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//! This file contains proofs related to modulo. These are internal
//! functions used within the math standard library.
//!
//! It's based on the following file from the Dafny math standard library:
//! `Source/DafnyStandardLibraries/src/Std/Arithmetic/Internal/ModInternals.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::super::prelude::*;
verus! {
use super::super::super::arithmetic::internals::general_internals::*;
use super::super::super::arithmetic::mul::*;
#[cfg(verus_keep_ghost)]
use super::mul_internals::group_mul_properties_internal;
#[cfg(verus_keep_ghost)]
use super::super::super::arithmetic::internals::mul_internals_nonlinear;
#[cfg(verus_keep_ghost)]
use super::super::super::arithmetic::internals::mod_internals_nonlinear::{
lemma_fundamental_div_mod,
lemma_mod_range,
lemma_small_mod,
};
#[cfg(verus_keep_ghost)]
use super::super::super::arithmetic::internals::div_internals_nonlinear;
#[cfg(verus_keep_ghost)]
use super::super::super::math::{add as add1, sub as sub1};
/// This function performs the modulus operation recursively.
#[verifier::opaque]
pub open spec fn mod_recursive(x: int, d: int) -> int
recommends
d > 0,
decreases
(if x < 0 {
(d - x)
} else {
x
}),
when d > 0
{
if x < 0 {
mod_recursive(d + x, d)
} else if x < d {
x
} else {
mod_recursive(x - d, d)
}
}
/// This utility function helps prove a mathematical property by
/// induction. The caller supplies an integer predicate, proves the
/// predicate holds in certain base cases, and proves correctness of
/// inductive steps both upward and downward from the base cases. This
/// lemma invokes induction to establish that the predicate holds for
/// all possible inputs.
///
/// `f`: The integer predicate
///
/// `n`: Upper bound on the base cases. Specifically, the caller
/// establishes `f(i)` for every value `i` satisfying `0 <= i < n`.
///
/// To prove inductive steps upward from the base cases, the caller
/// must establish that, for any `i >= 0`, `f(i) ==> f(add1(i, n))`.
/// `add1(i, n)` is just `i + n`, but written in a functional style
/// so that it can be used where functional triggers are required.
///
/// To prove inductive steps downward from the base cases, the caller
/// must establish that, for any `i < n`, `f(i) ==> f(sub1(i, n))`.
/// `sub1(i, n)` is just `i - n`, but written in a functional style
/// so that it can be used where functional triggers are required.
pub proof fn lemma_mod_induction_forall(n: int, f: spec_fn(int) -> bool)
requires
n > 0,
forall|i: int| 0 <= i < n ==> #[trigger] f(i),
forall|i: int| i >= 0 && #[trigger] f(i) ==> #[trigger] f(add1(i, n)),
forall|i: int| i < n && #[trigger] f(i) ==> #[trigger] f(sub1(i, n)),
ensures
forall|i| #[trigger] f(i),
{
assert forall|i: int| #[trigger] f(i) by {
lemma_induction_helper(n, f, i);
};
}
/// This utility function helps prove a mathematical property of a
/// pair of integers by induction. The caller supplies a predicate
/// over a pair of integers, proves the predicate holds in certain
/// base cases, and proves correctness of inductive steps both upward
/// and downward from the base cases. This lemma invokes induction to
/// establish that the predicate holds for all possible inputs.
///
/// `f`: The integer predicate
///
/// `n`: Upper bound on the base cases. Specifically, the caller
/// establishes `f(i, j)` for every pair of values `i, j` satisfying
/// `0 <= i < n` and `0 <= j < n`.
///
/// To prove inductive steps from the base cases, the caller must
/// establish that:
///
/// 1) For any `i >= 0`, `f(i, j) ==> f(add1(i, n), j)`. `add1(i, n)`
/// is just `i + n`, but written in a functional style so that it can
/// be used where functional triggers are required.
