Theory SubstAx

(*  Title:      HOL/UNITY/SubstAx.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

Weak LeadsTo relation (restricted to the set of reachable states)
*)

section‹Weak Progress›

theory SubstAx imports WFair Constrains begin

definition Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60) where
    "A Ensures B == {F. F ∈ (reachable F ∩ A) ensures B}"

definition LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60) where
    "A LeadsTo B == {F. F ∈ (reachable F ∩ A) leadsTo B}"

notation LeadsTo  (infixl "⟼w" 60)


text‹Resembles the previous definition of LeadsTo›
lemma LeadsTo_eq_leadsTo: 
     "A LeadsTo B = {F. F ∈ (reachable F ∩ A) leadsTo (reachable F ∩ B)}"
apply (unfold LeadsTo_def)
apply (blast dest: psp_stable2 intro: leadsTo_weaken)
done


subsection‹Specialized laws for handling invariants›

(** Conjoining an Always property **)

lemma Always_LeadsTo_pre:
     "F ∈ Always INV ==> (F ∈ (INV ∩ A) LeadsTo A') = (F ∈ A LeadsTo A')"
by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2 
              Int_assoc [symmetric])

lemma Always_LeadsTo_post:
     "F ∈ Always INV ==> (F ∈ A LeadsTo (INV ∩ A')) = (F ∈ A LeadsTo A')"
by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2 
              Int_assoc [symmetric])

(* [| F ∈ Always C;  F ∈ (C ∩ A) LeadsTo A' |] ==> F ∈ A LeadsTo A' *)
lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1]

(* [| F ∈ Always INV;  F ∈ A LeadsTo A' |] ==> F ∈ A LeadsTo (INV ∩ A') *)
lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2]


subsection‹Introduction rules: Basis, Trans, Union›

lemma leadsTo_imp_LeadsTo: "F ∈ A leadsTo B ==> F ∈ A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done

lemma LeadsTo_Trans:
     "[| F ∈ A LeadsTo B;  F ∈ B LeadsTo C |] ==> F ∈ A LeadsTo C"
apply (simp add: LeadsTo_eq_leadsTo)
apply (blast intro: leadsTo_Trans)
done

lemma LeadsTo_Union: 
     "(!!A. A ∈ S ==> F ∈ A LeadsTo B) ==> F ∈ (⋃S) LeadsTo B"
apply (simp add: LeadsTo_def)
apply (subst Int_Union)
apply (blast intro: leadsTo_UN)
done


subsection‹Derived rules›

lemma LeadsTo_UNIV [simp]: "F ∈ A LeadsTo UNIV"
by (simp add: LeadsTo_def)

text‹Useful with cancellation, disjunction›
lemma LeadsTo_Un_duplicate:
     "F ∈ A LeadsTo (A' ∪ A') ==> F ∈ A LeadsTo A'"
by (simp add: Un_ac)

lemma LeadsTo_Un_duplicate2:
     "F ∈ A LeadsTo (A' ∪ C ∪ C) ==> F ∈ A LeadsTo (A' ∪ C)"
by (simp add: Un_ac)

lemma LeadsTo_UN: 
     "(!!i. i ∈ I ==> F ∈ (A i) LeadsTo B) ==> F ∈ (⋃i ∈ I. A i) LeadsTo B"
apply (blast intro: LeadsTo_Union)
done

text‹Binary union introduction rule›
lemma LeadsTo_Un:
     "[| F ∈ A LeadsTo C; F ∈ B LeadsTo C |] ==> F ∈ (A ∪ B) LeadsTo C"
  using LeadsTo_UN [of "{A, B}" F id C] by auto

text‹Lets us look at the starting state›
lemma single_LeadsTo_I:
     "(!!s. s ∈ A ==> F ∈ {s} LeadsTo B) ==> F ∈ A LeadsTo B"
by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)

lemma subset_imp_LeadsTo: "A ⊆ B ==> F ∈ A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: subset_imp_leadsTo)
done

lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]

lemma LeadsTo_weaken_R:
     "[| F ∈ A LeadsTo A';  A' ⊆ B' |] ==> F ∈ A LeadsTo B'"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_R)
done

lemma LeadsTo_weaken_L:
     "[| F ∈ A LeadsTo A';  B ⊆ A |]   
      ==> F ∈ B LeadsTo A'"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done

lemma LeadsTo_weaken:
     "[| F ∈ A LeadsTo A';    
         B  ⊆ A;   A' ⊆ B' |]  
      ==> F ∈ B LeadsTo B'"
by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)

lemma Always_LeadsTo_weaken:
     "[| F ∈ Always C;  F ∈ A LeadsTo A';    
         C ∩ B ⊆ A;   C ∩ A' ⊆ B' |]  
      ==> F ∈ B LeadsTo B'"
by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)

