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theory Increasing(* Title: ZF/UNITY/Increasing ID: $Id ∈ Increasing.thy,v 1.2 2003/06/02 09:17:52 paulson Exp $ Author: Sidi O Ehmety, Cambridge University Computer Laboratory Copyright 2001 University of Cambridge Increasing's parameters are a state function f, a domain A and an order relation r over the domain A. *) header{*Charpentier's "Increasing" Relation*} theory Increasing imports Constrains Monotonicity begin definition increasing :: "[i, i, i=>i] => i" ("increasing[_]'(_, _')") where "increasing[A](r, f) == {F ∈ program. (∀k ∈ A. F ∈ stable({s ∈ state. <k, f(s)> ∈ r})) & (∀x ∈ state. f(x):A)}" definition Increasing :: "[i, i, i=>i] => i" ("Increasing[_]'(_, _')") where "Increasing[A](r, f) == {F ∈ program. (∀k ∈ A. F ∈ Stable({s ∈ state. <k, f(s)> ∈ r})) & (∀x ∈ state. f(x):A)}" abbreviation (input) IncWrt :: "[i=>i, i, i] => i" ("(_ IncreasingWrt _ '/ _)" [60, 0, 60] 60) where "f IncreasingWrt r/A == Increasing[A](r,f)" (** increasing **) lemma increasing_type: "increasing[A](r, f) <= program" by (unfold increasing_def, blast) lemma increasing_into_program: "F ∈ increasing[A](r, f) ==> F ∈ program" by (unfold increasing_def, blast) lemma increasing_imp_stable: "[| F ∈ increasing[A](r, f); x ∈ A |] ==>F ∈ stable({s ∈ state. <x, f(s)>:r})" by (unfold increasing_def, blast) lemma increasingD: "F ∈ increasing[A](r,f) ==> F ∈ program & (∃a. a ∈ A) & (∀s ∈ state. f(s):A)" apply (unfold increasing_def) apply (subgoal_tac "∃x. x ∈ state") apply (auto dest: stable_type [THEN subsetD] intro: st0_in_state) done lemma increasing_constant [simp]: "F ∈ increasing[A](r, %s. c) <-> F ∈ program & c ∈ A" apply (unfold increasing_def stable_def) apply (subgoal_tac "∃x. x ∈ state") apply (auto dest: stable_type [THEN subsetD] intro: st0_in_state) done lemma subset_increasing_comp: "[| mono1(A, r, B, s, g); refl(A, r); trans[B](s) |] ==> increasing[A](r, f) <= increasing[B](s, g comp f)" apply (unfold increasing_def stable_def part_order_def constrains_def mono1_def metacomp_def, clarify, simp) apply clarify apply (subgoal_tac "xa ∈ state") prefer 2 apply (blast dest!: ActsD) apply (subgoal_tac "<f (xb), f (xb) >:r") prefer 2 apply (force simp add: refl_def) apply (rotate_tac 5) apply (drule_tac x = "f (xb) " in bspec) apply (rotate_tac [2] -1) apply (drule_tac [2] x = act in bspec, simp_all) apply (drule_tac A = "act``?u" and c = xa in subsetD, blast) apply (drule_tac x = "f (xa) " and x1 = "f (xb) " in bspec [THEN bspec]) apply (rule_tac [3] b = "g (f (xb))" and A = B in trans_onD) apply simp_all done lemma imp_increasing_comp: "[| F ∈ increasing[A](r, f); mono1(A, r, B, s, g); refl(A, r); trans[B](s) |] ==> F ∈ increasing[B](s, g comp f)" by (rule subset_increasing_comp [THEN subsetD], auto) lemma strict_increasing: "increasing[nat](Le, f) <= increasing[nat](Lt, f)" by (unfold increasing_def Lt_def, auto) lemma strict_gt_increasing: "increasing[nat](Ge, f) <= increasing[nat](Gt, f)" apply (unfold increasing_def Gt_def Ge_def, auto) apply (erule natE) apply (auto simp add: stable_def) done (** Increasing **) lemma increasing_imp_Increasing: "F ∈ increasing[A](r, f) ==> F ∈ Increasing[A](r, f)" apply (unfold increasing_def Increasing_def) apply (auto intro: stable_imp_Stable) done lemma Increasing_type: "Increasing[A](r, f) <= program" by (unfold Increasing_def, auto) lemma Increasing_into_program: "F ∈ Increasing[A](r, f) ==> F ∈ program" by (unfold Increasing_def, auto) lemma Increasing_imp_Stable: "[| F ∈ Increasing[A](r, f); a ∈ A |] ==> F ∈ Stable({s ∈ state. <a,f(s)>:r})" by (unfold Increasing_def, blast) lemma IncreasingD: "F ∈ Increasing[A](r, f) ==> F ∈ program & (∃a. a ∈ A) & (∀s ∈ state. f(s):A)" apply (unfold Increasing_def) apply (subgoal_tac "∃x. x ∈ state") apply (auto intro: st0_in_state) done lemma Increasing_constant [simp]: "F ∈ Increasing[A](r, %s. c) <-> F ∈ program & (c ∈ A)" apply (subgoal_tac "∃x. x ∈ state") apply (auto dest!: IncreasingD intro: st0_in_state increasing_imp_Increasing) done lemma subset_Increasing_comp: "[| mono1(A, r, B, s, g); refl(A, r); trans[B](s) |] ==> Increasing[A](r, f) <= Increasing[B](s, g comp f)" apply (unfold Increasing_def Stable_def Constrains_def part_order_def constrains_def mono1_def metacomp_def, safe) apply (simp_all add: ActsD) apply (subgoal_tac "xb ∈ state & xa ∈ state") prefer 2 apply (simp add: ActsD) apply (subgoal_tac "<f (xb), f (xb) >:r") prefer 2 apply (force simp add: refl_def) apply (rotate_tac 5) apply (drule_tac x = "f (xb) " in bspec) apply simp_all apply clarify apply (rotate_tac -2) apply (drule_tac x = act in bspec) apply (drule_tac [2] A = "act``?u" and c = xa in subsetD, simp_all, blast) apply (drule_tac x = "f (xa) " and x1 = "f (xb) " in bspec [THEN bspec]) apply (rule_tac [3] b = "g (f (xb))" and A = B in trans_onD) apply simp_all done lemma imp_Increasing_comp: "[| F ∈ Increasing[A](r, f); mono1(A, r, B, s, g); refl(A, r); trans[B](s) |] ==> F ∈ Increasing[B](s, g comp f)" apply (rule subset_Increasing_comp [THEN subsetD], auto) done lemma strict_Increasing: "Increasing[nat](Le, f) <= Increasing[nat](Lt, f)" by (unfold Increasing_def Lt_def, auto) lemma strict_gt_Increasing: "Increasing[nat](Ge, f)<= Increasing[nat](Gt, f)" apply (unfold Increasing_def Ge_def Gt_def, auto) apply (erule natE) apply (auto simp add: Stable_def) done (** Two-place monotone operations **) lemma imp_increasing_comp2: "[| F ∈ increasing[A](r, f); F ∈ increasing[B](s, g); mono2(A, r, B, s, C, t, h); refl(A, r); refl(B, s); trans[C](t) |] ==> F ∈ increasing[C](t, %x. h(f(x), g(x)))" apply (unfold increasing_def stable_def part_order_def constrains_def mono2_def, clarify, simp) apply clarify apply (rename_tac xa xb) apply (subgoal_tac "xb ∈ state & xa ∈ state") prefer 2 apply (blast dest!: ActsD) apply (subgoal_tac "<f (xb), f (xb) >:r & <g (xb), g (xb) >:s") prefer 2 apply (force simp add: refl_def) apply (rotate_tac 6) apply (drule_tac x = "f (xb) " in bspec) apply (rotate_tac [2] 1) apply (drule_tac [2] x = "g (xb) " in bspec) apply simp_all apply (rotate_tac -1) apply (drule_tac x = act in bspec) apply (rotate_tac [2] -3) apply (drule_tac [2] x = act in bspec, simp_all) apply (drule_tac A = "act``?u" and c = xa in subsetD) apply (drule_tac [2] A = "act``?u" and c = xa in subsetD, blast, blast) apply (rotate_tac -4) apply (drule_tac x = "f (xa) " and x1 = "f (xb) " in bspec [THEN bspec]) apply (rotate_tac [3] -1) apply (drule_tac [3] x = "g (xa) " and x1 = "g (xb) " in bspec [THEN bspec]) apply simp_all apply (rule_tac b = "h (f (xb), g (xb))" and A = C in trans_onD) apply simp_all done lemma imp_Increasing_comp2: "[| F ∈ Increasing[A](r, f); F ∈ Increasing[B](s, g); mono2(A, r, B, s, C, t, h); refl(A, r); refl(B, s); trans[C](t) |] ==> F ∈ Increasing[C](t, %x. h(f(x), g(x)))" apply (unfold Increasing_def stable_def part_order_def constrains_def mono2_def Stable_def Constrains_def, safe) apply (simp_all add: ActsD) apply (subgoal_tac "xa ∈ state & x ∈ state") prefer 2 apply (blast dest!: ActsD) apply (subgoal_tac "<f (xa), f (xa) >:r & <g (xa), g (xa) >:s") prefer 2 apply (force simp add: refl_def) apply (rotate_tac 6) apply (drule_tac x = "f (xa) " in bspec) apply (rotate_tac [2] 1) apply (drule_tac [2] x = "g (xa) " in bspec) apply simp_all apply clarify apply (rotate_tac -2) apply (drule_tac x = act in bspec) apply (rotate_tac [2] -3) apply (drule_tac [2] x = act in bspec, simp_all) apply (drule_tac A = "act``?u" and c = x in subsetD) apply (drule_tac [2] A = "act``?u" and c = x in subsetD, blast, blast) apply (rotate_tac -9) apply (drule_tac x = "f (x) " and x1 = "f (xa) " in bspec [THEN bspec]) apply (rotate_tac [3] -1) apply (drule_tac [3] x = "g (x) " and x1 = "g (xa) " in bspec [THEN bspec]) apply simp_all apply (rule_tac b = "h (f (xa), g (xa))" and A = C in trans_onD) apply simp_all done end