); |
1531 | 300 |
qed "Inter_UNIV"; |
301 |
Addsimps[Inter_UNIV]; |
|
302 |
||
303 |
goal Set.thy "Inter(insert a B) = a Int Inter(B)"; |
|
1754
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changeset
|
304 |
by (Fast_tac 1); |
1531 | 305 |
qed "Inter_insert"; |
306 |
Addsimps[Inter_insert]; |
|
307 |
||
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Deleted faulty comment; proved new rule Inter_Un_subset
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diff
changeset
|
308 |
goal Set.thy "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
1754
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changeset
|
309 |
by (Fast_tac 1); |
1564
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Deleted faulty comment; proved new rule Inter_Un_subset
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diff
changeset
|
310 |
qed "Inter_Un_subset"; |
1531 | 311 |
|
923 | 312 |
goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)"; |
313 |
by (best_tac eq_cs 1); |
|
314 |
qed "Inter_Un_distrib"; |
|
315 |
||
1548 | 316 |
section "UN and INT"; |
923 | 317 |
|
318 |
(*Basic identities*) |
|
319 |
||
1179
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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diff
changeset
|
320 |
goal Set.thy "(UN x:{}. B x) = {}"; |
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diff
changeset
|
321 |
by (Fast_tac 1); |
1179
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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diff
changeset
|
322 |
qed "UN_empty"; |
1531 | 323 |
Addsimps[UN_empty]; |
324 |
||
325 |
goal Set.thy "(UN x:UNIV. B x) = (UN x. B x)"; |
|
1754
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diff
changeset
|
326 |
by (Fast_tac 1); |
1531 | 327 |
qed "UN_UNIV"; |
328 |
Addsimps[UN_UNIV]; |
|
329 |
||
330 |
goal Set.thy "(INT x:{}. B x) = UNIV"; |
|
1754
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diff
changeset
|
331 |
by (Fast_tac 1); |
1531 | 332 |
qed "INT_empty"; |
333 |
Addsimps[INT_empty]; |
|
334 |
||
335 |
goal Set.thy "(INT x:UNIV. B x) = (INT x. B x)"; |
|
1754
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diff
changeset
|
336 |
by (Fast_tac 1); |
1531 | 337 |
qed "INT_UNIV"; |
338 |
Addsimps[INT_UNIV]; |
|
1179
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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923
diff
changeset
|
339 |
|
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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923
diff
changeset
|
340 |
goal Set.thy "(UN x:insert a A. B x) = B a Un UNION A B"; |
1754
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parents:
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diff
changeset
|
341 |
by (Fast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
342 |
qed "UN_insert"; |
1531 | 343 |
Addsimps[UN_insert]; |
344 |
||
345 |
goal Set.thy "(INT x:insert a A. B x) = B a Int INTER A B"; |
|
1754
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parents:
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diff
changeset
|
346 |
by (Fast_tac 1); |
1531 | 347 |
qed "INT_insert"; |
348 |
Addsimps[INT_insert]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
349 |
|
923 | 350 |
goal Set.thy "Union(range(f)) = (UN x.f(x))"; |
1754
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parents:
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diff
changeset
|
351 |
by (Fast_tac 1); |
923 | 352 |
qed "Union_range_eq"; |
353 |
||
354 |
goal Set.thy "Inter(range(f)) = (INT x.f(x))"; |
|
1754
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parents:
1748
diff
changeset
|
355 |
by (Fast_tac 1); |
923 | 356 |
qed "Inter_range_eq"; |
357 |
||
358 |
goal Set.thy "Union(B``A) = (UN x:A. B(x))"; |
|
1754
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parents:
1748
diff
changeset
|
359 |
by (Fast_tac 1); |
923 | 360 |
qed "Union_image_eq"; |
361 |
||
362 |
goal Set.thy "Inter(B``A) = (INT x:A. B(x))"; |
|
1754
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berghofe
parents:
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diff
changeset
|
363 |
by (Fast_tac 1); |
923 | 364 |
qed "Inter_image_eq"; |
365 |
||
366 |
goal Set.thy "!!A. a: A ==> (UN y:A. c) = c"; |
|
1754
852093aeb0ab
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parents:
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diff
changeset
|
367 |
by (Fast_tac 1); |
923 | 368 |
qed "UN_constant"; |
369 |
||
370 |
goal Set.thy "!!A. a: A ==> (INT y:A. c) = c"; |
|
1754
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parents:
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diff
changeset
|
371 |
by (Fast_tac 1); |
923 | 372 |
qed "INT_constant"; |
373 |
||
374 |
goal Set.thy "(UN x.B) = B"; |
|
1754
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parents:
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diff
changeset
|
375 |
by (Fast_tac 1); |
923 | 376 |
qed "UN1_constant"; |
1531 | 377 |
Addsimps[UN1_constant]; |
923 | 378 |
|
379 |
goal Set.thy "(INT x.B) = B"; |
|
1754
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parents:
1748
diff
changeset
|
380 |
by (Fast_tac 1); |
923 | 381 |
qed "INT1_constant"; |
1531 | 382 |
Addsimps[INT1_constant]; |
