frap/Frap.v

Require Import String Arith Omega Program Sets Relations Map Var Invariant Bool ModelCheck.
Export String Arith Sets Relations Map Var Invariant Bool ModelCheck.
Require Import List.
Export List ListNotations.
Open Scope string_scope.
Open Scope list_scope.

Ltac inductN n :=
  match goal with
    | [ |- forall x : ?E, _ ] =>
      match type of E with
        | Prop =>
          let H := fresh in intro H;
            match n with
              | 1 => dependent induction H
              | S ?n' => inductN n'
            end
        | _ => intro; inductN n
      end
  end.

Ltac induct e := inductN e || dependent induction e.

Ltac invert' H := inversion H; clear H; subst.

Ltac invertN n :=
  match goal with
    | [ |- forall x : ?E, _ ] =>
      match type of E with
        | Prop =>
          let H := fresh in intro H;
            match n with
              | 1 => invert' H
              | S ?n' => invertN n'
            end
        | _ => intro; invertN n
      end
  end.

Ltac invert e := invertN e || invert' e.

Ltac invert0 e := invert e; fail.
Ltac invert1 e := invert0 e || (invert e; []).
Ltac invert2 e := invert1 e || (invert e; [|]).

Ltac maps_neq :=
  match goal with
  | [ H : ?m1 = ?m2 |- _ ] =>
    let rec recur E :=
        match E with
        | ?E' $+ (?k, _) => 
          (apply (f_equal (fun m => m $? k)) in H; simpl in *; autorewrite with core in *; simpl in *; congruence)
          || recur E'
        end in
    recur m1 || recur m2
end.

Ltac fancy_neq :=
  repeat match goal with
         | _ => maps_neq
         | [ H : _ = _ |- _ ] => invert H
         end.

Ltac removeDups :=
  match goal with
  | [ |- context[constant ?ls] ] =>
    someMatch ls;
    erewrite (@removeDups_ok _ ls)
      by repeat (apply RdNil
                 || (apply RdNew; [ simpl; intuition (congruence || solve [ fancy_neq ]) | ])
                 || (apply RdDup; [ simpl; intuition congruence | ]))
  end.

Ltac simplify := repeat (unifyTails; pose proof I);
                repeat match goal with
                       | [ H : True |- _ ] => clear H
                       end;
                repeat progress (simpl in *; intros; try autorewrite with core in *);
                                 repeat (removeDups || doSubtract).
Ltac propositional := intuition idtac.

Ltac linear_arithmetic := intros;
    repeat match goal with
           | [ |- context[max ?a ?b] ] =>
             let Heq := fresh "Heq" in destruct (Max.max_spec a b) as [[? Heq] | [? Heq]];
               rewrite Heq in *; clear Heq
           | [ _ : context[max ?a ?b] |- _ ] =>
             let Heq := fresh "Heq" in destruct (Max.max_spec a b) as [[? Heq] | [? Heq]];
               rewrite Heq in *; clear Heq
           | [ |- context[min ?a ?b] ] =>
             let Heq := fresh "Heq" in destruct (Min.min_spec a b) as [[? Heq] | [? Heq]];
               rewrite Heq in *; clear Heq
           | [ _ : context[min ?a ?b] |- _ ] =>
             let Heq := fresh "Heq" in destruct (Min.min_spec a b) as [[? Heq] | [? Heq]];
               rewrite Heq in *; clear Heq
           end; omega.

Ltac equality := intuition congruence.

Ltac cases E :=
  ((is_var E; destruct E)
   || match type of E with
      | {_} + {_} => destruct E
      | _ => let Heq := fresh "Heq" in destruct E eqn:Heq
      end);
  repeat match goal with
         | [ H : _ = left _ |- _ ] => clear H
         | [ H : _ = right _ |- _ ] => clear H
         end.

Global Opaque max min.

Infix "==n" := eq_nat_dec (no associativity, at level 50).

Export Frap.Map.

Ltac maps_equal := Frap.Map.M.maps_equal; simplify.

Ltac first_order := firstorder idtac.


(** * Model checking *)

Ltac model_check_done :=
  apply MscDone; apply prove_oneStepClosure; simplify; propositional; subst;
  repeat match goal with
         | [ H : ?P |- _ ] =>
           match type of P with
           | Prop => invert H; simplify
           end
         end; equality.

Ltac singletoner :=
  repeat match goal with
         (* | _ => apply singleton_in *)
         | [ |- _ ?S ] => idtac S; apply singleton_in
         | [ |- (_ \cup _) _ ] => apply singleton_in_other
         end.

Ltac model_check_step :=
  eapply MscStep; [
    repeat (apply oneStepClosure_empty
            || (apply oneStepClosure_split; [ simplify;
                                              repeat match goal with
                                                     | [ H : ?P |- _ ] =>
                                                       match type of P with
                                                       | Prop => invert H; simplify; try congruence
                                                       end
                                                     end; solve [ singletoner ] | ]))
  | simplify ].

Ltac model_check_steps1 := model_check_done || model_check_step.
Ltac model_check_steps := repeat model_check_steps1.

Ltac model_check_finish := simplify; propositional; subst; simplify; try equality; try linear_arithmetic.

Ltac model_check_infer :=
  apply multiStepClosure_ok; simplify; model_check_steps.

Ltac model_check_find_invariant :=
  simplify; eapply invariant_weaken; [ model_check_infer | ]; cbv beta in *.

Ltac model_check := model_check_find_invariant; model_check_finish.

Inductive ordering (n m : nat) :=
| Lt (_ : n < m)
| Eq (_ : n = m)
| Gt (_ : n > m).

Local Hint Constructors ordering.
Local Hint Extern 1 (_ < _) => omega.
Local Hint Extern 1 (_ > _) => omega.

Theorem totally_ordered : forall n m, ordering n m.
Proof.
  induction n; destruct m; simpl; eauto.
  destruct (IHn m); eauto.
Qed.

Ltac total_ordering N M := destruct (totally_ordered N M).