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6851d40 | 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 | // RUN: %verify "%s"
/*******************************************************************************
* Original: Copyright (c) Microsoft Corporation
* SPDX-License-Identifier: MIT
*
* Modifications and Extensions: Copyright by the contributors to the Dafny Project
* SPDX-License-Identifier: MIT
*******************************************************************************/
module {:options "-functionSyntax:4"} MulInternalsNonlinear {
/* WARNING: Think three times before adding to this file, as nonlinear
verification is highly unstable! */
/* multiplying two positive integers will result in a positive integer */
lemma LemmaMulStrictlyPositive(x: int, y: int)
ensures (0 < x && 0 < y) ==> (0 < x * y)
{}
/* multiplying two nonzero integers will never result in 0 as the poduct */
lemma LemmaMulNonzero(x: int, y: int)
ensures x * y != 0 <==> x != 0 && y != 0
{}
/* multiplication is associative */
lemma LemmaMulIsAssociative(x: int, y: int, z: int)
ensures x * (y * z) == (x * y) * z
{}
/* multiplication is distributive */
lemma LemmaMulIsDistributiveAdd(x: int, y: int, z: int)
ensures x * (y + z) == x * y + x * z
{}
/* the product of two integers is greater than the value of each individual integer */
lemma LemmaMulOrdering(x: int, y: int)
requires x != 0
requires y != 0
requires 0 <= x * y
ensures x * y >= x && x * y >= y
{}
/* multiplying by a positive integer preserves inequality */
lemma LemmaMulStrictInequality(x: int, y: int, z: int)
requires x < y
requires z > 0
ensures x * z < y * z
{}
}
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