[MLIR][Presburger] Use Identifiers outside Presburger library (#77316)

The pull request includes the following changes.
1. Refactors the interface to `PresburgerSpace::identifiers` to `setId` and a

[MLIR][Presburger] Use Identifiers outside Presburger library (#77316)

The pull request includes the following changes.
1. Refactors the interface to `PresburgerSpace::identifiers` to `setId` and a
const `getId`, instead of previous `getId` which returned a mutable
reference. `resetIds` does not need to be called to use identifiers, `setId`
calls `resetIds` if identifiers are not enabled.
2. Deprecates `FlatAffineRelation` by refactoring all usages of
`FlatAffineRelation` to `IntegerRelation`. To achieve this,
`FlatAffineRelation::compose` is refactored into
`IntegerRelation::mergeAndCompose`.
3. Deletes unneeded overrides of virtual functions `hasConsistentState`,
`clearAndCopyFrom` and `fourierMotzkinEliminate` from
`FlatLinearValueConstraints` as these were only used through
`FlatAffineRelation` and we now use `IntegerRelation`'s member functions
instead.
4. Fixes an existing bug in FlatLinearValueConstraints' constructor
which caused
identifiers set by superclass FlatLinearConstraints' constructor to be
erased.
5. Fixes `IntegerRelation::convertVarKind` not preserving identifiers.

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[MLIR][Presburger] Implement IntegerRelation::setId (#77872)

[MLIR][Presburger] Implement IntegerRelation::mergeAndAlignSymbols (#76736)

[MLIR][Presburger] Fix IntegerRelation::swapVar not swapping identifiers (#74407)

This commit fixes a bug where identifiers were not swapped when doing a
IntegerRelation::swapVar.

[MLIR][Presburger] Implement convertVarKind for PresburgerRelation

[MLIR][Presburger] Implement findSymbolicIntegerLexMax for IntegerRelation

This patch implements findSymbolicIntegerLexMax for IntegerRelation

Reviewed By: Groverkss

Differential Revision: https:/

[MLIR][Presburger] Implement findSymbolicIntegerLexMax for IntegerRelation

This patch implements findSymbolicIntegerLexMax for IntegerRelation

Reviewed By: Groverkss

Differential Revision: https://reviews.llvm.org/D156023

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[MLIR][Presburger] Improve unittest parsing

This patch adds better functions for parsing MultiAffineFunctions and
PWMAFunctions in Presburger unittests.

A PWMAFunction can now be parsed as:

```
PW

[MLIR][Presburger] Improve unittest parsing

This patch adds better functions for parsing MultiAffineFunctions and
PWMAFunctions in Presburger unittests.

A PWMAFunction can now be parsed as:

```
PWMAFunction result = parsePWMAF({
{"(x, y) : (x >= 10, x <= 20, y >= 1)", "(x, y) -> (x + y)"},
{"(x, y) : (x >= 21)", "(x, y) -> (x + y)"},
{"(x, y) : (x <= 9)", "(x, y) -> (x - y)"},
{"(x, y) : (x >= 10, x <= 20, y <= 0)", "(x, y) -> (x - y)"},
});
```

which is much more readable than the old format since the output can be
described as an AffineMap, instead of coefficients.

This patch also adds support for parsing divisions in MultiAffineFunctions
and PWMAFunctions which was previously not possible.

Reviewed By: arjunp

Differential Revision: https://reviews.llvm.org/D133654

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Revert "[MLIR][Presburger] Improve unittest parsing"

This reverts commit 84d07d021333f7b5716f0444d5c09105557272e0.

Reverted to fix a compilation issue on gcc8.

[MLIR][Presburger] Improve unittest parsing

This patch adds better functions for parsing MultiAffineFunctions and
PWMAFunctions in Presburger unittests.

A PWMAFunction can now be parsed as:

```
PW

[MLIR][Presburger] Improve unittest parsing

This patch adds better functions for parsing MultiAffineFunctions and
PWMAFunctions in Presburger unittests.

