# Theory Follows

Up to index of Isabelle/HOL/UNITY

theory Follows
imports SubstAx ListOrder Multiset
begin

    (*  Title:      HOL/UNITY/Follows
        ID:         $Id: Follows.thy,v 1.14 2005/06/17 14:13:10 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1998  University of Cambridge
    *)

    header{*The Follows Relation of Charpentier and Sivilotte*}

    theory Follows imports SubstAx ListOrder Multiset begin

    constdefs

      Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set"
                     (infixl "Fols" 65)
       "f Fols g == Increasing g ∩ Increasing f Int
                    Always {s. f s ≤ g s} Int
                    (\<Inter>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)
          ==> (\<Inter>j. {s. j ≤ g s} LeadsTo {s. j ≤ f s}) ⊆
              (\<Inter>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 [iff]: "F ∈ (%s. c) Fols (%s. c)"
    by (simp add: Follows_def)

    lemma mono_Follows_o: "mono h ==> f Fols g ⊆ (h o f) Fols (h o g)"
    by (auto simp add: Follows_def mono_Increasing_o [THEN [2] rev_subsetD]
                       mono_Always_o [THEN [2] rev_subsetD]
                       mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D])

    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 (simp add: Follows_def)
    apply (simp add: Always_eq_includes_reachable)
    apply (blast intro: order_trans LeadsTo_Trans)
    done

    subsection{*Destruction rules*}

    lemma Follows_Increasing1: "F ∈ f Fols g ==> F ∈ Increasing f"
    by (simp add: Follows_def)

    lemma Follows_Increasing2: "F ∈ f Fols g ==> F ∈ Increasing g"
    by (simp add: Follows_def)

    lemma Follows_Bounded: "F ∈ f Fols g ==> F ∈ Always {s. f s ⊆ g s}"
    by (simp add: Follows_def)

    lemma Follows_LeadsTo:
         "F ∈ f Fols g ==> F ∈ {s. k ≤ g s} LeadsTo {s. k ≤ f s}"
    by (simp add: Follows_def)

    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 (simp add: 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 (simp add: 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

    subsection{*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) ∪ (g s))"
    apply (simp add: 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) ∪ (g s))"
    apply (auto simp add: Increasing_def Stable_def Constrains_def
                          stable_def constrains_def)
    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 ∪ g' s ≤ f s ∪ g s}"
    by (simp add: Always_eq_includes_reachable, blast)

    (*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};
             ∀k. 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, blast+)
    done

    lemma Follows_Un:
        "[| F ∈ f' Fols f;  F ∈ g' Fols g |]
         ==> F ∈ (%s. (f' s) ∪ (g' s)) Fols (%s. (f s) ∪ (g s))"
    apply (simp add: Follows_def Increasing_Un Always_Un del: Un_subset_iff, 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

    subsection{*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 (simp add: 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 (auto simp add: Increasing_def Stable_def Constrains_def
                          stable_def constrains_def)
    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};
             ∀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, 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 (simp add: 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.
            [| ∀i ∈ I. F ∈ f' i Fols f i;  finite I |]
            ==> F ∈ (%s. ∑i ∈ I. f' i s) Fols (%s. ∑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:
         "∀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

lemma mono_Always_o:

      mono h ==> Always {s. f s ≤ g s} ⊆ Always {s. h (f s) ≤ h (g s)}

lemma mono_LeadsTo_o:

      mono h
      ==> (INT j. {s. j ≤ g s} LeadsTo {s. j ≤ f s})
          ⊆ (INT k. {s. k ≤ h (g s)} LeadsTo {s. k ≤ h (f s)})

lemma Follows_constant:

      F ∈ (%s. c) Fols (%s. c)

lemma mono_Follows_o:

      mono h ==> f Fols g ⊆ (h o f) Fols (h o g)

lemma mono_Follows_apply:

      mono h ==> f Fols g ⊆ (%x. h (f x)) Fols (%x. h (g x))

lemma Follows_trans:

      [| F ∈ f Fols g; F ∈ g Fols h |] ==> F ∈ f Fols h

### Destruction rules

lemma Follows_Increasing1:

      F ∈ f Fols g ==> F ∈ Increasing f

lemma Follows_Increasing2:

      F ∈ f Fols g ==> F ∈ Increasing g

lemma Follows_Bounded:

      F ∈ f Fols g ==> F ∈ Always {s. f s ⊆ g s}

lemma Follows_LeadsTo:

      F ∈ f Fols g ==> F ∈ {s. k ≤ g s} LeadsTo {s. k ≤ f s}

lemma Follows_LeadsTo_pfixLe:

      F ∈ f Fols g ==> F ∈ {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}

lemma Follows_LeadsTo_pfixGe:

      F ∈ f Fols g ==> F ∈ {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}

lemma Always_Follows1:

      [| F ∈ Always {s. f s = f' s}; F ∈ f Fols g |] ==> F ∈ f' Fols g

lemma Always_Follows2:

      [| F ∈ Always {s. g s = g' s}; F ∈ f Fols g |] ==> F ∈ f Fols g'

### Union properties (with the subset ordering)

lemma increasing_Un:

      [| F ∈ increasing f; F ∈ increasing g |] ==> F ∈ increasing (%s. f s ∪ g s)

lemma Increasing_Un:

      [| F ∈ Increasing f; F ∈ Increasing g |] ==> F ∈ Increasing (%s. f s ∪ g s)

lemma Always_Un:

      [| 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}

lemma Follows_Un_lemma:

      [| F ∈ Increasing f; F ∈ Increasing g; F ∈ Increasing g';
         F ∈ Always {s. f' s ⊆ f s}; ∀k. F ∈ {s. k ⊆ f s} LeadsTo {s. k ⊆ f' s} |]
      ==> F ∈ {s. k ⊆ f s ∪ g s} LeadsTo {s. k ⊆ f' s ∪ g s}

lemma Follows_Un:

      [| F ∈ f' Fols f; F ∈ g' Fols g |]
      ==> F ∈ (%s. f' s ∪ g' s) Fols (%s. f s ∪ g s)

### Multiset union properties (with the multiset ordering)

lemma increasing_union:

      [| F ∈ increasing f; F ∈ increasing g |] ==> F ∈ increasing (%s. f s + g s)

lemma Increasing_union:

      [| F ∈ Increasing f; F ∈ Increasing g |] ==> F ∈ Increasing (%s. f s + g s)

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}

lemma Follows_union_lemma:

      [| F ∈ Increasing f; F ∈ Increasing g; F ∈ Increasing g';
         F ∈ Always {s. f' s ≤ f s}; ∀k. F ∈ {s. k ≤ f s} LeadsTo {s. k ≤ f' s} |]
      ==> F ∈ {s. k ≤ f s + g s} LeadsTo {s. k ≤ f' s + g s}

lemma Follows_union:

      [| F ∈ f' Fols f; F ∈ g' Fols g |]
      ==> F ∈ (%s. f' s + g' s) Fols (%s. f s + g s)

lemma Follows_setsum:

      [| ∀i∈I. F ∈ f' i Fols f i; finite I |]
      ==> F ∈ (%s. ∑i∈I. f' i s) Fols (%s. ∑i∈I. f i s)

lemma Increasing_imp_Stable_pfixGe:

      F ∈ Increasing func ==> F ∈ Stable {s. h pfixGe func s}

lemma LeadsTo_le_imp_pfixGe:

      ∀z. F ∈ {s. z ≤ f s} LeadsTo {s. z ≤ g s}
      ==> F ∈ {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}