src/HOL/equalities.ML
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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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section "{}";
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goal Set.thy "{x.False} = {}";
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by (Blast_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 (Blast_tac 1);
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qed "subset_empty";
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Addsimps [subset_empty];
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section "insert";
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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 (Blast_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 (Blast_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 (Blast_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 (insert y A) = insert y (insert x A)";
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by (Blast_tac 1);
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qed "insert_commute";
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goal Set.thy "(insert x A <= B) = (x:B & A <= B)";
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by (Blast_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 (Blast_tac 1);
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qed "mk_disjoint_insert";
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goal Set.thy
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    "!!A. A~={} ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)";
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by (Blast_tac 1);
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qed "UN_insert_distrib";
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goal Set.thy "(UN x. insert a (B x)) = insert a (UN x. B x)";
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by (Blast_tac 1);
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qed "UN1_insert_distrib";
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section "``";
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goal Set.thy "f``{} = {}";
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by (Blast_tac 1);
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qed "image_empty";
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Addsimps[image_empty];
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goal Set.thy "f``insert a B = insert (f a) (f``B)";
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by (Blast_tac 1);
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qed "image_insert";
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Addsimps[image_insert];
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qed_goal "ball_image" Set.thy "(!y:F``S. P y) = (!x:S. P (F x))"
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 (fn _ => [Blast_tac 1]);
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goal Set.thy "!!x. x:A ==> insert (f x) (f``A) = f``A";
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by (Blast_tac 1);
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qed "insert_image";
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Addsimps [insert_image];
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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 (Blast_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 _ => [Blast_tac 1]);
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qed_goalw "image_range" Set.thy [image_def]
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 "f``range g = range (%x. f (g x))" 
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 (fn _ => [rtac Collect_cong 1, Blast_tac 1]);
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section "Int";
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goal Set.thy "A Int A = A";
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by (Blast_tac 1);
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qed "Int_absorb";
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Addsimps[Int_absorb];
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goal Set.thy "A Int B  =  B Int A";
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by (Blast_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 (Blast_tac 1);
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qed "Int_assoc";
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goal Set.thy "{} Int B = {}";
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by (Blast_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 (Blast_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 (Blast_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 (Blast_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 (Blast_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 (Blast_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";
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goal Set.thy "A Un A = A";
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by (Blast_tac 1);
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qed "Un_absorb";
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Addsimps[Un_absorb];
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goal Set.thy "A Un B  =  B Un A";
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by (Blast_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 (Blast_tac 1);
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qed "Un_assoc";
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goal Set.thy "{} Un B = B";
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by (Blast_tac 1);
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qed "Un_empty_left";
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Addsimps[Un_empty_left];
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goal Set.thy "A Un {} = A";
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by (Blast_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 (Blast_tac 1);
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qed "Un_UNIV_left";
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Addsimps[Un_UNIV_left];
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goal Set.thy "A Un UNIV = UNIV";
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by (Blast_tac 1);
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qed "Un_UNIV_right";
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Addsimps[Un_UNIV_right];
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1843
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goal Set.thy "(insert a B) Un C = insert a (B Un C)";
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by (Blast_tac 1);
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qed "Un_insert_left";
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goal Set.thy "A Un (insert a B) = insert a (A Un B)";
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by (Blast_tac 1);
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qed "Un_insert_right";
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goal Set.thy "(A Int B) Un C  =  (A Un C) Int (B Un C)";
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by (Blast_tac 1);
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qed "Un_Int_distrib";
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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 (Blast_tac 1);
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qed "Un_Int_crazy";
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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";
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goal Set.thy "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)";
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by (Blast_tac 1);
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qed "subset_insert_iff";
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goal Set.thy "(A Un B = {}) = (A = {} & B = {})";
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by (fast_tac (!claset addEs [equalityCE]) 1);
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qed "Un_empty";
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Addsimps[Un_empty];
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section "Compl";
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goal Set.thy "A Int Compl(A) = {}";
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by (Blast_tac 1);
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qed "Compl_disjoint";
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Addsimps[Compl_disjoint];
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goal Set.thy "A Un Compl(A) = UNIV";
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d8f254ad1ab9 Calls Blast_tac
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by (Blast_tac 1);
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qed "Compl_partition";
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goal Set.thy "Compl(Compl(A)) = A";
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d8f254ad1ab9 Calls Blast_tac
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by (Blast_tac 1);
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qed "double_complement";
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Addsimps[double_complement];
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goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)";
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by (Blast_tac 1);
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qed "Compl_Un";