A PWMAFunction can now be parsed as:

```
PWMAFunction result = parsePWMAF({
{"(x, y) : (x >= 10, x <= 20, y >= 1)", "(x, y) -> (x + y)"},
{"(x, y) : (x >= 21)", "(x, y) -> (x + y)"},
{"(x, y) : (x <= 9)", "(x, y) -> (x - y)"},
{"(x, y) : (x >= 10, x <= 20, y <= 0)", "(x, y) -> (x - y)"},
});
```

which is much more readable than the old format since the output can be
described as an AffineMap, instead of coefficients.

This patch also adds support for parsing divisions in MultiAffineFunctions
and PWMAFunctions which was previously not possible.

Reviewed By: arjunp

Differential Revision: https://reviews.llvm.org/D133654

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[MLIR][Presburger] Refactor MultiAffineFunction to be defined over universe

This patch refactors MAF to be defined over the universe in a given space
instead of being defined over a restricted domai

[MLIR][Presburger] Refactor MultiAffineFunction to be defined over universe

This patch refactors MAF to be defined over the universe in a given space
instead of being defined over a restricted domain.

The reasoning for this refactor is to store division representation for local
variables explicitly for the function outputs. This change is required for
unionLexMax/Min to support local variables which will be upstreamed after this
patch. Another reason for this refactor is to have a flattened form of
AffineMap as MultiAffineFunction.

Reviewed By: arjunp

Differential Revision: https://reviews.llvm.org/D131864

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[MLIR][Presburger] support symbolicLexMin for IntegerRelation

This also changes the space of the returned lexmin for IntegerPolyhedrons;
the symbols in the poly now correspond to symbols in the resu

[MLIR][Presburger] support symbolicLexMin for IntegerRelation

This also changes the space of the returned lexmin for IntegerPolyhedrons;
the symbols in the poly now correspond to symbols in the result rather than dims.

Reviewed By: Groverkss

Differential Revision: https://reviews.llvm.org/D128933

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[MLIR][Presburger] Rename variable/identifier -> variable

Currently, in the Presburger library, we use the words "variables" and
"identifiers" interchangeably. This patch changes this to only use "v

[MLIR][Presburger] Rename variable/identifier -> variable

Currently, in the Presburger library, we use the words "variables" and
"identifiers" interchangeably. This patch changes this to only use "variables" to
refer to the variables of PresburgerSpace.

The reasoning behind this change is that the current usage of the word "identifier"
is misleading. variables do not "identify" anything. The information attached to them is the
actual "identifier" for the variable. The word "identifier", will later be used
to refer to the information attached to each variable in space.

Reviewed By: ftynse

Differential Revision: https://reviews.llvm.org/D128585

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[MLIR][Presburger] Add applyDomain/Range to IntegerRelation

This patch adds support for applying a relation on domain/range of a relation.

Reviewed By: arjunp, ftynse

Differential Revision: https:

[MLIR][Presburger] Add applyDomain/Range to IntegerRelation

This patch adds support for applying a relation on domain/range of a relation.

Reviewed By: arjunp, ftynse

Differential Revision: https://reviews.llvm.org/D126339

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[MLIR][Presburger] Add intersectDomain/Range to IntegerRelation

This patch adds support for intersection a set with a relation.

Reviewed By: arjunp

Differential Revision: https://reviews.llvm.org/

[MLIR][Presburger] Add intersectDomain/Range to IntegerRelation

This patch adds support for intersection a set with a relation.

Reviewed By: arjunp

Differential Revision: https://reviews.llvm.org/D126328

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[MLIR][Presburger] Add inverse to IntegerRelation

This patch adds support for obtaining inverse of a relation.

Reviewed By: ftynse

Differential Revision: https://reviews.llvm.org/D126327

[MLIR][Presburger] Add getDomainSet, getRangeSet to IntegerRelation

This patch adds support for obtaining a set corresponding to the domain/range
of the relation.

Reviewed By: ftynse

Differential

[MLIR][Presburger] Add getDomainSet, getRangeSet to IntegerRelation

This patch adds support for obtaining a set corresponding to the domain/range
of the relation.

Reviewed By: ftynse

Differential Revision: https://reviews.llvm.org/D126326

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