(* Title: HOL/UNITY/Follows
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The "Follows" relation of Charpentier and Sivilotte
*)
theory Follows = SubstAx + ListOrder + Multiset:
constdefs
Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set"
(infixl "Fols" 65)
"f Fols g == Increasing g Int Increasing f Int
Always {s. f s <= g s} Int
(INT k. {s. k <= g s} LeadsTo {s. k <= f s})"
(*Does this hold for "invariant"?*)
lemma mono_Always_o:
"mono h ==> Always {s. f s <= g s} <= Always {s. h (f s) <= h (g s)}"
apply (simp add: Always_eq_includes_reachable)
apply (blast intro: monoD)
done
lemma mono_LeadsTo_o:
"mono (h::'a::order => 'b::order)
==> (INT j. {s. j <= g s} LeadsTo {s. j <= f s}) <=
(INT k. {s. k <= h (g s)} LeadsTo {s. k <= h (f s)})"
apply auto
apply (rule single_LeadsTo_I)
apply (drule_tac x = "g s" in spec)
apply (erule LeadsTo_weaken)
apply (blast intro: monoD order_trans)+
done
lemma Follows_constant: "F : (%s. c) Fols (%s. c)"
by (unfold Follows_def, auto)
declare Follows_constant [iff]
lemma mono_Follows_o: "mono h ==> f Fols g <= (h o f) Fols (h o g)"
apply (unfold Follows_def, clarify)
apply (simp add: mono_Increasing_o [THEN [2] rev_subsetD]
mono_Always_o [THEN [2] rev_subsetD]
mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D])
done
lemma mono_Follows_apply:
"mono h ==> f Fols g <= (%x. h (f x)) Fols (%x. h (g x))"
apply (drule mono_Follows_o)
apply (force simp add: o_def)
done
lemma Follows_trans:
"[| F : f Fols g; F: g Fols h |] ==> F : f Fols h"
apply (unfold Follows_def)
apply (simp add: Always_eq_includes_reachable)
apply (blast intro: order_trans LeadsTo_Trans)
done
(** Destructiom rules **)
lemma Follows_Increasing1:
"F : f Fols g ==> F : Increasing f"
apply (unfold Follows_def, blast)
done
lemma Follows_Increasing2:
"F : f Fols g ==> F : Increasing g"
apply (unfold Follows_def, blast)
done
lemma Follows_Bounded:
"F : f Fols g ==> F : Always {s. f s <= g s}"
apply (unfold Follows_def, blast)
done
lemma Follows_LeadsTo:
"F : f Fols g ==> F : {s. k <= g s} LeadsTo {s. k <= f s}"
apply (unfold Follows_def, blast)
done
lemma Follows_LeadsTo_pfixLe:
"F : f Fols g ==> F : {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}"
apply (rule single_LeadsTo_I, clarify)
apply (drule_tac k="g s" in Follows_LeadsTo)
apply (erule LeadsTo_weaken)
apply blast
apply (blast intro: pfixLe_trans prefix_imp_pfixLe)
done
lemma Follows_LeadsTo_pfixGe:
"F : f Fols g ==> F : {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}"
apply (rule single_LeadsTo_I, clarify)
apply (drule_tac k="g s" in Follows_LeadsTo)
apply (erule LeadsTo_weaken)
apply blast
apply (blast intro: pfixGe_trans prefix_imp_pfixGe)
done
lemma Always_Follows1:
"[| F : Always {s. f s = f' s}; F : f Fols g |] ==> F : f' Fols g"
apply (unfold Follows_def Increasing_def Stable_def, auto)
apply (erule_tac [3] Always_LeadsTo_weaken)
apply (erule_tac A = "{s. z <= f s}" and A' = "{s. z <= f s}" in Always_Constrains_weaken, auto)
apply (drule Always_Int_I, assumption)
apply (force intro: Always_weaken)
done
lemma Always_Follows2:
"[| F : Always {s. g s = g' s}; F : f Fols g |] ==> F : f Fols g'"
apply (unfold Follows_def Increasing_def Stable_def, auto)
apply (erule_tac [3] Always_LeadsTo_weaken)
apply (erule_tac A = "{s. z <= g s}" and A' = "{s. z <= g s}" in Always_Constrains_weaken, auto)
apply (drule Always_Int_I, assumption)
apply (force intro: Always_weaken)
done
(** Union properties (with the subset ordering) **)
(*Can replace "Un" by any sup. But existing max only works for linorders.*)
lemma increasing_Un:
"[| F : increasing f; F: increasing g |]
==> F : increasing (%s. (f s) Un (g s))"
apply (unfold increasing_def stable_def constrains_def, auto)
apply (drule_tac x = "f xa" in spec)
apply (drule_tac x = "g xa" in spec)
apply (blast dest!: bspec)
done
lemma Increasing_Un:
"[| F : Increasing f; F: Increasing g |]
==> F : Increasing (%s. (f s) Un (g s))"
apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
apply (drule_tac x = "f xa" in spec)
apply (drule_tac x = "g xa" in spec)
apply (blast dest!: bspec)
done
lemma Always_Un:
"[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
