Theory FP

theory FP
imports UNITY
(*  Title:      ZF/UNITY/FP.thy
    Author:     Sidi O Ehmety, Computer Laboratory
    Copyright   2001  University of Cambridge

From Misra, "A Logic for Concurrent Programming", 1994

Theory ported from HOL.

section‹Fixed Point of a Program›

theory FP imports UNITY begin

  FP_Orig :: "i=>i"  where
    "FP_Orig(F) == ⋃({A ∈ Pow(state). ∀B. F ∈ stable(A ∩ B)})"

  FP :: "i=>i"  where
    "FP(F) == {s∈state. F ∈ stable({s})}"

lemma FP_Orig_type: "FP_Orig(F) ⊆ state"
by (unfold FP_Orig_def, blast)

lemma st_set_FP_Orig [iff]: "st_set(FP_Orig(F))"
apply (unfold st_set_def)
apply (rule FP_Orig_type)

lemma FP_type: "FP(F) ⊆ state"
by (unfold FP_def, blast)

lemma st_set_FP [iff]: "st_set(FP(F))"
apply (unfold st_set_def)
apply (rule FP_type)

lemma stable_FP_Orig_Int: "F ∈ program ==> F ∈ stable(FP_Orig(F) ∩ B)"
apply (simp only: FP_Orig_def stable_def Int_Union2)
apply (blast intro: constrains_UN)

lemma FP_Orig_weakest2: 
    "[| ∀B. F ∈ stable (A ∩ B); st_set(A) |]  ==> A ⊆ FP_Orig(F)"
by (unfold FP_Orig_def stable_def st_set_def, blast)

lemmas FP_Orig_weakest = allI [THEN FP_Orig_weakest2]

lemma stable_FP_Int: "F ∈ program ==> F ∈ stable (FP(F) ∩ B)"
apply (subgoal_tac "FP (F) ∩ B = (⋃x∈B. FP (F) ∩ {x}) ")
 prefer 2 apply blast
apply (simp (no_asm_simp) add: Int_cons_right)
apply (unfold FP_def stable_def)
apply (rule constrains_UN)
apply (auto simp add: cons_absorb)

lemma FP_subset_FP_Orig: "F ∈ program ==> FP(F) ⊆ FP_Orig(F)"
by (rule stable_FP_Int [THEN FP_Orig_weakest], auto)

lemma FP_Orig_subset_FP: "F ∈ program ==> FP_Orig(F) ⊆ FP(F)"
apply (unfold FP_Orig_def FP_def, clarify)
apply (drule_tac x = "{x}" in spec)
apply (simp add: Int_cons_right)
apply (frule stableD2)
apply (auto simp add: cons_absorb st_set_def)

lemma FP_equivalence: "F ∈ program ==> FP(F) = FP_Orig(F)"
by (blast intro!: FP_Orig_subset_FP FP_subset_FP_Orig)

lemma FP_weakest [rule_format]:
     "[| ∀B. F ∈ stable(A ∩ B); F ∈ program; st_set(A) |] ==> A ⊆ FP(F)"
by (simp add: FP_equivalence FP_Orig_weakest)

lemma Diff_FP: 
     "[| F ∈ program;  st_set(A) |] 
      ==> A-FP(F) = (⋃act ∈ Acts(F). A - {s ∈ state. act``{s} ⊆ {s}})"
by (unfold FP_def stable_def constrains_def st_set_def, blast)