src/HOL/UNITY/UNITY.ML

converted theory Transitive_Closure;

(*  Title:      HOL/UNITY/UNITY
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

The basic UNITY theory (revised version, based upon the "co" operator)

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

(*Perhaps equalities.ML shouldn't add this in the first place!*)
Delsimps [image_Collect];

(*** The abstract type of programs ***)

val rep_ss = simpset() addsimps 
                [Init_def, Acts_def, AllowedActs_def, 
		 mk_program_def, Program_def, Rep_Program, 
		 Rep_Program_inverse, Abs_Program_inverse];


Goal "Id : Acts F";
by (cut_inst_tac [("x", "F")] Rep_Program 1);
by (auto_tac (claset(), rep_ss));
qed "Id_in_Acts";
AddIffs [Id_in_Acts];

Goal "insert Id (Acts F) = Acts F";
by (simp_tac (simpset() addsimps [insert_absorb, Id_in_Acts]) 1);
qed "insert_Id_Acts";
AddIffs [insert_Id_Acts];

Goal "Id : AllowedActs F";
by (cut_inst_tac [("x", "F")] Rep_Program 1);
by (auto_tac (claset(), rep_ss));
qed "Id_in_AllowedActs";
AddIffs [Id_in_AllowedActs];

Goal "insert Id (AllowedActs F) = AllowedActs F";
by (simp_tac (simpset() addsimps [insert_absorb, Id_in_AllowedActs]) 1);
qed "insert_Id_AllowedActs";
AddIffs [insert_Id_AllowedActs];

(** Inspectors for type "program" **)

Goal "Init (mk_program (init,acts,allowed)) = init";
by (auto_tac (claset(), rep_ss));
qed "Init_eq";

Goal "Acts (mk_program (init,acts,allowed)) = insert Id acts";
by (auto_tac (claset(), rep_ss));
qed "Acts_eq";

Goal "AllowedActs (mk_program (init,acts,allowed)) = insert Id allowed";
by (auto_tac (claset(), rep_ss));
qed "AllowedActs_eq";

Addsimps [Init_eq, Acts_eq, AllowedActs_eq];

(** Equality for UNITY programs **)

Goal "mk_program (Init F, Acts F, AllowedActs F) = F";
by (cut_inst_tac [("x", "F")] Rep_Program 1);
by (auto_tac (claset(), rep_ss));
by (REPEAT (dres_inst_tac [("f", "Abs_Program")] arg_cong 1));
by (asm_full_simp_tac (rep_ss addsimps [insert_absorb]) 1);
qed "surjective_mk_program";

Goal "[| Init F = Init G; Acts F = Acts G; AllowedActs F = AllowedActs G |] \
\     ==> F = G";
by (res_inst_tac [("t", "F")] (surjective_mk_program RS subst) 1);
by (res_inst_tac [("t", "G")] (surjective_mk_program RS subst) 1);
by (Asm_simp_tac 1);
qed "program_equalityI";

val [major,minor] =
Goal "[| F = G; \
\        [| Init F = Init G; Acts F = Acts G; AllowedActs F = AllowedActs G |]\
\        ==> P |] ==> P";
by (rtac minor 1);
by (auto_tac (claset(), simpset() addsimps [major]));
qed "program_equalityE";

Goal "(F=G) =  \
\     (Init F = Init G & Acts F = Acts G &AllowedActs F = AllowedActs G)";
by (blast_tac (claset() addIs [program_equalityI, program_equalityE]) 1);
qed "program_equality_iff";

Addsimps [surjective_mk_program];


(*** These rules allow "lazy" definition expansion 
     They avoid expanding the full program, which is a large expression
***)

Goal "F == mk_program (init,acts,allowed) ==> Init F = init";
by Auto_tac;
qed "def_prg_Init";

Goal "F == mk_program (init,acts,allowed) ==> Acts F = insert Id acts";
by Auto_tac;
qed "def_prg_Acts";

Goal "F == mk_program (init,acts,allowed) \
\     ==> AllowedActs F = insert Id allowed";
by Auto_tac;
qed "def_prg_AllowedActs";

(*The program is not expanded, but its Init and Acts are*)
val [rew] = goal thy
    "[| F == mk_program (init,acts,allowed) |] \
\    ==> Init F = init & Acts F = insert Id acts";
by (rewtac rew);
by Auto_tac;
qed "def_prg_simps";

(*An action is expanded only if a pair of states is being tested against it*)
Goal "[| act == {(s,s'). P s s'} |] ==> ((s,s') : act) = P s s'";
by Auto_tac;
qed "def_act_simp";

fun simp_of_act def = def RS def_act_simp;

(*A set is expanded only if an element is being tested against it*)
Goal "A == B ==> (x : A) = (x : B)";
by Auto_tac;
qed "def_set_simp";

fun simp_of_set def = def RS def_set_simp;


