This PR adds a normalizer for non-commutative semirings to `grind`.
Examples:
```lean
open Lean.Grind
variable (R : Type u) [Semiring R]
example (a b c : R) : a * (b + c) = a * c + a * b := by grind
example (a b : R) : (a + 2 * b)^2 = a^2 + 2 * a * b + 2 * b * a + 4 * b^2 := by grind
example (a b : R) : b^2 + (a + 2 * b)^2 = a^2 + 2 * a * b + b * (1+1) * a * 1 + 5 * b^2 := by grind
example (a b : R) : a^3 + a^2*b + a*b*a + b*a^2 + a*b^2 + b*a*b + b^2*a + b^3 = (a+b)^3 := by grind
```
This PR implements `NatModule` normalization when the `AddRightCancel`
instance is not available. Note that in this case, the embedding into
`IntModule` is not injective. Therefore, we use a custom normalizer,
similar to the `CommSemiring` normalizer used in the `grind ring`
module. Example:
```lean
open Lean Grind
example [NatModule α] (a b c : α)
: 2•a + 2•(b + 2•c) + 3•a = 4•a + c + 2•b + 3•c + a := by
grind
```
This PR adds support for `NatModule` equalities and inequalities in
`grind linarith`. Examples:
```lean
open Lean Grind Std
example [NatModule α] [LE α] [LT α]
[LawfulOrderLT α] [IsLinearOrder α] [OrderedAdd α]
(x y : α) : x ≤ y → 2 • x + y ≤ 3 • y := by
grind
example [NatModule α] [AddRightCancel α] [LE α] [LT α]
[LawfulOrderLT α] [IsLinearOrder α] [OrderedAdd α]
(a b c d : α) : a ≤ b → a ≥ c + d → d ≤ 0 → d ≥ 0 → b = c → a = b := by
grind
```
This PR adds some test cases for `grind` working with `Fin`. There are
many still failing tests in `tests/lean/grind/grind_fin.lean` which I'm
intending to triage and work on.
This PR reviews the expected-to-fail-right-now tests for `grind`, moving
some (now passing) tests to the main test suite, updating some tests,
and adding some tests about normalisation of exponents.
This PR eliminates uses of `intros x y z` (with arguments) and updates
the `intros` docstring to suggest that `intro x y z` should be used
instead. The `intros` tactic is historical, and can be traced all the
way back to Lean 2, when `intro` could only introduce a single
hypothesis. Since 2020, the `intro` tactic has superceded it. The
`intros` tactic (without arguments) is currently still useful.
This PR adds a version of `CommRing.Expr.toPoly` optimized for kernel
reduction. We use this function not only to implement `grind ring`, but
also to interface the ring module with `grind cutsat`.
This PR adds a `HPow \a Int \a` field to `Lean.Grind.Field`, and
sufficient axioms to connect it to the operations, so that in future we
can reason about exponents in `grind`. To avoid collisions, we also move
the `HPow \a Nat \a` field in `Semiring` from the extends clause to a
field. Finally, we add some failing tests about normalizing exponents.
This PR resolves a defeq diamond, which caused a problem in Mathlib:
```
import Mathlib
example (R : Type) [I : Ring R] :
@AddCommGroup.toGrindIntModule R (@Ring.toAddCommGroup R I) =
@Lean.Grind.Ring.instIntModule R (@Ring.toGrindRing R I) := rfl -- fails
```
This PR uses the commutative ring module to normalize nonlinear
polynomials in `grind cutsat`. Examples:
```lean
example (a b : Nat) (h₁ : a + 1 ≠ a * b * a) (h₂ : a * a * b ≤ a + 1) : b * a^2 < a + 1 := by
grind
example (a b c : Int) (h₁ : a + 1 + c = b * a) (h₂ : c + 2*b*a = 0) : 6 * a * b - 2 * a ≤ 2 := by
grind
```
This PR implements support for equations `<num> = 0` in rings and fields
of unknown characteristic. Examples:
```lean
example [Field α] (a : α) : (2 * a)⁻¹ = a⁻¹ / 2 := by grind
example [Field α] (a : α) : (2 : α) ≠ 0 → 1 / a + 1 / (2 * a) = 3 / (2 * a) := by grind
example [CommRing α] (a b : α) (h₁ : a + 2 = a) (h₂ : 2*b + a = 0) : a = 0 := by
grind
example [CommRing α] (a b : α) (h₁ : a + 6 = a) (h₂ : b + 9 = b) (h₂ : 3*b + a = 0) : a = 0 := by
grind
example [CommRing α] (a b : α) (h₁ : a + 6 = a) (h₂ : b + 9 = b) (h₂ : 3*b + a = 0) : a = 0 := by
grind
example [CommRing α] (a b : α) (h₁ : a + 2 = a) (h₂ : b = 0) : 4*a + b = 0 := by
grind
example [CommRing α] (a b c : α) (h₁ : a + 6 = a) (h₂ : c = c + 9) (h : b + 3*c = 0) : 27*a + b = 0 := by
grind
```
This PR embeds a NatModule into its IntModule completion, which is
injective when we have AddLeftCancel, and monotone when the modules are
ordered. Also adds some (failing) grind test cases that can be verified
once `grind` uses this embedding.
This PR implements support for normalization for commutative semirings
that do not implement `AddRightCancel`. Examples:
```lean
variable (R : Type u) [CommSemiring R]
example (a b c : R) : a * (b + c) = a * c + b * a := by grind
example (a b : R) : (a + b)^2 = a^2 + 2 * a * b + b^2 := by grind
example (a b : R) : (a + 2 * b)^2 = a^2 + 4 * a * b + 4 * b^2 := by grind
example (a b : R) : (a + 2 * b)^2 = 4 * b^2 + b * 4 * a + a^2 := by grind
```
This PR implements support for (commutative) semirings in `grind`. It
uses the Grothendieck completion to construct a (commutative) ring
`Lean.Grind.Ring.OfSemiring.Q α` from a (commutative) semiring `α`. This
construction is mostly useful for semirings that implement
`AddRightCancel α`. Otherwise, the function `toQ` is not injective.
Examples:
```lean
example (x y : Nat) : x^2*y = 1 → x*y^2 = y → y*x = 1 := by
grind
example [CommSemiring α] [AddRightCancel α] (x y : α) : x^2*y = 1 → x*y^2 = y → y*x = 1 := by
grind
example (a b : Nat) : 3 * a * b = a * b * 3 := by grind
example (k z : Nat) : k * (z * 2 * (z * 2 + 1)) = z * (k * (2 * (z * 2 + 1))) := by grind
example [CommSemiring α] [AddRightCancel α] [IsCharP α 0] (x y : α)
: x^2*y = 1 → x*y^2 = y → x + y = 1 → False := by
grind
```
This PR implements equality elimination in `grind linarith`. The current
implementation supports only `IntModule` and `IntModule` +
`NoNatZeroDivisors`
This PR implements the Rabinowitsch transformation for `Field`
disequalities in `grind`. For example, this transformation is necessary
for solving:
```lean
example [Field α] (a : α) : a^2 = 0 → a = 0 := by
grind
```
