-
Notifications
You must be signed in to change notification settings - Fork 298
/
positivity.lean
174 lines (113 loc) · 6.75 KB
/
positivity.lean
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
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
import algebra.order.smul
import analysis.normed.group.basic
import analysis.special_functions.pow
import data.complex.exponential
import data.rat.nnrat
import data.real.ereal
import data.real.hyperreal
import data.real.sqrt
import tactic.positivity
/-! # Tests for the `positivity` tactic
This tactic proves goals of the form `0 ≤ a` and `0 < a`.
-/
open_locale ennreal nnreal
universe u
variables {α β : Type*}
/- ## Numeric goals -/
example : 0 ≤ 0 := by positivity
example : 0 ≤ 3 := by positivity
example : 0 < 3 := by positivity
/- ## Goals working directly from a hypothesis -/
example {a : ℤ} (ha : 0 ≤ a) : 0 ≤ a := by positivity
example {a : ℤ} (ha : 0 < a) : 0 ≤ a := by positivity
example {a : ℤ} (ha : 0 < a) : 0 < a := by positivity
example {a : ℤ} (ha : 3 ≤ a) : 0 < a := by positivity
example {a : ℤ} (ha : 3 < a) : 0 ≤ a := by positivity
example {a : ℤ} (ha : 3 < a) : 0 < a := by positivity
example {a b : ℤ} (h : 0 ≤ a + b) : 0 ≤ a + b := by positivity
/- ## Tests of the @[positivity] plugin tactics (addition, multiplication, division) -/
example {a : ℤ} (ha : 3 < a) : 0 ≤ a + a := by positivity
example {a b : ℤ} (ha : 3 < a) (hb : 4 ≤ b) : 0 ≤ 3 + a + b + b + 14 := by positivity
example {H : Type*} [linear_ordered_add_comm_group H] {a b : H} (ha : 0 < a) (hb : 0 ≤ b) :
0 ≤ a + a + b :=
by positivity
example {a : ℤ} (ha : 3 < a) : 0 < a + a := by positivity
example {a b : ℚ} (ha : 3 < a) (hb : 4 ≤ b) : 0 < 3 + a * b / 7 + b + 7 + 14 := by positivity
example {a b : ℤ} (ha : 3 < a) (hb : 4 ≤ b) : 0 < 3 + a * b / 7 + b + 7 + 14 := by positivity
example {a : ℤ} (ha : 0 < a) : 0 < a / a := by positivity
/-! ### Exponentiation -/
example [ordered_semiring α] [nontrivial α] (a : α) : 0 < a ^ 0 := by positivity
example [linear_ordered_ring α] (a : α) (n : ℕ) : 0 ≤ a ^ (bit0 n) := by positivity
example [ordered_semiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) : 0 ≤ a ^ n := by positivity
example [ordered_semiring α] {a : α} {n : ℕ} (ha : 0 < a) : 0 < a ^ n := by positivity
example [linear_ordered_semifield α] (a : α) : 0 < a ^ (0 : ℤ) := by positivity
example [linear_ordered_field α] (a : α) (n : ℤ) : 0 ≤ a ^ (bit0 n) := by positivity
example [linear_ordered_semifield α] {a : α} {n : ℤ} (ha : 0 ≤ a) : 0 ≤ a ^ n := by positivity
example [linear_ordered_semifield α] {a : α} {n : ℤ} (ha : 0 < a) : 0 < a ^ n := by positivity
example {a b : cardinal.{u}} (ha : 0 < a) : 0 < a ^ b := by positivity
example {a b : ordinal.{u}} (ha : 0 < a) : 0 < a ^ b := by positivity
example {a b : ℝ} (ha : 0 ≤ a) : 0 ≤ a ^ b := by positivity
example {a b : ℝ} (ha : 0 < a) : 0 < a ^ b := by positivity
example {a : ℝ≥0} {b : ℝ} (ha : 0 < a) : 0 < a ^ b := by positivity
example {a : ℝ≥0∞} {b : ℝ} (ha : 0 < a) (hb : 0 ≤ b) : 0 < a ^ b := by positivity
example {a : ℝ≥0∞} {b : ℝ} (ha : 0 < a) (hb : 0 < b) : 0 < a ^ b := by positivity
example {a : ℤ} (ha : 3 < a) : 0 ≤ a ^ 2 + a := by positivity
example {a : ℤ} (ha : 3 < a) : 0 ≤ a ^ 3 + a := by positivity
example {a : ℤ} (ha : 3 < a) : 0 < a ^ 2 + a := by positivity
example {a b : ℤ} (ha : 3 < a) (hb : b ≥ 4) : 0 ≤ 3 * a ^ 2 * b + b * 7 + 14 := by positivity
