(* Title: ZF/UNITY/Monotonicity.thy
ID: $Id \<in> Monotonicity.thy,v 1.1 2003/05/28 16:13:42 paulson Exp $
Author: Sidi O Ehmety, Cambridge University Computer Laboratory
Copyright 2002 University of Cambridge
Monotonicity of an operator (meta-function) with respect to arbitrary set relations.
*)
header{*Monotonicity of an Operator WRT a Relation*}
theory Monotonicity imports GenPrefix MultisetSum
begin
definition
mono1 :: "[i, i, i, i, i=>i] => o" where
"mono1(A, r, B, s, f) ==
(\<forall>x \<in> A. \<forall>y \<in> A. <x,y> \<in> r --> <f(x), f(y)> \<in> s) & (\<forall>x \<in> A. f(x) \<in> B)"
(* monotonicity of a 2-place meta-function f *)
definition
mono2 :: "[i, i, i, i, i, i, [i,i]=>i] => o" where
"mono2(A, r, B, s, C, t, f) ==
(\<forall>x \<in> A. \<forall>y \<in> A. \<forall>u \<in> B. \<forall>v \<in> B.
<x,y> \<in> r & <u,v> \<in> s --> <f(x,u), f(y,v)> \<in> t) &
(\<forall>x \<in> A. \<forall>y \<in> B. f(x,y) \<in> C)"
(* Internalized relations on sets and multisets *)
definition
SetLe :: "i =>i" where
"SetLe(A) == {<x,y> \<in> Pow(A)*Pow(A). x <= y}"
definition
MultLe :: "[i,i] =>i" where
"MultLe(A, r) == multirel(A, r - id(A)) Un id(Mult(A))"
lemma mono1D:
"[| mono1(A, r, B, s, f); <x, y> \<in> r; x \<in> A; y \<in> A |] ==> <f(x), f(y)> \<in> s"
by (unfold mono1_def, auto)
lemma mono2D:
"[| mono2(A, r, B, s, C, t, f);
<x, y> \<in> r; <u,v> \<in> s; x \<in> A; y \<in> A; u \<in> B; v \<in> B |]
==> <f(x, u), f(y,v)> \<in> t"
by (unfold mono2_def, auto)
(** Monotonicity of take **)
lemma take_mono_left_lemma:
"[| i le j; xs \<in> list(A); i \<in> nat; j \<in> nat |]
==> <take(i, xs), take(j, xs)> \<in> prefix(A)"
apply (case_tac "length (xs) le i")
apply (subgoal_tac "length (xs) le j")
apply (simp)
apply (blast intro: le_trans)
apply (drule not_lt_imp_le, auto)
apply (case_tac "length (xs) le j")
apply (auto simp add: take_prefix)
apply (drule not_lt_imp_le, auto)
apply (drule_tac m = i in less_imp_succ_add, auto)
apply (subgoal_tac "i #+ k le length (xs) ")
apply (simp add: take_add prefix_iff take_type drop_type)
apply (blast intro: leI)
done
lemma take_mono_left:
"[| i le j; xs \<in> list(A); j \<in> nat |]
==> <take(i, xs), take(j, xs)> \<in> prefix(A)"
by (blast intro: le_in_nat take_mono_left_lemma)
lemma take_mono_right:
"[| <xs,ys> \<in> prefix(A); i \<in> nat |]
==> <take(i, xs), take(i, ys)> \<in> prefix(A)"
by (auto simp add: prefix_iff)
lemma take_mono:
"[| i le j; <xs, ys> \<in> prefix(A); j \<in> nat |]
==> <take(i, xs), take(j, ys)> \<in> prefix(A)"
apply (rule_tac b = "take (j, xs) " in prefix_trans)
apply (auto dest: prefix_type [THEN subsetD] intro: take_mono_left take_mono_right)
done
lemma mono_take [iff]:
"mono2(nat, Le, list(A), prefix(A), list(A), prefix(A), take)"
apply (unfold mono2_def Le_def, auto)
apply (blast intro: take_mono)
done
(** Monotonicity of length **)
lemmas length_mono = prefix_length_le
lemma mono_length [iff]:
"mono1(list(A), prefix(A), nat, Le, length)"
apply (unfold mono1_def)
apply (auto dest: prefix_length_le simp add: Le_def)
done
(** Monotonicity of Un **)
lemma mono_Un [iff]:
"mono2(Pow(A), SetLe(A), Pow(A), SetLe(A), Pow(A), SetLe(A), op Un)"
by (unfold mono2_def SetLe_def, auto)
(* Monotonicity of multiset union *)
lemma mono_munion [iff]:
"mono2(Mult(A), MultLe(A,r), Mult(A), MultLe(A, r), Mult(A), MultLe(A, r), munion)"
apply (unfold mono2_def MultLe_def)
apply (auto simp add: Mult_iff_multiset)
apply (blast intro: munion_multirel_mono munion_multirel_mono1 munion_multirel_mono2 multiset_into_Mult)+
done
lemma mono_succ [iff]: "mono1(nat, Le, nat, Le, succ)"
by (unfold mono1_def Le_def, auto)
end