(*** co ***)

(*These operators are not overloaded, but their operands are sets, and 
  ultimately there's a risk of reaching equality, which IS overloaded*)
overload_1st_set "UNITY.constrains";
overload_1st_set "UNITY.stable";
overload_1st_set "UNITY.unless";

val prems = Goalw [constrains_def]
    "(!!act s s'. [| act: Acts F;  (s,s') : act;  s: A |] ==> s': A') \
\    ==> F : A co A'";
by (blast_tac (claset() addIs prems) 1);
qed "constrainsI";

Goalw [constrains_def]
    "[| F : A co A'; act: Acts F;  (s,s'): act;  s: A |] ==> s': A'";
by (Blast_tac 1);
qed "constrainsD";

Goalw [constrains_def] "F : {} co B";
by (Blast_tac 1);
qed "constrains_empty";

Goalw [constrains_def] "(F : A co {}) = (A={})";
by (Blast_tac 1);
qed "constrains_empty2";

Goalw [constrains_def] "(F : UNIV co B) = (B = UNIV)";
by (Blast_tac 1);
qed "constrains_UNIV";

Goalw [constrains_def] "F : A co UNIV";
by (Blast_tac 1);
qed "constrains_UNIV2";

AddIffs [constrains_empty, constrains_empty2, 
	 constrains_UNIV, constrains_UNIV2];

(*monotonic in 2nd argument*)
Goalw [constrains_def]
    "[| F : A co A'; A'<=B' |] ==> F : A co B'";
by (Blast_tac 1);
qed "constrains_weaken_R";

(*anti-monotonic in 1st argument*)
Goalw [constrains_def]
    "[| F : A co A'; B<=A |] ==> F : B co A'";
by (Blast_tac 1);
qed "constrains_weaken_L";

Goalw [constrains_def]
   "[| F : A co A'; B<=A; A'<=B' |] ==> F : B co B'";
by (Blast_tac 1);
qed "constrains_weaken";

(** Union **)

Goalw [constrains_def]
    "[| F : A co A'; F : B co B' |] ==> F : (A Un B) co (A' Un B')";
by (Blast_tac 1);
qed "constrains_Un";

Goalw [constrains_def]
    "ALL i:I. F : (A i) co (A' i) ==> F : (UN i:I. A i) co (UN i:I. A' i)";
by (Blast_tac 1);
bind_thm ("constrains_UN", ballI RS result());

Goalw [constrains_def] "(A Un B) co C = (A co C) Int (B co C)";
by (Blast_tac 1);
qed "constrains_Un_distrib";

Goalw [constrains_def] "(UN i:I. A i) co B = (INT i:I. A i co B)";
by (Blast_tac 1);
qed "constrains_UN_distrib";

Goalw [constrains_def] "C co (A Int B) = (C co A) Int (C co B)";
by (Blast_tac 1);
qed "constrains_Int_distrib";

Goalw [constrains_def] "A co (INT i:I. B i) = (INT i:I. A co B i)";
by (Blast_tac 1);
qed "constrains_INT_distrib";

(** Intersection **)

Goalw [constrains_def]
    "[| F : A co A'; F : B co B' |] ==> F : (A Int B) co (A' Int B')";
by (Blast_tac 1);
qed "constrains_Int";

Goalw [constrains_def]
    "ALL i:I. F : (A i) co (A' i) ==> F : (INT i:I. A i) co (INT i:I. A' i)";
by (Blast_tac 1);
bind_thm ("constrains_INT", ballI RS result());

Goalw [constrains_def] "F : A co A' ==> A <= A'";
by Auto_tac;
qed "constrains_imp_subset";

(*The reasoning is by subsets since "co" refers to single actions
  only.  So this rule isn't that useful.*)
Goalw [constrains_def]
    "[| F : A co B; F : B co C |] ==> F : A co C";
by (Blast_tac 1);
qed "constrains_trans";

Goalw [constrains_def]
   "[| F : A co (A' Un B); F : B co B' |] ==> F : A co (A' Un B')";
by (Clarify_tac 1);
by (Blast_tac 1);
qed "constrains_cancel";


(*** unless ***)

Goalw [unless_def] "F : (A-B) co (A Un B) ==> F : A unless B";
by (assume_tac 1);
qed "unlessI";

Goalw [unless_def] "F : A unless B ==> F : (A-B) co (A Un B)";
by (assume_tac 1);
qed "unlessD";


(*** stable ***)

Goalw [stable_def] "F : A co A ==> F : stable A";
by (assume_tac 1);
qed "stableI";

Goalw [stable_def] "F : stable A ==> F : A co A";
by (assume_tac 1);
qed "stableD";

Goalw [stable_def, constrains_def] "stable UNIV = UNIV";
by Auto_tac;
qed "stable_UNIV";
Addsimps [stable_UNIV];

