# Theory Increasing

theory Increasing
imports Constrains Monotonicity
```(*  Title:      ZF/UNITY/Increasing.thy
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.
*)

section‹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 for u 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 for u 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 for u in subsetD)
apply (drule_tac [2] A = "act``u" and c = xa for u 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 for u in subsetD)
apply (drule_tac [2] A = "act``u" and c = x for u 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
```