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|
Require Import ZArith Nnat Omega.
Open Scope Z_scope.
(** Test of the zify preprocessor for (R)Omega *)
(* More details in file PreOmega.v
(r)omega with Z : starts with zify_op
(r)omega with nat : starts with zify_nat
(r)omega with positive : starts with zify_positive
(r)omega with N : starts with uses zify_N
(r)omega with * : starts zify (a saturation of the others)
*)
(* zify_op *)
Goal forall a:Z, Z.max a a = a.
intros.
omega with *.
Qed.
Goal forall a b:Z, Z.max a b = Z.max b a.
intros.
omega with *.
Qed.
Goal forall a b c:Z, Z.max a (Z.max b c) = Z.max (Z.max a b) c.
intros.
omega with *.
Qed.
Goal forall a b:Z, Z.max a b + Z.min a b = a + b.
intros.
omega with *.
Qed.
Goal forall a:Z, (Z.abs a)*(Z.sgn a) = a.
intros.
zify.
intuition; subst; omega. (* pure multiplication: omega alone can't do it *)
Qed.
Goal forall a:Z, Z.abs a = a -> a >= 0.
intros.
omega with *.
Qed.
Goal forall a:Z, Z.sgn a = a -> a = 1 \/ a = 0 \/ a = -1.
intros.
omega with *.
Qed.
(* zify_nat *)
Goal forall m: nat, (m<2)%nat -> (0<= m+m <=2)%nat.
intros.
omega with *.
Qed.
Goal forall m:nat, (m<1)%nat -> (m=0)%nat.
intros.
omega with *.
Qed.
Goal forall m: nat, (m<=100)%nat -> (0<= m+m <=200)%nat.
intros.
omega with *.
Qed.
(* 2000 instead of 200: works, but quite slow *)
Goal forall m: nat, (m*m>=0)%nat.
intros.
omega with *.
Qed.
(* zify_positive *)
Goal forall m: positive, (m<2)%positive -> (2 <= m+m /\ m+m <= 2)%positive.
intros.
omega with *.
Qed.
Goal forall m:positive, (m<2)%positive -> (m=1)%positive.
intros.
omega with *.
Qed.
Goal forall m: positive, (m<=1000)%positive -> (2<=m+m/\m+m <=2000)%positive.
intros.
omega with *.
Qed.
Goal forall m: positive, (m*m>=1)%positive.
intros.
omega with *.
Qed.
(* zify_N *)
Goal forall m:N, (m<2)%N -> (0 <= m+m /\ m+m <= 2)%N.
intros.
omega with *.
Qed.
Goal forall m:N, (m<1)%N -> (m=0)%N.
intros.
omega with *.
Qed.
Goal forall m:N, (m<=1000)%N -> (0<=m+m/\m+m <=2000)%N.
intros.
omega with *.
Qed.
Goal forall m:N, (m*m>=0)%N.
intros.
omega with *.
Qed.
(* mix of datatypes *)
Goal forall p, Z.of_N (N.of_nat (N.to_nat (Npos p))) = Zpos p.
intros.
omega with *.
Qed.
|