(** Union **)

Goalw [stable_def]
    "[| F : stable A; F : stable A' |] ==> F : stable (A Un A')";
by (blast_tac (claset() addIs [constrains_Un]) 1);
qed "stable_Un";

Goalw [stable_def]
    "ALL i:I. F : stable (A i) ==> F : stable (UN i:I. A i)";
by (blast_tac (claset() addIs [constrains_UN]) 1);
bind_thm ("stable_UN", ballI RS result());

(** Intersection **)

Goalw [stable_def]
    "[| F : stable A;  F : stable A' |] ==> F : stable (A Int A')";
by (blast_tac (claset() addIs [constrains_Int]) 1);
qed "stable_Int";

Goalw [stable_def]
    "ALL i:I. F : stable (A i) ==> F : stable (INT i:I. A i)";
by (blast_tac (claset() addIs [constrains_INT]) 1);
bind_thm ("stable_INT", ballI RS result());

Goalw [stable_def, constrains_def]
    "[| F : stable C; F : A co (C Un A') |] ==> F : (C Un A) co (C Un A')";
by (Blast_tac 1);
qed "stable_constrains_Un";

Goalw [stable_def, constrains_def]
  "[| F : stable C; F :  (C Int A) co A' |] ==> F : (C Int A) co (C Int A')";
by (Blast_tac 1);
qed "stable_constrains_Int";

(*[| F : stable C; F :  (C Int A) co A |] ==> F : stable (C Int A) *)
bind_thm ("stable_constrains_stable", stable_constrains_Int RS stableI);


(*** invariant ***)

Goal "[| Init F<=A;  F: stable A |] ==> F : invariant A";
by (asm_simp_tac (simpset() addsimps [invariant_def]) 1);
qed "invariantI";

(*Could also say "invariant A Int invariant B <= invariant (A Int B)"*)
Goal "[| F : invariant A;  F : invariant B |] ==> F : invariant (A Int B)";
by (auto_tac (claset(), simpset() addsimps [invariant_def, stable_Int]));
qed "invariant_Int";


(*** increasing ***)

Goalw [increasing_def]
     "F : increasing f ==> F : stable {s. z <= f s}";
by (Blast_tac 1);
qed "increasingD";

Goalw [increasing_def, stable_def] "F : increasing (%s. c)";
by Auto_tac; 
qed "increasing_constant";
AddIffs [increasing_constant];

Goalw [increasing_def, stable_def, constrains_def]
     "mono g ==> increasing f <= increasing (g o f)";
by Auto_tac;
by (blast_tac (claset() addIs [monoD, order_trans]) 1);
qed "mono_increasing_o";

(*Holds by the theorem (Suc m <= n) = (m < n) *)
Goalw [increasing_def]
     "!!z::nat. F : increasing f ==> F: stable {s. z < f s}";
by (simp_tac (simpset() addsimps [Suc_le_eq RS sym]) 1);
by (Blast_tac 1);
qed "strict_increasingD";


(** The Elimination Theorem.  The "free" m has become universally quantified!
    Should the premise be !!m instead of ALL m ?  Would make it harder to use
    in forward proof. **)

Goalw [constrains_def]
    "[| ALL m:M. F : {s. s x = m} co (B m) |] \
\    ==> F : {s. s x : M} co (UN m:M. B m)";
by (Blast_tac 1);
qed "elimination";

(*As above, but for the trivial case of a one-variable state, in which the
  state is identified with its one variable.*)
Goalw [constrains_def]
    "(ALL m:M. F : {m} co (B m)) ==> F : M co (UN m:M. B m)";
by (Blast_tac 1);
qed "elimination_sing";



(*** Theoretical Results from Section 6 ***)

Goalw [constrains_def, strongest_rhs_def]
    "F : A co (strongest_rhs F A )";
by (Blast_tac 1);
qed "constrains_strongest_rhs";

Goalw [constrains_def, strongest_rhs_def]
    "F : A co B ==> strongest_rhs F A <= B";
by (Blast_tac 1);
qed "strongest_rhs_is_strongest";


(** Ad-hoc set-theory rules **)

Goal "A Un B - (A - B) = B";
by (Blast_tac 1);
qed "Un_Diff_Diff";
Addsimps [Un_Diff_Diff];

Goal "Union(B) Int A = Union((%C. C Int A)`B)";
by (Blast_tac 1);
qed "Int_Union_Union";

(** Needed for WF reasoning in WFair.ML **)

Goal "less_than `` {k} = greaterThan k";
by (Blast_tac 1);
qed "Image_less_than";

Goal "less_than^-1 `` {k} = lessThan k";
by (Blast_tac 1);
qed "Image_inverse_less_than";

Addsimps [Image_less_than, Image_inverse_less_than];