[llvm] [DA] Add initial support for monotonicity check (PR #162280)

[Ryotaro Kasuga via llvm-commits]

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@@ -177,13 +189,192 @@ void DependenceAnalysisWrapperPass::getAnalysisUsage(AnalysisUsage &AU) const {
   AU.addRequiredTransitive<LoopInfoWrapperPass>();
 }
 
+namespace {
+
+/// The property of monotonicity of a SCEV. To define the monotonicity, assume
+/// a SCEV defined within N-nested loops. Let i_k denote the iteration number
+/// of the k-th loop. Then we can regard the SCEV as an N-ary function:
+///
+///   F(i_1, i_2, ..., i_N)
+///
+/// The domain of i_k is the closed range [0, BTC_k], where BTC_k is the
+/// backedge-taken count of the k-th loop.
+///
+/// A function F is said to be "monotonically increasing with respect to the
+/// k-th loop" if x <= y implies the following condition:
+///
+///   F(i_1, ..., i_{k-1}, x, i_{k+1}, ..., i_N) <=
+///   F(i_1, ..., i_{k-1}, y, i_{k+1}, ..., i_N)
+///
+/// where i_1, ..., i_{k-1}, i_{k+1}, ..., i_N, x, y in their domains.
+///
+/// Likewise F is "monotonically decreasing with respect to the k-th loop"
+/// if x <= y implies
+///
+///   F(i_1, ..., i_{k-1}, x, i_{k+1}, ..., i_N) >=
+///   F(i_1, ..., i_{k-1}, y, i_{k+1}, ..., i_N)
+///
+/// A function F with either monotonically increasing or decreasing with
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kasuga-fj wrote:

Fixed.

Excuse: I mimicked this phrase from [wikipedia](https://en.wikipedia.org/wiki/Monotonic_function). 

> A function with either property is called strictly monotone.

Probably I need to learn English harder...

https://github.com/llvm/llvm-project/pull/162280