///
/// 2) For any `j >= 0`, `f(i, j) ==> f(i, add1(j, n))`
///
/// 3) For any `i < n`, `f(i) ==> f(sub1(i, n))`. `sub1(i, n)` is just
/// `i - n`, but written in a functional style so that it can be used
/// where functional triggers are required.
///
/// 4) For any `j < n`, `f(j) ==> f(i, sub1(j, n))`.
pub proof fn lemma_mod_induction_forall2(n: int, f: spec_fn(int, int) -> bool)
requires
n > 0,
forall|i: int, j: int| 0 <= i < n && 0 <= j < n ==> #[trigger] f(i, j),
forall|i: int, j: int| i >= 0 && #[trigger] f(i, j) ==> #[trigger] f(add1(i, n), j),
forall|i: int, j: int| j >= 0 && #[trigger] f(i, j) ==> #[trigger] f(i, add1(j, n)),
forall|i: int, j: int| i < n && #[trigger] f(i, j) ==> #[trigger] f(sub1(i, n), j),
forall|i: int, j: int| j < n && #[trigger] f(i, j) ==> #[trigger] f(i, sub1(j, n)),
ensures
forall|i: int, j: int| #[trigger] f(i, j),
{
assert forall|x: int, y: int| #[trigger] f(x, y) by {
assert forall|i: int| 0 <= i < n implies #[trigger] f(i, y) by {
let fj = |j| f(i, j);
lemma_mod_induction_forall(n, fj);
assert(fj(y));
};
let fi = |i| f(i, y);
lemma_mod_induction_forall(n, fi);
assert(fi(x));
};
}
/// Proof that when dividing, adding the denominator to the numerator
/// increases the result by 1. Specifically, for the given `n` and `x`,
/// `(x + n) / n == x / n + 1`.
#[verifier::spinoff_prover]
pub proof fn lemma_div_add_denominator(n: int, x: int)
requires
n > 0,
ensures
(x + n) / n == x / n + 1,
{
lemma_fundamental_div_mod(x, n);
lemma_fundamental_div_mod(x + n, n);
let zp = (x + n) / n - x / n - 1;
assert(0 == n * zp + ((x + n) % n) - (x % n)) by {
broadcast use group_mul_properties_internal;
};
if (zp > 0) {
lemma_mul_inequality(1, zp, n);
}
if (zp < 0) {
lemma_mul_inequality(zp, -1, n);
}
}
/// Proof that when dividing, subtracting the denominator from the numerator
/// decreases the result by 1. Specifically, for the given `n` and `x`,
/// `(x - n) / n == x / n - 1`.
pub proof fn lemma_div_sub_denominator(n: int, x: int)
requires
n > 0,
ensures
(x - n) / n == x / n - 1,
{
lemma_fundamental_div_mod(x, n);
lemma_fundamental_div_mod(x - n, n);
let zm = (x - n) / n - x / n + 1;
assert(0 == n * zm + ((x - n) % n) - (x % n)) by {
broadcast use group_mul_properties_internal;
}
if (zm > 0) {
lemma_mul_inequality(1, zm, n);
}
if (zm < 0) {
lemma_mul_inequality(zm, -1, n);
}
}
/// Proof that when dividing, adding the denominator to the numerator
/// doesn't change the remainder. Specifically, for the given `n` and
/// `x`, `(x + n) % n == x % n`.
#[verifier::spinoff_prover]
pub proof fn lemma_mod_add_denominator(n: int, x: int)
requires
n > 0,
ensures
(x + n) % n == x % n,
{
lemma_fundamental_div_mod(x, n);
lemma_fundamental_div_mod(x + n, n);
let zp = (x + n) / n - x / n - 1;
assert(n * zp == n * ((x + n) / n - x / n) - n) by {
assert(n * (((x + n) / n - x / n) - 1) == n * ((x + n) / n - x / n) - n) by {
broadcast use group_mul_is_commutative_and_distributive;
};
};
assert(0 == n * zp + ((x + n) % n) - (x % n)) by {
broadcast use group_mul_properties_internal;
}
if (zp > 0) {
lemma_mul_inequality(1, zp, n);
} else if (zp < 0) {
lemma_mul_inequality(zp, -1, n);
} else {
broadcast use group_mul_properties_internal;
}
}
/// Proof that when dividing, subtracting the denominator from the
/// numerator doesn't change the remainder. Specifically, for the
/// given `n` and `x`, `(x - n) % n == x % n`.