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goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)";
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d8f254ad1ab9 Calls Blast_tac
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by (Blast_tac 1);
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qed "Compl_Int";
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goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))";
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d8f254ad1ab9 Calls Blast_tac
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   253
by (Blast_tac 1);
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qed "Compl_UN";
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goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))";
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d8f254ad1ab9 Calls Blast_tac
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   257
by (Blast_tac 1);
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qed "Compl_INT";
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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)";
1754
852093aeb0ab Replaced fast_tac by Fast_tac (which uses default claset)
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parents: 1748
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by (fast_tac (!claset addSEs [equalityE]) 1);
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qed "Un_Int_assoc_eq";
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section "Union";
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goal Set.thy "Union({}) = {}";
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d8f254ad1ab9 Calls Blast_tac
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   270
by (Blast_tac 1);
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qed "Union_empty";
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Addsimps[Union_empty];
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   274
goal Set.thy "Union(UNIV) = UNIV";
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d8f254ad1ab9 Calls Blast_tac
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   275
by (Blast_tac 1);
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qed "Union_UNIV";
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Addsimps[Union_UNIV];
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goal Set.thy "Union(insert a B) = a Un Union(B)";
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d8f254ad1ab9 Calls Blast_tac
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   280
by (Blast_tac 1);
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qed "Union_insert";
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Addsimps[Union_insert];
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   284
goal Set.thy "Union(A Un B) = Union(A) Un Union(B)";
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d8f254ad1ab9 Calls Blast_tac
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parents: 2519
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   285
by (Blast_tac 1);
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qed "Union_Un_distrib";
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Addsimps[Union_Un_distrib];
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goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)";
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d8f254ad1ab9 Calls Blast_tac
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   290
by (Blast_tac 1);
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qed "Union_Int_subset";
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   292
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val prems = goal Set.thy
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   "(Union(C) Int A = {}) = (! B:C. B Int A = {})";
1754
852093aeb0ab Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1748
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   295
by (fast_tac (!claset addSEs [equalityE]) 1);
923
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qed "Union_disjoint";
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   297
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   298
section "Inter";
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   299
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   300
goal Set.thy "Inter({}) = UNIV";
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d8f254ad1ab9 Calls Blast_tac
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   301
by (Blast_tac 1);
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qed "Inter_empty";
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   303
Addsimps[Inter_empty];
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   304
e5eb247ad13c Added a constant UNIV == {x.True}
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   305
goal Set.thy "Inter(UNIV) = {}";
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d8f254ad1ab9 Calls Blast_tac
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parents: 2519
diff changeset
   306
by (Blast_tac 1);
1531
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   307
qed "Inter_UNIV";
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   308
Addsimps[Inter_UNIV];
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   309
e5eb247ad13c Added a constant UNIV == {x.True}
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   310
goal Set.thy "Inter(insert a B) = a Int Inter(B)";
2891
d8f254ad1ab9 Calls Blast_tac
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parents: 2519
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   311
by (Blast_tac 1);
1531
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   312
qed "Inter_insert";
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   313
Addsimps[Inter_insert];
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parents: 1465
diff changeset
   314
1564
822575c737bd Deleted faulty comment; proved new rule Inter_Un_subset
paulson
parents: 1553
diff changeset
   315
goal Set.thy "Inter(A) Un Inter(B) <= Inter(A Int B)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   316
by (Blast_tac 1);
1564
822575c737bd Deleted faulty comment; proved new rule Inter_Un_subset
paulson
parents: 1553
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   317
qed "Inter_Un_subset";
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   318
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diff changeset
   319
goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   320
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   321
qed "Inter_Un_distrib";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   322
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   323
section "UN and INT";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   324
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   325
(*Basic identities*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   326
1179
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   327
goal Set.thy "(UN x:{}. B x) = {}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   328
by (Blast_tac 1);
1179
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   329
qed "UN_empty";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   330
Addsimps[UN_empty];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   331
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   332
goal Set.thy "(UN x:UNIV. B x) = (UN x. B x)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   333
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   334
qed "UN_UNIV";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   335
Addsimps[UN_UNIV];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   336
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   337
goal Set.thy "(INT x:{}. B x) = UNIV";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   338
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   339
qed "INT_empty";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   340
Addsimps[INT_empty];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   341
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   342
goal Set.thy "(INT x:UNIV. B x) = (INT x. B x)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   343
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   344
qed "INT_UNIV";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   345
Addsimps[INT_UNIV];
1179
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   346
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   347
goal Set.thy "(UN x:insert a A. B x) = B a Un UNION A B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   348
by (Blast_tac 1);
1179
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   349
qed "UN_insert";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   350
Addsimps[UN_insert];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   351
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   352
goal Set.thy "(INT x:insert a A. B x) = B a Int INTER A B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   353
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   354
qed "INT_insert";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   355
Addsimps[INT_insert];
1179
7678408f9751 Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents: 923
diff changeset
   356
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   357
goal Set.thy
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   358
    "!!A. A~={} ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   359
by (Blast_tac 1);
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   360
qed "INT_insert_distrib";
