This PR removes the `NatCast (Fin n)` global instance (both the direct
instance, and the indirect one via `Lean.Grind.Semiring`), as that
instance causes causes `x < n` (for `x : Fin k`, `n : Nat`) to be
elaborated as `x < ↑n` rather than `↑x < n`, which is undesirable. Note
however that in Mathlib this happens anyway!
This PR adds `Lean.Grind.Ring.IsOrdered`, and cleans up the ring/module
grind API. These typeclasses are at present unused, but will support
future algorithmic improvements in `grind`.
This PR splits `Lean.Grind.CommRing` into 4 typeclasses, for semirings
and noncommutative rings. This does not yet change the behaviour of
`grind`, which expects to find all 4 typeclasses. Later we will make
some generalizations.
This PR changes `Lean.Grind.CommRing` to inline the `NatCast` instance
(i.e. to be provided by the user) rather than constructing one from the
existing data. Without this change we can't construct instances in
Mathlib that `grind` can use.
This PR implements the generation of compact proof terms for
Nullstellensatz certificates in the new commutative ring procedure in
`grind`. Some examples:
```lean
example [CommRing α] (x y : α) : x = 1 → y = 2 → 2*x + y = 4 := by
grind +ring
example [CommRing α] [IsCharP α 7] (x y : α) : 3*x = 1 → 3*y = 2 → x + y = 1 := by
grind +ring
example [CommRing α] [NoZeroNatDivisors α] (x y : α) : 3*x = 1 → 3*y = 2 → x + y = 1 := by
grind +ring
example (x y z : BitVec 8) : z = y → (x + 1)*(x - 1)*y + y = z*x^2 + 1 → False := by
grind +ring
```
This PR adds the helper type class `NoZeroNatDivisors` for the
commutative ring procedure in `grind`. Core only implements it for
`Int`. It can be instantiated in Mathlib for any type `A` that
implements `NoZeroSMulDivisors Nat A`.
See `findSimp?` and `PolyDerivation` for details on how this instance
impacts the commutative ring procedure.
This PR makes `IntCast` a field of `Lean.Grind.CommRing`, along with
additional axioms relating it to negation of `OfNat`. This allows use to
use existing instances which are not definitionally equal to the
previously given construction.
---------
Co-authored-by: Leonardo de Moura <leomoura@amazon.com>
This PR adds support to `grind` for detecting unsatisfiable commutative
ring equations when the ring characteristic is known. Examples:
```lean
example (x : Int) : (x + 1)*(x - 1) = x^2 → False := by
grind +ring
example (x y : Int) : (x + 1)*(x - 1)*y + y = y*x^2 + 1 → False := by
grind +ring
example (x : UInt8) : (x + 1)*(x - 1) = x^2 → False := by
grind +ring
example (x y : BitVec 8) : (x + 1)*(x - 1)*y + y = y*x^2 + 1 → False := by
grind +ring
```
This PR implements basic support for `CommRing` in `grind`. Terms are
already being reified and normalized. We still need to process the
equations, but `grind` can already prove simple examples such as:
```lean
open Lean.Grind in
example [CommRing α] (x : α) : (x + 1)*(x - 1) = x^2 - 1 := by
grind +ring
open Lean.Grind in
example [CommRing α] [IsCharP α 256] (x : α) : (x + 16)*(x - 16) = x^2 := by
grind +ring
example (x : Int) : (x + 1)*(x - 1) = x^2 - 1 := by
grind +ring
example (x : UInt8) : (x + 16)*(x - 16) = x^2 := by
grind +ring
example (x : Int) : (x + 1)^2 - 1 = x^2 + 2*x := by
grind +ring
example (x : BitVec 8) : (x + 16)*(x - 16) = x^2 := by
grind +ring
example (x : BitVec 8) : (x + 1)^2 - 1 = x^2 + 2*x := by
grind +ring
```
This PR fixes several issues in the `CommRing` multivariate polynomial
library:
1. Replaces the previous array type with the universe polymorphic
`RArray`.
2. Properly eliminates cancelled monomials.
3. Sorts monomials in decreasing order.
4. Marks the parameter `p` of the `IsCharP` class as an output
parameter.
5. Adds `LawfulBEq` instances for the types `Power`, `Mon`, and `Poly`.
This PR add a function for converting `CommRing` expressions into
multivariate polynomials.
Co-authored-by: Leonardo de Moura <leonardodemoura@Leonardos-MacBook-Pro.local>
This PR upstreams many of the results from `Mathlib/Data/Int/Init.lean`.
Notably, we upstream the `simp` tag on `Int.natCast_pow`. While this is
desirable as a `simp` lemma, it is non-confluent with other good `simp`
lemmas like `Int.emod_bmod_congr`, and this will need to be addressed in
the future.
This PR adds a mixin typeclass for `Lean.Grind.CommRing` recording the
characteristic of the ring, and constructs instances for `Int`, `IntX`,
`UIntX`, and `BitVec`.
