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
175

//! This file contains proofs related to powers of 2. These are part
//! of the math standard library.
//!
//! It's based on the following file from the Dafny math standard library:
//! `Source/DafnyStandardLibraries/src/Std/Arithmetic/Power2.dfy`.
//! That file has the following copyright notice:
//! /*******************************************************************************
//! *  Original: Copyright (c) Microsoft Corporation
//! *  SPDX-License-Identifier: MIT
//! *
//! *  Modifications and Extensions: Copyright by the contributors to the Dafny Project
//! *  SPDX-License-Identifier: MIT
//! *******************************************************************************/
#[allow(unused_imports)]
use super::super::prelude::*;

verus! {

#[cfg(verus_keep_ghost)]
use super::power::{pow, lemma_pow_positive, lemma_pow_adds, lemma_pow_strictly_increases};

/// This function computes 2 to the power of the given natural number
/// `e`. It's opaque so that the SMT solver doesn't waste time
/// repeatedly recursively unfolding it.
#[verifier::opaque]
pub open spec fn pow2(e: nat) -> nat
    decreases
            e  // ensures pow2(e) > 0
            // cannot have ensurs clause in spec functions
            // a workaround is the lemma_pow2_pos below
            ,
{
    // you cannot reveal in a spec function, which cause more reveals clauses
    // for the proof
    // reveal(pow);
    pow(2, e) as nat
}

/// Proof that 2 to the power of any natural number (specifically,
/// `e`) is positive
pub broadcast proof fn lemma_pow2_pos(e: nat)
    ensures
        #[trigger] pow2(e) > 0,
{
    reveal(pow2);
    lemma_pow_positive(2, e);
}

/// Proof that `pow2(e)` is equivalent to `pow(2, e)`
pub broadcast proof fn lemma_pow2(e: nat)
    ensures
        #[trigger] pow2(e) == pow(2, e) as int,
    decreases e,
{
    reveal(pow);
    reveal(pow2);
    if e != 0 {
        lemma_pow2((e - 1) as nat);
    }
}

/// Proof relating 2^e to 2^(e-1).
pub broadcast proof fn lemma_pow2_unfold(e: nat)
    requires
        e > 0,
    ensures
        #[trigger] pow2(e) == 2 * pow2((e - 1) as nat),
{
    lemma_pow2(e);
    lemma_pow2((e - 1) as nat);
}

/// Proof that `2^(e1 + e2)` is equivalent to `2^e1 * 2^e2`.
pub broadcast proof fn lemma_pow2_adds(e1: nat, e2: nat)
    ensures
        #[trigger] pow2(e1 + e2) == pow2(e1) * pow2(e2),
{
    lemma_pow2(e1);
    lemma_pow2(e2);
    lemma_pow2(e1 + e2);
    lemma_pow_adds(2, e1, e2);
}

/// Proof that if `e1 < e2` then `2^e1 < 2^e2`.
pub broadcast proof fn lemma_pow2_strictly_increases(e1: nat, e2: nat)
    requires
        e1 < e2,
    ensures
        #[trigger] pow2(e1) < #[trigger] pow2(e2),
{
    lemma_pow2(e1);
    lemma_pow2(e2);
    lemma_pow_strictly_increases(2, e1, e2);
}

/// Proof establishing the concrete values for all powers of 2 from 0 to 32 and also 2^64
pub proof fn lemma2_to64()
    ensures
        pow2(0) == 0x1,
        pow2(1) == 0x2,
        pow2(2) == 0x4,
        pow2(3) == 0x8,
        pow2(4) == 0x10,
        pow2(5) == 0x20,
        pow2(6) == 0x40,
        pow2(7) == 0x80,
        pow2(8) == 0x100,
        pow2(9) == 0x200,
        pow2(10) == 0x400,
        pow2(11) == 0x800,
        pow2(12) == 0x1000,
        pow2(13) == 0x2000,
        pow2(14) == 0x4000,
        pow2(15) == 0x8000,
        pow2(16) == 0x10000,
        pow2(17) == 0x20000,
        pow2(18) == 0x40000,
        pow2(19) == 0x80000,
        pow2(20) == 0x100000,
        pow2(21) == 0x200000,
        pow2(22) == 0x400000,
        pow2(23) == 0x800000,
        pow2(24) == 0x1000000,
        pow2(25) == 0x2000000,
        pow2(26) == 0x4000000,
        pow2(27) == 0x8000000,
        pow2(28) == 0x10000000,
        pow2(29) == 0x20000000,
        pow2(30) == 0x40000000,
        pow2(31) == 0x80000000,
        pow2(32) == 0x100000000,
        pow2(64) == 0x10000000000000000,
{
    reveal(pow2);
    reveal(pow);
    #[verusfmt::skip]
    assert(
        pow2(0) == 0x1 &&
        pow2(1) == 0x2 &&
        pow2(2) == 0x4 &&
        pow2(3) == 0x8 &&
        pow2(4) == 0x10 &&
        pow2(5) == 0x20 &&
        pow2(6) == 0x40 &&
        pow2(7) == 0x80 &&
        pow2(8) == 0x100 &&
        pow2(9) == 0x200 &&
        pow2(10) == 0x400 &&
        pow2(11) == 0x800 &&
        pow2(12) == 0x1000 &&
        pow2(13) == 0x2000 &&
        pow2(14) == 0x4000 &&
        pow2(15) == 0x8000 &&
        pow2(16) == 0x10000 &&
        pow2(17) == 0x20000 &&
        pow2(18) == 0x40000 &&
        pow2(19) == 0x80000 &&
        pow2(20) == 0x100000 &&
        pow2(21) == 0x200000 &&
        pow2(22) == 0x400000 &&
        pow2(23) == 0x800000 &&
        pow2(24) == 0x1000000 &&
        pow2(25) == 0x2000000 &&
        pow2(26) == 0x4000000 &&
        pow2(27) == 0x8000000 &&
        pow2(28) == 0x10000000 &&
        pow2(29) == 0x20000000 &&
        pow2(30) == 0x40000000 &&
        pow2(31) == 0x80000000 &&
        pow2(32) == 0x100000000 &&
        pow2(64) == 0x10000000000000000
    ) by(compute_only);
}

} // verus!