pub proof fn lemma_mod_sub_denominator(n: int, x: int)
requires
n > 0,
ensures
(x - n) % n == x % n,
{
lemma_fundamental_div_mod(x, n);
lemma_fundamental_div_mod(x - n, n);
let zm = (x - n) / n - x / n + 1;
broadcast use group_mul_is_distributive; // OBSERVE
assert(0 == n * zm + ((x - n) % n) - (x % n)) by {
broadcast use group_mul_properties_internal;
}
if (zm > 0) {
lemma_mul_inequality(1, zm, n);
}
if (zm < 0) {
lemma_mul_inequality(zm, -1, n);
}
}
/// Proof that for the given `n` and `x`, `x % n == x` if and only if
/// `0 <= x < n`.
pub proof fn lemma_mod_below_denominator(n: int, x: int)
requires
n > 0,
ensures
0 <= x < n <==> x % n == x,
{
assert forall|x: int| 0 <= x < n <==> #[trigger] (x % n) == x by {
if (0 <= x < n) {
lemma_small_mod(x as nat, n as nat);
}
lemma_mod_range(x, n);
}
}
/// Proof of basic properties of the division given the divisor `n`:
///
/// 1) Adding the denominator to the numerator increases the quotient
/// by 1 and doesn't change the remainder.
///
/// 2) Subtracting the denominator from the numerator decreases the
/// quotient by 1 and doesn't change the remainder.
///
/// 3) The numerator is the same as the result if and only if the
/// numerator is in the half-open range `[0, n)`.
pub proof fn lemma_mod_basics(n: int)
requires
n > 0,
ensures
forall|x: int| #[trigger] ((x + n) % n) == x % n,
forall|x: int| #[trigger] ((x - n) % n) == x % n,
forall|x: int| #[trigger] ((x + n) / n) == x / n + 1,
forall|x: int| #[trigger] ((x - n) / n) == x / n - 1,
forall|x: int| 0 <= x < n <==> #[trigger] (x % n) == x,
{
assert forall|x: int| #[trigger] ((x + n) % n) == x % n by {
lemma_mod_add_denominator(n, x);
};
assert forall|x: int| #[trigger] ((x - n) % n) == x % n by {
lemma_mod_sub_denominator(n, x);
assert((x - n) % n == x % n);
};
assert forall|x: int| #[trigger] ((x + n) / n) == x / n + 1 by {
lemma_div_add_denominator(n, x);
};
assert forall|x: int| #[trigger] ((x - n) / n) == x / n - 1 by {
lemma_div_sub_denominator(n, x);
};
assert forall|x: int| 0 <= x < n <==> #[trigger] (x % n) == x by {
lemma_mod_below_denominator(n, x);
};
}
/// Proof that if `x == q * r + n` and `0 <= r < n`, then `q == x / n`
/// and `r == x % n`. Essentially, this is the converse of the
/// fundamental theorem of division and modulo.
pub proof fn lemma_quotient_and_remainder(x: int, q: int, r: int, n: int)
requires
n > 0,
0 <= r < n,
x == q * n + r,
ensures
q == x / n,
r == x % n,
decreases
(if q > 0 {
q
} else {
-q
}),
{
lemma_mod_basics(n);
if q > 0 {
mul_internals_nonlinear::lemma_mul_is_distributive_add(n, q - 1, 1);
broadcast use lemma_mul_is_commutative;
assert(q * n + r == (q - 1) * n + n + r);
lemma_quotient_and_remainder(x - n, q - 1, r, n);
} else if q < 0 {
lemma_mul_is_distributive_sub(n, q + 1, 1);
broadcast use lemma_mul_is_commutative;
assert(q * n + r == (q + 1) * n - n + r);
lemma_quotient_and_remainder(x + n, q + 1, r, n);
} else {
div_internals_nonlinear::lemma_small_div();
assert(r / n == 0);
}
}
/// This function says that for any `x` and `y`, there are two
/// possibilities for the sum `x % n + y % n`: (1) It's in the range
/// `[0, n)` and it's equal to `(x + y) % n`. (2) It's in the range
/// `[n, n + n)` and it's equal to `(x + y) % n + n`.