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   361
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   362
goal Set.thy "(INT x. insert a (B x)) = insert a (INT x. B x)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   363
by (Blast_tac 1);
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   364
qed "INT1_insert_distrib";
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   365
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   366
goal Set.thy "Union(range(f)) = (UN x.f(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   367
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   368
qed "Union_range_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   370
goal Set.thy "Inter(range(f)) = (INT x.f(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   371
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   372
qed "Inter_range_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   373
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
goal Set.thy "Union(B``A) = (UN x:A. B(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   375
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
qed "Union_image_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
goal Set.thy "Inter(B``A) = (INT x:A. B(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   379
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
qed "Inter_image_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
goal Set.thy "!!A. a: A ==> (UN y:A. c) = c";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   383
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
qed "UN_constant";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
goal Set.thy "!!A. a: A ==> (INT y:A. c) = c";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   387
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   388
qed "INT_constant";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
goal Set.thy "(UN x.B) = B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   391
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
qed "UN1_constant";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   393
Addsimps[UN1_constant];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   394
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   395
goal Set.thy "(INT x.B) = B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   396
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   397
qed "INT1_constant";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   398
Addsimps[INT1_constant];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   401
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
qed "UN_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
(*Look: it has an EXISTENTIAL quantifier*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   406
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
qed "INT_eq";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   408
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
(*Distributive laws...*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   411
goal Set.thy "A Int Union(B) = (UN C:B. A Int C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   412
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
qed "Int_Union";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 2891
diff changeset
   415
(* Devlin, Setdamentals of Contemporary Set Theory, page 12, exercise 5: 
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   416
   Union of a family of unions **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C)  Un  Union(B``C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   418
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
qed "Un_Union_image";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   420
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
(*Equivalent version*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   422
goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i))  Un  (UN i:I. B(i))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   423
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
qed "UN_Un_distrib";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   427
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
qed "Un_Inter";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   429
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   430
goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   431
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
qed "Int_Inter_image";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   434
(*Equivalent version*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   435
goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   436
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
qed "INT_Int_distrib";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   439
(*Halmos, Naive Set Theory, page 35.*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   441
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   442
qed "Int_UN_distrib";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   443
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   445
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   446
qed "Un_INT_distrib";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   448
goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   449
    "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   450
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   451
qed "Int_UN_distrib2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   452
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   453
goal Set.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   454
    "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   455
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   456
qed "Un_INT_distrib2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   457
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   458
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   459
section"Bounded quantifiers";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   460
2519
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   461
(** These are not added to the default simpset because (a) they duplicate the
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   462
    body and (b) there are no similar rules for Int. **)
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   463
2519
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   464
goal Set.thy "(ALL x:A Un B.P x) = ((ALL x:A.P x) & (ALL x:B.P x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   465
by (Blast_tac 1);
2519
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   466
qed "ball_Un";
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   467
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   468
goal Set.thy "(EX x:A Un B.P x) = ((EX x:A.P x) | (EX x:B.P x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   469
by (Blast_tac 1);
2519
761e3094e32f New rewrites for bounded quantifiers
paulson
parents: 2513
diff changeset
   470
qed "bex_Un";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   471
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 2031
diff changeset
   472
1548
afe750876848 Added 'section' commands
nipkow
parents: 1531
diff changeset
   473
section "-";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   474
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   475
goal Set.thy "A-A = {}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   476
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   477
qed "Diff_cancel";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   478
Addsimps[Diff_cancel];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   480
goal Set.thy "{}-A = {}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   481
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   482
qed "empty_Diff";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   483
Addsimps[empty_Diff];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   484
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   485
goal Set.thy "A-{} = A";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   486
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   487
qed "Diff_empty";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   488
Addsimps[Diff_empty];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   489
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   490
goal Set.thy "A-UNIV = {}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   491
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   492
qed "Diff_UNIV";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   493
Addsimps[Diff_UNIV];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   494
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   495
goal Set.thy "!!x. x~:A ==> A - insert x B = A-B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   496
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   497
qed "Diff_insert0";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   498
Addsimps [Diff_insert0];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   499
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   500
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   501
goal Set.thy "A - insert a B = A - B - {a}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   502
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   503
qed "Diff_insert";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   504
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   505
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   506
