diff -r 000000000000 -r 7949f97df77a mono.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mono.ML Thu Sep 16 12:21:07 1993 +0200 @@ -0,0 +1,72 @@ +(* Title: HOL/mono + ID: $Id$ + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Copyright 1991 University of Cambridge + +Monotonicity of various operations +*) + +val [prem] = goal Set.thy + "[| !!x. P(x) ==> Q(x) |] ==> Collect(P) <= Collect(Q)"; +by (fast_tac (set_cs addIs [prem]) 1); +val Collect_mono = result(); + +goal Set.thy "!!A B. A<=B ==> f``A <= f``B"; +by (fast_tac set_cs 1); +val image_mono = result(); + +goal Set.thy "!!A B. A<=B ==> Union(A) <= Union(B)"; +by (fast_tac set_cs 1); +val Union_mono = result(); + +goal Set.thy "!!A B. B<=A ==> Inter(A) <= Inter(B)"; +by (fast_tac set_cs 1); +val Inter_anti_mono = result(); + +val prems = goal Set.thy + "[| A<=B; !!x. x:A ==> f(x)<=g(x) |] ==> \ +\ (UN x:A. f(x)) <= (UN x:B. g(x))"; +by (fast_tac (set_cs addIs (prems RL [subsetD])) 1); +val UN_mono = result(); + +val [prem] = goal Set.thy + "[| !!x. f(x)<=g(x) |] ==> (UN x. f(x)) <= (UN x. g(x))"; +by (fast_tac (set_cs addIs [prem RS subsetD]) 1); +val UN1_mono = result(); + +val prems = goal Set.thy + "[| B<=A; !!x. x:A ==> f(x)<=g(x) |] ==> \ +\ (INT x:A. f(x)) <= (INT x:A. g(x))"; +by (fast_tac (set_cs addIs (prems RL [subsetD])) 1); +val INT_anti_mono = result(); + +(*The inclusion is POSITIVE! *) +val [prem] = goal Set.thy + "[| !!x. f(x)<=g(x) |] ==> (INT x. f(x)) <= (INT x. g(x))"; +by (fast_tac (set_cs addIs [prem RS subsetD]) 1); +val INT1_mono = result(); + +goal Set.thy "!!A B. [| A<=C; B<=D |] ==> A Un B <= C Un D"; +by (fast_tac set_cs 1); +val Un_mono = result(); + +goal Set.thy "!!A B. [| A<=C; B<=D |] ==> A Int B <= C Int D"; +by (fast_tac set_cs 1); +val Int_mono = result(); + +goal Set.thy "!!A::'a set. [| A<=C; D<=B |] ==> A-B <= C-D"; +by (fast_tac set_cs 1); +val Diff_mono = result(); + +goal Set.thy "!!A B. A<=B ==> Compl(B) <= Compl(A)"; +by (fast_tac set_cs 1); +val Compl_anti_mono = result(); + +val prems = goal Prod.thy + "[| A<=C; !!x. x:A ==> B<=D |] ==> Sigma(A,%x.B) <= Sigma(C,%x.D)"; +by (cut_facts_tac prems 1); +by (fast_tac (set_cs addIs (prems RL [subsetD]) + addSIs [SigmaI] + addSEs [SigmaE]) 1); +val Sigma_mono = result(); +