==> F : Always {s. f' s Un g' s <= f s Un g s}"
apply (simp add: Always_eq_includes_reachable, blast)
done
(*Lemma to re-use the argument that one variable increases (progress)
while the other variable doesn't decrease (safety)*)
lemma Follows_Un_lemma:
"[| F : Increasing f; F : Increasing g;
F : Increasing g'; F : Always {s. f' s <= f s};
ALL k. F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
==> F : {s. k <= f s Un g s} LeadsTo {s. k <= f' s Un g s}"
apply (rule single_LeadsTo_I)
apply (drule_tac x = "f s" in IncreasingD)
apply (drule_tac x = "g s" in IncreasingD)
apply (rule LeadsTo_weaken)
apply (rule PSP_Stable)
apply (erule_tac x = "f s" in spec)
apply (erule Stable_Int, assumption)
apply blast
apply blast
done
lemma Follows_Un:
"[| F : f' Fols f; F: g' Fols g |]
==> F : (%s. (f' s) Un (g' s)) Fols (%s. (f s) Un (g s))"
apply (unfold Follows_def)
apply (simp add: Increasing_Un Always_Un, auto)
apply (rule LeadsTo_Trans)
apply (blast intro: Follows_Un_lemma)
(*Weakening is used to exchange Un's arguments*)
apply (blast intro: Follows_Un_lemma [THEN LeadsTo_weaken])
done
(** Multiset union properties (with the multiset ordering) **)
lemma increasing_union:
"[| F : increasing f; F: increasing g |]
==> F : increasing (%s. (f s) + (g s :: ('a::order) multiset))"
apply (unfold increasing_def stable_def constrains_def, auto)
apply (drule_tac x = "f xa" in spec)
apply (drule_tac x = "g xa" in spec)
apply (drule bspec, assumption)
apply (blast intro: union_le_mono order_trans)
done
lemma Increasing_union:
"[| F : Increasing f; F: Increasing g |]
==> F : Increasing (%s. (f s) + (g s :: ('a::order) multiset))"
apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
apply (drule_tac x = "f xa" in spec)
apply (drule_tac x = "g xa" in spec)
apply (drule bspec, assumption)
apply (blast intro: union_le_mono order_trans)
done
lemma Always_union:
"[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
==> F : Always {s. f' s + g' s <= f s + (g s :: ('a::order) multiset)}"
apply (simp add: Always_eq_includes_reachable)
apply (blast intro: union_le_mono)
done
(*Except the last line, IDENTICAL to the proof script for Follows_Un_lemma*)
lemma Follows_union_lemma:
"[| F : Increasing f; F : Increasing g;
F : Increasing g'; F : Always {s. f' s <= f s};
ALL k::('a::order) multiset.
F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
==> F : {s. k <= f s + g s} LeadsTo {s. k <= f' s + g s}"
apply (rule single_LeadsTo_I)
apply (drule_tac x = "f s" in IncreasingD)
apply (drule_tac x = "g s" in IncreasingD)
apply (rule LeadsTo_weaken)
apply (rule PSP_Stable)
apply (erule_tac x = "f s" in spec)
apply (erule Stable_Int, assumption)
apply blast
apply (blast intro: union_le_mono order_trans)
done
(*The !! is there to influence to effect of permutative rewriting at the end*)
lemma Follows_union:
"!!g g' ::'b => ('a::order) multiset.
[| F : f' Fols f; F: g' Fols g |]
==> F : (%s. (f' s) + (g' s)) Fols (%s. (f s) + (g s))"
apply (unfold Follows_def)
apply (simp add: Increasing_union Always_union, auto)
apply (rule LeadsTo_Trans)
apply (blast intro: Follows_union_lemma)
(*now exchange union's arguments*)
apply (simp add: union_commute)
apply (blast intro: Follows_union_lemma)
done
lemma Follows_setsum:
"!!f ::['c,'b] => ('a::order) multiset.
[| ALL i: I. F : f' i Fols f i; finite I |]
==> F : (%s. \<Sum>i:I. f' i s) Fols (%s. \<Sum>i:I. f i s)"
apply (erule rev_mp)
apply (erule finite_induct, simp)
apply (simp add: Follows_union)
done
(*Currently UNUSED, but possibly of interest*)
lemma Increasing_imp_Stable_pfixGe:
"F : Increasing func ==> F : Stable {s. h pfixGe (func s)}"
apply (simp add: Increasing_def Stable_def Constrains_def constrains_def)
apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
prefix_imp_pfixGe)
done
(*Currently UNUSED, but possibly of interest*)
lemma LeadsTo_le_imp_pfixGe:
"ALL z. F : {s. z <= f s} LeadsTo {s. z <= g s}
==> F : {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}"
apply (rule single_LeadsTo_I)
apply (drule_tac x = "f s" in spec)
apply (erule LeadsTo_weaken)
prefer 2
apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
prefix_imp_pfixGe, blast)
done
end