goal Set.thy "A - insert a B = A - {a} - B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   507
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   508
qed "Diff_insert2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   509
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   510
goal Set.thy "insert x A - B = (if x:B then A-B else insert x (A-B))";
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   511
by (simp_tac (!simpset setloop split_tac[expand_if]) 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   512
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   513
qed "insert_Diff_if";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   514
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   515
goal Set.thy "!!x. x:B ==> insert x A - B = A-B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   516
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   517
qed "insert_Diff1";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   518
Addsimps [insert_Diff1];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   519
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   520
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
   521
by (fast_tac (!claset addSIs prems) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   522
qed "insert_Diff";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   523
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   524
goal Set.thy "A Int (B-A) = {}";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   525
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   526
qed "Diff_disjoint";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   527
Addsimps[Diff_disjoint];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   528
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   529
goal Set.thy "!!A. A<=B ==> A Un (B-A) = B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   530
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   531
qed "Diff_partition";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   532
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   533
goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   534
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   535
qed "double_diff";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   536
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   537
goal Set.thy "A - (B Un C) = (A-B) Int (A-C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   538
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   539
qed "Diff_Un";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   540
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   541
goal Set.thy "A - (B Int C) = (A-B) Un (A-C)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   542
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   543
qed "Diff_Int";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   544
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   545
Addsimps[subset_UNIV, empty_subsetI, subset_refl];
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   546
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   547
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   548
(** Miniscoping: pushing in big Unions and Intersections **)
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   549
local
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2519
diff changeset
   550
  fun prover s = prove_goal Set.thy s (fn _ => [Blast_tac 1])
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   551
in
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   552
val UN1_simps = map prover 
2031
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   553
                ["(UN x. insert a (B x)) = insert a (UN x. B x)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   554
                 "(UN x. A x Int B)  = ((UN x.A x) Int B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   555
                 "(UN x. A Int B x)  = (A Int (UN x.B x))",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   556
                 "(UN x. A x Un B)   = ((UN x.A x) Un B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   557
                 "(UN x. A Un B x)   = (A Un (UN x.B x))",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   558
                 "(UN x. A x - B)    = ((UN x.A x) - B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   559
                 "(UN x. A - B x)    = (A - (INT x.B x))"];
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   560
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   561
val INT1_simps = map prover
2031
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   562
                ["(INT x. insert a (B x)) = insert a (INT x. B x)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   563
                 "(INT x. A x Int B) = ((INT x.A x) Int B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   564
                 "(INT x. A Int B x) = (A Int (INT x.B x))",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   565
                 "(INT x. A x Un B)  = ((INT x.A x) Un B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   566
                 "(INT x. A Un B x)  = (A Un (INT x.B x))",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   567
                 "(INT x. A x - B)   = ((INT x.A x) - B)",
03a843f0f447 Ran expandshort
paulson
parents: 2024
diff changeset
   568
                 "(INT x. A - B x)   = (A - (UN x.B x))"];
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   569
2513
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   570
val UN_simps = map prover 
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   571
                ["(UN x:C. A x Int B)  = ((UN x:C.A x) Int B)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   572
                 "(UN x:C. A Int B x)  = (A Int (UN x:C.B x))",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   573
                 "(UN x:C. A x - B)    = ((UN x:C.A x) - B)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   574
                 "(UN x:C. A - B x)    = (A - (INT x:C.B x))"];
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   575
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   576
val INT_simps = map prover
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   577
                ["(INT x:C. insert a (B x)) = insert a (INT x:C. B x)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   578
                 "(INT x:C. A x Un B)  = ((INT x:C.A x) Un B)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   579
                 "(INT x:C. A Un B x)  = (A Un (INT x:C.B x))"];
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   580
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   581
(*The missing laws for bounded Unions and Intersections are conditional
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   582
  on the index set's being non-empty.  Thus they are probably NOT worth 
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   583
  adding as default rewrites.*)
2513
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   584
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   585
val ball_simps = map prover
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   586
    ["(ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   587
     "(ALL x:A. P | Q x) = (P | (ALL x:A. Q x))",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   588
     "(ALL x:{}. P x) = True",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   589
     "(ALL x:insert a B. P x) = (P(a) & (ALL x:B. P x))",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   590
     "(ALL x:Union(A). P x) = (ALL y:A. ALL x:y. P x)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   591
     "(ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"];
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   592
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   593
val ball_conj_distrib = 
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   594
    prover "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))";
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   595
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   596
val bex_simps = map prover
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   597
    ["(EX x:A. P x & Q) = ((EX x:A. P x) & Q)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   598
     "(EX x:A. P & Q x) = (P & (EX x:A. Q x))",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   599
     "(EX x:{}. P x) = False",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   600
     "(EX x:insert a B. P x) = (P(a) | (EX x:B. P x))",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   601
     "(EX x:Union(A). P x) = (EX y:A. EX x:y.  P x)",
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   602
     "(EX x:Collect Q. P x) = (EX x. Q x & P x)"];
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   603
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   604
val bex_conj_distrib = 
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   605
    prover "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))";
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   606
2021
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   607
end;
dd5866263153 Added miniscoping for UN and INT
paulson
parents: 1917
diff changeset
   608
2513
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   609
Addsimps (UN1_simps @ INT1_simps @ UN_simps @ INT_simps @ 
d708d8cdc8e8 New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents: 2512
diff changeset
   610
	  ball_simps @ bex_simps);