(* Title: CCL/subset
ID: $Id$
Modified version of
Title: HOL/subset
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Derived rules involving subsets
Union and Intersection as lattice operations
*)
(*** Big Union -- least upper bound of a set ***)
val prems = goal Set.thy
"B:A ==> B <= Union(A)";
by (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1));
qed "Union_upper";
val prems = goal Set.thy
"[| !!X. X:A ==> X<=C |] ==> Union(A) <= C";
by (REPEAT (ares_tac [subsetI] 1
ORELSE eresolve_tac ([UnionE] @ (prems RL [subsetD])) 1));
qed "Union_least";
(*** Big Intersection -- greatest lower bound of a set ***)
val prems = goal Set.thy
"B:A ==> Inter(A) <= B";
by (REPEAT (resolve_tac (prems@[subsetI]) 1
ORELSE etac InterD 1));
qed "Inter_lower";
val prems = goal Set.thy
"[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)";
by (REPEAT (ares_tac [subsetI,InterI] 1
ORELSE eresolve_tac (prems RL [subsetD]) 1));
qed "Inter_greatest";
(*** Finite Union -- the least upper bound of 2 sets ***)
goal Set.thy "A <= A Un B";
by (REPEAT (ares_tac [subsetI,UnI1] 1));
qed "Un_upper1";
goal Set.thy "B <= A Un B";
by (REPEAT (ares_tac [subsetI,UnI2] 1));
qed "Un_upper2";
val prems = goal Set.thy "[| A<=C; B<=C |] ==> A Un B <= C";
by (cut_facts_tac prems 1);
by (DEPTH_SOLVE (ares_tac [subsetI] 1
ORELSE eresolve_tac [UnE,subsetD] 1));
qed "Un_least";
(*** Finite Intersection -- the greatest lower bound of 2 sets *)
goal Set.thy "A Int B <= A";
by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
qed "Int_lower1";
goal Set.thy "A Int B <= B";
by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
qed "Int_lower2";
val prems = goal Set.thy "[| C<=A; C<=B |] ==> C <= A Int B";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [subsetI,IntI] 1
ORELSE etac subsetD 1));
qed "Int_greatest";
(*** Monotonicity ***)
val [prem] = goalw Set.thy [mono_def]
"[| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)";
by (REPEAT (ares_tac [allI, impI, prem] 1));
qed "monoI";
val [major,minor] = goalw Set.thy [mono_def]
"[| mono(f); A <= B |] ==> f(A) <= f(B)";
by (rtac (major RS spec RS spec RS mp) 1);
by (rtac minor 1);
qed "monoD";
val [prem] = goal Set.thy "mono(f) ==> f(A) Un f(B) <= f(A Un B)";
by (rtac Un_least 1);
by (rtac (Un_upper1 RS (prem RS monoD)) 1);
by (rtac (Un_upper2 RS (prem RS monoD)) 1);
qed "mono_Un";
val [prem] = goal Set.thy "mono(f) ==> f(A Int B) <= f(A) Int f(B)";
by (rtac Int_greatest 1);
by (rtac (Int_lower1 RS (prem RS monoD)) 1);
by (rtac (Int_lower2 RS (prem RS monoD)) 1);
qed "mono_Int";
(****)
val set_cs = FOL_cs
addSIs [ballI, subsetI, InterI, INT_I, CollectI,
ComplI, IntI, UnCI, singletonI]
addIs [bexI, UnionI, UN_I]
addSEs [bexE, UnionE, UN_E,
CollectE, ComplE, IntE, UnE, emptyE, singletonE]
addEs [ballE, InterD, InterE, INT_D, INT_E, subsetD, subsetCE];
fun cfast_tac prems = cut_facts_tac prems THEN' fast_tac set_cs;
fun prover s = prove_goal Set.thy s (fn _=>[fast_tac set_cs 1]);
val mem_rews = [trivial_set,empty_eq] @ (map prover
[ "(a : A Un B) <-> (a:A | a:B)",
"(a : A Int B) <-> (a:A & a:B)",
"(a : Compl(B)) <-> (~a:B)",
"(a : {b}) <-> (a=b)",
"(a : {}) <-> False",
"(a : {x. P(x)}) <-> P(a)" ]);
val set_congs = [ball_cong, bex_cong, INT_cong, UN_cong];
val set_ss = FOL_ss addcongs set_congs
addsimps mem_rews;