923 | 383 |
|
384 |
goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
385 |
by (Fast_tac 1); |
923 | 386 |
qed "UN_eq"; |
387 |
||
388 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
389 |
goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
|
1754
852093aeb0ab
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berghofe
parents:
1748
diff
changeset
|
390 |
by (Fast_tac 1); |
923 | 391 |
qed "INT_eq"; |
392 |
||
393 |
(*Distributive laws...*) |
|
394 |
||
395 |
goal Set.thy "A Int Union(B) = (UN C:B. A Int C)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
396 |
by (Fast_tac 1); |
923 | 397 |
qed "Int_Union"; |
398 |
||
399 |
(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: |
|
400 |
Union of a family of unions **) |
|
401 |
goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
402 |
by (Fast_tac 1); |
923 | 403 |
qed "Un_Union_image"; |
404 |
||
405 |
(*Equivalent version*) |
|
406 |
goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
407 |
by (Fast_tac 1); |
923 | 408 |
qed "UN_Un_distrib"; |
409 |
||
410 |
goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
411 |
by (Fast_tac 1); |
923 | 412 |
qed "Un_Inter"; |
413 |
||
414 |
goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
|
415 |
by (best_tac eq_cs 1); |
|
416 |
qed "Int_Inter_image"; |
|
417 |
||
418 |
(*Equivalent version*) |
|
419 |
goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
420 |
by (Fast_tac 1); |
923 | 421 |
qed "INT_Int_distrib"; |
422 |
||
423 |
(*Halmos, Naive Set Theory, page 35.*) |
|
424 |
goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
425 |
by (Fast_tac 1); |
923 | 426 |
qed "Int_UN_distrib"; |
427 |
||
428 |
goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
429 |
by (Fast_tac 1); |
923 | 430 |
qed "Un_INT_distrib"; |
431 |
||
432 |
goal Set.thy |
|
433 |
"(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
434 |
by (Fast_tac 1); |
923 | 435 |
qed "Int_UN_distrib2"; |
436 |
||
437 |
goal Set.thy |
|
438 |
"(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
439 |
by (Fast_tac 1); |
923 | 440 |
qed "Un_INT_distrib2"; |
441 |
||
1548 | 442 |
section "-"; |
923 | 443 |
|
444 |
goal Set.thy "A-A = {}"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
445 |
by (Fast_tac 1); |
923 | 446 |
qed "Diff_cancel"; |
1531 | 447 |
Addsimps[Diff_cancel]; |
923 | 448 |
|
449 |
goal Set.thy "{}-A = {}"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
450 |
by (Fast_tac 1); |
923 | 451 |
qed "empty_Diff"; |
1531 | 452 |
Addsimps[empty_Diff]; |
923 | 453 |
|
454 |
goal Set.thy "A-{} = A"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
455 |
by (Fast_tac 1); |
923 | 456 |
qed "Diff_empty"; |
1531 | 457 |
Addsimps[Diff_empty]; |
458 |
||
459 |
goal Set.thy "A-UNIV = {}"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
460 |
by (Fast_tac 1); |
1531 | 461 |
qed "Diff_UNIV"; |
462 |
Addsimps[Diff_UNIV]; |
|
463 |
||
464 |
goal Set.thy "!!x. x~:A ==> A - insert x B = A-B"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
465 |
by (Fast_tac 1); |
1531 | 466 |
qed "Diff_insert0"; |
467 |
Addsimps [Diff_insert0]; |
|
923 | 468 |
|
469 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
470 |
goal Set.thy "A - insert a B = A - B - {a}"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
471 |
by (Fast_tac 1); |
923 | 472 |
qed "Diff_insert"; |
473 |
||
474 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
475 |
goal Set.thy "A - insert a B = A - {a} - B"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
476 |
by (Fast_tac 1); |
923 | 477 |
qed "Diff_insert2"; |
478 |
||
1531 | 479 |
goal Set.thy "insert x A - B = (if x:B then A-B else insert x (A-B))"; |
1553 | 480 |
by (simp_tac (!simpset setloop split_tac[expand_if]) 1); |
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
481 |
by (Fast_tac 1); |
1531 | 482 |
qed "insert_Diff_if"; |
483 |
||
484 |
goal Set.thy "!!x. x:B ==> insert x A - B = A-B"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
485 |
by (Fast_tac 1); |
1531 | 486 |
qed "insert_Diff1"; |
487 |
Addsimps [insert_Diff1]; |
|
488 |
||
923 | 489 |
val prems = goal Set.thy "a:A ==> insert a (A-{a}) = A"; |
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
490 |
by (fast_tac (!claset addSIs prems) 1); |
923 | 491 |
qed "insert_Diff"; |
492 |
||
493 |
goal Set.thy "A Int (B-A) = {}"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
494 |
by (Fast_tac 1); |
923 | 495 |
qed "Diff_disjoint"; |
1531 | 496 |
Addsimps[Diff_disjoint]; |
923 | 497 |
|
498 |
goal Set.thy "!!A. A<=B ==> A Un (B-A) = B"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
499 |
by (Fast_tac 1); |
923 | 500 |
qed "Diff_partition"; |
501 |
||
502 |
goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
503 |
by (Fast_tac 1); |
923 | 504 |
qed "double_diff"; |
505 |
||
506 |
goal Set.thy "A - (B Un C) = (A-B) Int (A-C)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
507 |
by (Fast_tac 1); |
923 | 508 |
qed "Diff_Un"; |
509 |
||
510 |
goal Set.thy "A - (B Int C) = (A-B) Un (A-C)"; |
|
1754
852093aeb0ab
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1748
diff
changeset
|
511 |
by (Fast_tac 1); |
923 | 512 |
qed "Diff_Int"; |
513 |
||
1531 | 514 |
Addsimps[subset_UNIV, empty_subsetI, subset_refl]; |