example {a b : ℤ} (ha : 3 < a) (hb : b ≥ 4) : 0 < 3 * a ^ 2 * b + b * 7 + 14 := by positivity
example {x : ℚ} (hx : 0 ≤ x) : 0 ≤ x⁻¹ := by positivity
example {a : ℤ} : 0 ≤ |a| := by positivity
example {a : ℤ} : 0 < |a| + 3 := by positivity
example {a : ℤ} (ha : 1 < a) : 0 < |(3:ℤ) + a| := by positivity
example {a : ℝ} (ha : 0 ≤ a) : 0 ≤ real.sqrt a := by positivity
example {a : ℝ} (ha : 0 ≤ a) : 0 < real.sqrt (a + 3) := by positivity
example {a b : ℤ} (ha : 3 < a) : 0 ≤ min a (b ^ 2) := by positivity
-- test that the tactic can ignore arithmetic operations whose associated extension tactic requires
-- more typeclass assumptions than are available
example {R : Type*} [has_zero R] [has_div R] [linear_order R] {a b c : R} (h1 : 0 < a) (h2 : 0 < b)
(h3 : 0 < c) :
0 < max (a / b) c :=
by positivity
example : 0 ≤ max 3 4 := by positivity
example {b : ℤ} : 0 ≤ max (-3) (b ^ 2) := by positivity
example {b : ℤ} : 0 ≤ max (b ^ 2) 0 := by positivity
example : 0 ≤ max (0:ℤ) (-3) := by positivity
example : 0 ≤ max (-3 : ℤ) 5 := by positivity
example [ordered_semiring α] [ordered_add_comm_monoid β] [smul_with_zero α β]
[ordered_smul α β] {a : α} (ha : 0 < a) {b : β} (hb : 0 < b) : 0 ≤ a • b := by positivity
example {r : ℝ} : 0 < real.exp r := by positivity
example {V : Type*} [normed_add_comm_group V] (x : V) : 0 ≤ ∥x∥ := by positivity
example {X : Type*} [metric_space X] (x y : X) : 0 ≤ dist x y := by positivity
example {E : Type*} [add_group E] {p : add_group_seminorm E} {x : E} : 0 ≤ p x := by positivity
example {E : Type*} [group E] {p : group_seminorm E} {x : E} : 0 ≤ p x := by positivity
/- ### Canonical orders -/
example {a : ℕ} : 0 ≤ a := by positivity
example {a : ℚ≥0} : 0 ≤ a := by positivity
example {a : ℝ≥0} : 0 ≤ a := by positivity
example {a : ℝ≥0∞} : 0 ≤ a := by positivity
/- ### Coercions -/
example {a : ℕ} : (0 : ℤ) ≤ a := by positivity
example {a : ℕ} (ha : 0 < a) : (0 : ℤ) < a := by positivity
example {a : ℤ} (ha : 0 ≤ a) : (0 : ℚ) ≤ a := by positivity
example {a : ℤ} (ha : 0 < a) : (0 : ℚ) < a := by positivity
example {a : ℚ} (ha : 0 ≤ a) : (0 : ℝ) ≤ a := by positivity
example {a : ℚ} (ha : 0 < a) : (0 : ℝ) < a := by positivity
example {r : ℝ≥0} : (0 : ℝ) ≤ r := by positivity
example {r : ℝ≥0} (hr : 0 < r) : (0 : ℝ) < r := by positivity
example {r : ℝ≥0} (hr : 0 < r) : (0 : ℝ≥0∞) < r := by positivity
-- example {r : ℝ≥0} : (0 : ereal) ≤ r := by positivity -- TODO: Handle `coe_trans`
-- example {r : ℝ≥0} (hr : 0 < r) : (0 : ereal) < r := by positivity
example {r : ℝ} (hr : 0 ≤ r) : (0 : ereal) ≤ r := by positivity
example {r : ℝ} (hr : 0 < r) : (0 : ereal) < r := by positivity
example {r : ℝ} (hr : 0 ≤ r) : (0 : hyperreal) ≤ r := by positivity
example {r : ℝ} (hr : 0 < r) : (0 : hyperreal) < r := by positivity
example {r : ℝ≥0∞} : (0 : ereal) ≤ r := by positivity
example {r : ℝ≥0∞} (hr : 0 < r) : (0 : ereal) < r := by positivity
example {α : Type*} [ordered_ring α] {n : ℤ} : 0 ≤ ((n ^ 2 : ℤ) : α) := by positivity
example {r : ℝ≥0} : 0 ≤ ((r : ℝ) : ereal) := by positivity
example {r : ℝ≥0} : 0 < ((r + 1 : ℝ) : ereal) := by positivity
/- ## Tests that the tactic is agnostic on reversed inequalities -/
example {a : ℤ} (ha : a > 0) : 0 ≤ a := by positivity
example {a : ℤ} (ha : 0 < a) : a ≥ 0 := by positivity
example {a : ℤ} (ha : a > 0) : a ≥ 0 := by positivity