pub open spec fn mod_auto_plus(n: int) -> bool
recommends
n > 0,
{
forall|x: int, y: int|
{
let z = (x % n) + (y % n);
((0 <= z < n && #[trigger] ((x + y) % n) == z) || (n <= z < n + n && ((x + y) % n) == z
- n))
}
}
/// This function says that for any `x` and `y`, there are two
/// possibilities for the difference `x % n - y % n`: (1) It's in the
/// range `[0, n)` and it's equal to `(x - y) % n`. (2) It's in the
/// range `[-n, 0)` and it's equal to `(x + y) % n - n`.
pub open spec fn mod_auto_minus(n: int) -> bool
recommends
n > 0,
{
forall|x: int, y: int|
{
let z = (x % n) - (y % n);
((0 <= z < n && #[trigger] ((x - y) % n) == z) || (-n <= z < 0 && ((x - y) % n) == z
+ n))
}
}
/// This function states various useful properties about the modulo
/// operator when the divisor is `n`.
pub open spec fn mod_auto(n: int) -> bool
recommends
n > 0,
{
&&& (n % n == 0 && (-n) % n == 0)
&&& (forall|x: int| #[trigger] ((x % n) % n) == x % n)
&&& (forall|x: int| 0 <= x < n <==> #[trigger] (x % n) == x)
&&& mod_auto_plus(n)
&&& mod_auto_minus(n)
}
/// Proof of `mod_auto(n)`, which states various useful properties
/// about the modulo operator when the divisor is the positive number
/// `n`
pub proof fn lemma_mod_auto(n: int)
requires
n > 0,
ensures
mod_auto(n),
{
lemma_mod_basics(n);
broadcast use group_mul_properties_internal;
assert forall|x: int, y: int|
{
let z = (x % n) + (y % n);
((0 <= z < n && #[trigger] ((x + y) % n) == z) || (n <= z < n + n && ((x + y) % n) == z
- n))
} by {
let xq = x / n;
let xr = x % n;
lemma_fundamental_div_mod(x, n);
assert(x == xq * n + xr);
let yq = y / n;
let yr = y % n;
lemma_fundamental_div_mod(y, n);
assert(y == yq * n + yr);
if xr + yr < n {
lemma_quotient_and_remainder(x + y, xq + yq, xr + yr, n);
} else {
lemma_quotient_and_remainder(x + y, xq + yq + 1, xr + yr - n, n);
}
}
assert forall|x: int, y: int|
{
let z = (x % n) - (y % n);
((0 <= z < n && #[trigger] ((x - y) % n) == z) || (-n <= z < 0 && ((x - y) % n) == z
+ n))
} by {
let xq = x / n;
let xr = x % n;
lemma_fundamental_div_mod(x, n);
assert(x == n * (x / n) + (x % n));
let yq = y / n;
let yr = y % n;
lemma_fundamental_div_mod(y, n);
assert(y == yq * n + yr);
if xr - yr >= 0 {
lemma_quotient_and_remainder(x - y, xq - yq, xr - yr, n);
} else { // xr - yr < 0
lemma_quotient_and_remainder(x - y, xq - yq - 1, xr - yr + n, n);
}
}
}
/// This utility function helps prove a mathematical property by
/// induction. The caller supplies an integer predicate, proves the
/// predicate holds in certain base cases, and proves correctness of
/// inductive steps both upward and downward from the base cases. This
/// lemma invokes induction to establish that the predicate holds for
/// the given arbitrary input `x`.