(** Two theorems for "proof lattices" **)

lemma LeadsTo_Un_post: "F ∈ A LeadsTo B ==> F ∈ (A ∪ B) LeadsTo B"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo)

lemma LeadsTo_Trans_Un:
     "[| F ∈ A LeadsTo B;  F ∈ B LeadsTo C |]  
      ==> F ∈ (A ∪ B) LeadsTo C"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)


(** Distributive laws **)

lemma LeadsTo_Un_distrib:
     "(F ∈ (A ∪ B) LeadsTo C)  = (F ∈ A LeadsTo C & F ∈ B LeadsTo C)"
by (blast intro: LeadsTo_Un LeadsTo_weaken_L)

lemma LeadsTo_UN_distrib:
     "(F ∈ (⋃i ∈ I. A i) LeadsTo B)  =  (∀i ∈ I. F ∈ (A i) LeadsTo B)"
by (blast intro: LeadsTo_UN LeadsTo_weaken_L)

lemma LeadsTo_Union_distrib:
     "(F ∈ (⋃S) LeadsTo B)  =  (∀A ∈ S. F ∈ A LeadsTo B)"
by (blast intro: LeadsTo_Union LeadsTo_weaken_L)


(** More rules using the premise "Always INV" **)

lemma LeadsTo_Basis: "F ∈ A Ensures B ==> F ∈ A LeadsTo B"
by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)

lemma EnsuresI:
     "[| F ∈ (A-B) Co (A ∪ B);  F ∈ transient (A-B) |]    
      ==> F ∈ A Ensures B"
apply (simp add: Ensures_def Constrains_eq_constrains)
apply (blast intro: ensuresI constrains_weaken transient_strengthen)
done

lemma Always_LeadsTo_Basis:
     "[| F ∈ Always INV;       
         F ∈ (INV ∩ (A-A')) Co (A ∪ A');  
         F ∈ transient (INV ∩ (A-A')) |]    
  ==> F ∈ A LeadsTo A'"
apply (rule Always_LeadsToI, assumption)
apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
done

text‹Set difference: maybe combine with ‹leadsTo_weaken_L›??
  This is the most useful form of the "disjunction" rule›
lemma LeadsTo_Diff:
     "[| F ∈ (A-B) LeadsTo C;  F ∈ (A ∩ B) LeadsTo C |]  
      ==> F ∈ A LeadsTo C"
by (blast intro: LeadsTo_Un LeadsTo_weaken)


lemma LeadsTo_UN_UN: 
     "(!! i. i ∈ I ==> F ∈ (A i) LeadsTo (A' i))  
      ==> F ∈ (⋃i ∈ I. A i) LeadsTo (⋃i ∈ I. A' i)"
apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
done


text‹Version with no index set›
lemma LeadsTo_UN_UN_noindex: 
     "(!!i. F ∈ (A i) LeadsTo (A' i)) ==> F ∈ (⋃i. A i) LeadsTo (⋃i. A' i)"
by (blast intro: LeadsTo_UN_UN)

text‹Version with no index set›
lemma all_LeadsTo_UN_UN:
     "∀i. F ∈ (A i) LeadsTo (A' i)  
      ==> F ∈ (⋃i. A i) LeadsTo (⋃i. A' i)"
by (blast intro: LeadsTo_UN_UN)

text‹Binary union version›
lemma LeadsTo_Un_Un:
     "[| F ∈ A LeadsTo A'; F ∈ B LeadsTo B' |]  
            ==> F ∈ (A ∪ B) LeadsTo (A' ∪ B')"
by (blast intro: LeadsTo_Un LeadsTo_weaken_R)


