summaryrefslogtreecommitdiff
path: root/test-suite/success/OmegaPre.v
blob: f4996734bf020dfcd0a2c7992ef3740eafcb9e80 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
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, Zmax a a = a.
intros.
omega with *.
Qed.

Goal forall a b:Z, Zmax a b = Zmax b a.
intros.
omega with *.
Qed.

Goal forall a b c:Z, Zmax a (Zmax b c) = Zmax (Zmax a b) c.
intros.
omega with *.
Qed.

Goal forall a b:Z, Zmax a b + Zmin a b = a + b.
intros.
omega with *.
Qed.

Goal forall a:Z, (Zabs a)*(Zsgn a) = a.
intros.
zify.
intuition; subst; omega. (* pure multiplication: omega alone can't do it *)
Qed.

Goal forall a:Z, Zabs a = a -> a >= 0.
intros.
omega with *.
Qed.

Goal forall a:Z, Zsgn 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 (nat_of_N (Npos p))) = Zpos p.
intros.
omega with *.
Qed.