///
/// `f`: The integer predicate
///
/// `n`: Upper bound on the base cases. Specifically, the caller
/// establishes `f(i)` for every value `i` satisfying `is_le(0, i) &&
/// i < n`.
///
/// `x`: The desired case established by this lemma. Its postcondition
/// thus includes `f(x)`.
///
/// To prove inductive steps upward from the base cases, the caller
/// must establish that, for any `i`, `is_le(0, i) && f(i) ==> f(i +
/// n)`. `is_le(0, i)` is just `0 <= i`, but written in a functional
/// style so that it can be used where functional triggers are
/// required.
///
/// To prove inductive steps downward from the base cases, the caller
/// must establish that, for any `i`, `is_le(i + 1, n) && f(i) ==> f(i
/// - n)`. `is_le(i + 1, n)` is just `i + 1 <= n`, but written in a
/// functional style so that it can be used where functional triggers
/// are required.
pub proof fn lemma_mod_induction_auto(n: int, x: int, f: spec_fn(int) -> bool)
requires
n > 0,
mod_auto(n) ==> {
&&& (forall|i: int| #[trigger] is_le(0, i) && i < n ==> f(i))
&&& (forall|i: int| #[trigger] is_le(0, i) && f(i) ==> f(i + n))
&&& (forall|i: int| #[trigger] is_le(i + 1, n) && f(i) ==> f(i - n))
},
ensures
mod_auto(n),
f(x),
{
lemma_mod_auto(n);
assert(forall|i: int| is_le(0, i) && #[trigger] f(i) ==> #[trigger] f(add1(i, n)));
assert(forall|i: int| is_le(i + 1, n) && #[trigger] f(i) ==> #[trigger] f(sub1(i, n)));
assert forall|i: int| 0 <= i < n implies #[trigger] f(i) by {
assert(forall|i: int| is_le(0, i) && i < n ==> f(i));
assert(is_le(0, i) && i < n);
};
lemma_mod_induction_forall(n, f);
assert(f(x));
}
/// This utility function helps prove a mathematical property by
/// induction. The caller supplies an integer predicate, proves the
/// predicate holds in certain base cases, and proves correctness of
/// inductive steps both upward and downward from the base cases. This
/// lemma invokes induction to establish that the predicate holds for
/// all integer values.
///
/// `f`: The integer predicate
///
/// `n`: Upper bound on the base cases. Specifically, the caller
/// establishes `f(i)` for every value `i` satisfying `is_le(0, i) &&
/// i < n`.
///
/// To prove inductive steps upward from the base cases, the caller
/// must establish that, for any `i`, `is_le(0, i) && f(i) ==> f(i +
/// n)`. `is_le(0, i)` is just `0 <= i`, but written in a functional
/// style so that it can be used where functional triggers are
/// required.
///
/// To prove inductive steps downward from the base cases, the caller
/// must establish that, for any `i`, `is_le(i + 1, n) && f(i) ==> f(i
/// - n)`. `is_le(i + 1, n)` is just `i + 1 <= n`, but written in a
/// functional style so that it can be used where functional triggers
/// are required.
pub proof fn lemma_mod_induction_auto_forall(n: int, f: spec_fn(int) -> bool)
requires
n > 0,
mod_auto(n) ==> {
&&& (forall|i: int| #[trigger] is_le(0, i) && i < n ==> f(i))
&&& (forall|i: int| #[trigger] is_le(0, i) && f(i) ==> f(i + n))
&&& (forall|i: int| #[trigger] is_le(i + 1, n) && f(i) ==> f(i - n))
},
ensures
mod_auto(n),
forall|i| #[trigger] f(i),
{
assert(mod_auto(n)) by {
lemma_mod_induction_auto(n, 0, f);
}
assert forall|i| #[trigger] f(i) by {
lemma_mod_induction_auto(n, i, f);
}
}
} // verus!