(** The cancellation law **)

lemma LeadsTo_cancel2:
     "[| F ∈ A LeadsTo (A' ∪ B); F ∈ B LeadsTo B' |]     
      ==> F ∈ A LeadsTo (A' ∪ B')"
by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)

lemma LeadsTo_cancel_Diff2:
     "[| F ∈ A LeadsTo (A' ∪ B); F ∈ (B-A') LeadsTo B' |]  
      ==> F ∈ A LeadsTo (A' ∪ B')"
apply (rule LeadsTo_cancel2)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done

lemma LeadsTo_cancel1:
     "[| F ∈ A LeadsTo (B ∪ A'); F ∈ B LeadsTo B' |]  
      ==> F ∈ A LeadsTo (B' ∪ A')"
apply (simp add: Un_commute)
apply (blast intro!: LeadsTo_cancel2)
done

lemma LeadsTo_cancel_Diff1:
     "[| F ∈ A LeadsTo (B ∪ A'); F ∈ (B-A') LeadsTo B' |]  
      ==> F ∈ A LeadsTo (B' ∪ A')"
apply (rule LeadsTo_cancel1)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done


text‹The impossibility law›

text‹The set "A" may be non-empty, but it contains no reachable states›
lemma LeadsTo_empty: "[|F ∈ A LeadsTo {}; all_total F|] ==> F ∈ Always (-A)"
apply (simp add: LeadsTo_def Always_eq_includes_reachable)
apply (drule leadsTo_empty, auto)
done


subsection‹PSP: Progress-Safety-Progress›

text‹Special case of PSP: Misra's "stable conjunction"›
lemma PSP_Stable:
     "[| F ∈ A LeadsTo A';  F ∈ Stable B |]  
      ==> F ∈ (A ∩ B) LeadsTo (A' ∩ B)"
apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
apply (drule psp_stable, assumption)
apply (simp add: Int_ac)
done

lemma PSP_Stable2:
     "[| F ∈ A LeadsTo A'; F ∈ Stable B |]  
      ==> F ∈ (B ∩ A) LeadsTo (B ∩ A')"
by (simp add: PSP_Stable Int_ac)

lemma PSP:
     "[| F ∈ A LeadsTo A'; F ∈ B Co B' |]  
      ==> F ∈ (A ∩ B') LeadsTo ((A' ∩ B) ∪ (B' - B))"
apply (simp add: LeadsTo_def Constrains_eq_constrains)
apply (blast dest: psp intro: leadsTo_weaken)
done

lemma PSP2:
     "[| F ∈ A LeadsTo A'; F ∈ B Co B' |]  
      ==> F ∈ (B' ∩ A) LeadsTo ((B ∩ A') ∪ (B' - B))"
by (simp add: PSP Int_ac)

lemma PSP_Unless: 
     "[| F ∈ A LeadsTo A'; F ∈ B Unless B' |]  
      ==> F ∈ (A ∩ B) LeadsTo ((A' ∩ B) ∪ B')"
apply (unfold Unless_def)
apply (drule PSP, assumption)
apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
done


lemma Stable_transient_Always_LeadsTo:
     "[| F ∈ Stable A;  F ∈ transient C;   
         F ∈ Always (-A ∪ B ∪ C) |] ==> F ∈ A LeadsTo B"
apply (erule Always_LeadsTo_weaken)
apply (rule LeadsTo_Diff)
   prefer 2
   apply (erule
          transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
   apply (blast intro: subset_imp_LeadsTo)+
done


subsection‹Induction rules›

(** Meta or object quantifier ????? **)
lemma LeadsTo_wf_induct:
     "[| wf r;      
         ∀m. F ∈ (A ∩ f-`{m}) LeadsTo                      
                    ((A ∩ f-`(r¯ `` {m})) ∪ B) |]  
      ==> F ∈ A LeadsTo B"
apply (simp add: LeadsTo_eq_leadsTo)
apply (erule leadsTo_wf_induct)
apply (blast intro: leadsTo_weaken)
done


lemma Bounded_induct:
     "[| wf r;      
         ∀m ∈ I. F ∈ (A ∩ f-`{m}) LeadsTo                    
                      ((A ∩ f-`(r¯ `` {m})) ∪ B) |]  
      ==> F ∈ A LeadsTo ((A - (f-`I)) ∪ B)"
apply (erule LeadsTo_wf_induct, safe)
apply (case_tac "m ∈ I")
apply (blast intro: LeadsTo_weaken)
apply (blast intro: subset_imp_LeadsTo)
done


