| // RUN: %verify --disable-nonlinear-arithmetic "%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"} GeneralInternals { |
|
|
| /* this predicate is primarily used as a trigger */ |
| ghost predicate IsLe(x: int, y: int) |
| { |
| x <= y |
| } |
|
|
| /* aids in the process of induction for modulus */ |
| lemma LemmaInductionHelper(n: int, f: int -> bool, x: int) |
| requires n > 0 |
| requires forall i :: 0 <= i < n ==> f(i) |
| requires forall i {:trigger f(i), f(i + n)} :: i >= 0 && f(i) ==> f(i + n) |
| requires forall i {:trigger f(i), f(i - n)} :: i < n && f(i) ==> f(i - n) |
| ensures f(x) |
| decreases if x >= n then x else -x |
| { |
| if (x >= n) |
| { |
| LemmaInductionHelper(n, f, x - n); |
| assert f((x - n) + n); |
| } |
| else if (x < 0) |
| { |
| LemmaInductionHelper(n, f, x + n); |
| } |
| } |
| } |
| |