lemma LessThan_induct:
     "(!!m::nat. F ∈ (A ∩ f-`{m}) LeadsTo ((A ∩ f-`(lessThan m)) ∪ B))
      ==> F ∈ A LeadsTo B"
by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)

text‹Integer version.  Could generalize from 0 to any lower bound›
lemma integ_0_le_induct:
     "[| F ∈ Always {s. (0::int) ≤ f s};   
         !! z. F ∈ (A ∩ {s. f s = z}) LeadsTo                      
                   ((A ∩ {s. f s < z}) ∪ B) |]  
      ==> F ∈ A LeadsTo B"
apply (rule_tac f = "nat o f" in LessThan_induct)
apply (simp add: vimage_def)
apply (rule Always_LeadsTo_weaken, assumption+)
apply (auto simp add: nat_eq_iff nat_less_iff)
done

lemma LessThan_bounded_induct:
     "!!l::nat. ∀m ∈ greaterThan l. 
                   F ∈ (A ∩ f-`{m}) LeadsTo ((A ∩ f-`(lessThan m)) ∪ B)
            ==> F ∈ A LeadsTo ((A ∩ (f-`(atMost l))) ∪ B)"
apply (simp only: Diff_eq [symmetric] vimage_Compl 
                  Compl_greaterThan [symmetric])
apply (rule wf_less_than [THEN Bounded_induct], simp)
done

lemma GreaterThan_bounded_induct:
     "!!l::nat. ∀m ∈ lessThan l. 
                 F ∈ (A ∩ f-`{m}) LeadsTo ((A ∩ f-`(greaterThan m)) ∪ B)
      ==> F ∈ A LeadsTo ((A ∩ (f-`(atLeast l))) ∪ B)"
apply (rule_tac f = f and f1 = "%k. l - k" 
       in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
apply (simp add: Image_singleton, clarify)
apply (case_tac "m<l")
 apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
apply (blast intro: not_le_imp_less subset_imp_LeadsTo)
done


subsection‹Completion: Binary and General Finite versions›

lemma Completion:
     "[| F ∈ A LeadsTo (A' ∪ C);  F ∈ A' Co (A' ∪ C);  
         F ∈ B LeadsTo (B' ∪ C);  F ∈ B' Co (B' ∪ C) |]  
      ==> F ∈ (A ∩ B) LeadsTo ((A' ∩ B') ∪ C)"
apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
apply (blast intro: completion leadsTo_weaken)
done

lemma Finite_completion_lemma:
     "finite I  
      ==> (∀i ∈ I. F ∈ (A i) LeadsTo (A' i ∪ C)) -->   
          (∀i ∈ I. F ∈ (A' i) Co (A' i ∪ C)) -->  
          F ∈ (⋂i ∈ I. A i) LeadsTo ((⋂i ∈ I. A' i) ∪ C)"
apply (erule finite_induct, auto)
apply (rule Completion)
   prefer 4
   apply (simp only: INT_simps [symmetric])
   apply (rule Constrains_INT, auto)
done

lemma Finite_completion: 
     "[| finite I;   
         !!i. i ∈ I ==> F ∈ (A i) LeadsTo (A' i ∪ C);  
         !!i. i ∈ I ==> F ∈ (A' i) Co (A' i ∪ C) |]    
      ==> F ∈ (⋂i ∈ I. A i) LeadsTo ((⋂i ∈ I. A' i) ∪ C)"
by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])

lemma Stable_completion: 
     "[| F ∈ A LeadsTo A';  F ∈ Stable A';    
         F ∈ B LeadsTo B';  F ∈ Stable B' |]  
      ==> F ∈ (A ∩ B) LeadsTo (A' ∩ B')"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
apply (force+)
done

lemma Finite_stable_completion: 
     "[| finite I;   
         !!i. i ∈ I ==> F ∈ (A i) LeadsTo (A' i);  
         !!i. i ∈ I ==> F ∈ Stable (A' i) |]    
      ==> F ∈ (⋂i ∈ I. A i) LeadsTo (⋂i ∈ I. A' i)"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
apply (simp_all, blast+)
done

end