c3a1669398
This PR implements the main loop of the new commutative ring procedure in `grind`. In the main loop, for each polynomial `p` in the todo queue, the procedure: - Simplifies it using the current basis. - Computes critical pairs with polynomials already in the basis and adds them to the queue. After the queue is empty, the disequalities are re-simplified using the new basis. `grind` can now solve examples such as: ```lean example [CommRing α] (x y : α) : x*y*x = 1 → x*y*y = y → y = 1 := by grind +ring example [CommRing α] (x y : α) : x^2*y = 1 → x*y^2 = y → y*x = 1 := by grind +ring example (x y : BitVec 16) : x^2*y = 1 → x*y^2 = y → y*x = 1 := by grind +ring ```
64 lines
2.4 KiB
Text
64 lines
2.4 KiB
Text
import Lean.Meta.Tactic.Grind.Arith.CommRing.Poly
|
|
open Lean.Grind.CommRing
|
|
|
|
def w : Expr := .var 0
|
|
def x : Expr := .var 1
|
|
def y : Expr := .var 2
|
|
def z : Expr := .var 3
|
|
|
|
instance : Add Expr where
|
|
add a b := .add a b
|
|
instance : Sub Expr where
|
|
sub a b := .sub a b
|
|
instance : Neg Expr where
|
|
neg a := .neg a
|
|
instance : Mul Expr where
|
|
mul a b := .mul a b
|
|
instance : HPow Expr Nat Expr where
|
|
hPow a k := .pow a k
|
|
instance : OfNat Expr n where
|
|
ofNat := .num n
|
|
|
|
def spol' (p₁ p₂ : Poly) : Poly :=
|
|
p₁.spol p₂ |>.spol
|
|
|
|
def check_spoly (e₁ e₂ r : Expr) : Bool :=
|
|
let p₁ := e₁.toPoly
|
|
let p₂ := e₂.toPoly
|
|
let r := r.toPoly
|
|
let s := p₁.spol p₂
|
|
spol' p₁ p₂ == r &&
|
|
spol' p₂ p₁ == r.mulConst (-1) &&
|
|
s.spol == r &&
|
|
r == (p₁.mulMon s.k₁ s.m₁).combine (p₂.mulMon s.k₂ s.m₂)
|
|
|
|
example : check_spoly (y^2 - x + 1) (x*y - 1 + y) (-x^2 + y + x - y^2) := by native_decide
|
|
example : check_spoly (y - z + 1) (x*y - 1) (-x*z + 1 + x) := by native_decide
|
|
example : check_spoly (z^3 - x*y) (z*y - 1) (z^2 - x*y^2) := by native_decide
|
|
example : check_spoly (x + 1) (z + 1) (z - x) := by native_decide
|
|
example : check_spoly (w^2*x - y) (w*x^2 - z) (-y*x + z*w) := by native_decide
|
|
example : check_spoly (2*z^3 - x*y) (3*z*y - 1) (2*z^2 - 3*x*y^2) := by native_decide
|
|
example : check_spoly (2*x + 3) (3*z + 1) (9*z - 2*x) := by native_decide
|
|
|
|
example : check_spoly (2*y^2 - x + 1) (2*x*y - 1 + y) (-x^2 + y + x - y^2) := by native_decide
|
|
example : check_spoly (2*y^2 - x + 1) (4*x*y - 1 + y) (-2*x^2 + y + 2*x - y^2) := by native_decide
|
|
example : check_spoly (6*y^2 - x + 1) (4*x*y - 1 + y) (-2*x^2 + 3*y + 2*x - 3*y^2) := by native_decide
|
|
|
|
def simp? (p₁ p₂ : Poly) : Option Poly :=
|
|
(·.p) <$> p₁.simp? p₂
|
|
|
|
partial def simp' (p₁ p₂ : Poly) : Poly :=
|
|
if let some r := p₁.simp? p₂ then
|
|
assert! r.p == (p₂.mulMon r.k₂ r.m₂).combine (p₁.mulConst r.k₁)
|
|
simp' r.p p₂
|
|
else
|
|
p₁
|
|
|
|
def check_simp' (e₁ e₂ r : Expr) : Bool :=
|
|
r.toPoly == simp' e₁.toPoly e₂.toPoly
|
|
|
|
example : check_simp' (x^2*y - 1) (x*y - y) (y - 1) := by native_decide
|
|
example : check_simp' (x^2 + x + 1) (2*x + 1) 3 := by native_decide
|
|
example : check_simp' (3*x^2 + x + y + 1) (2*x + 1) (4*y + 5) := by native_decide
|
|
example : check_simp' (3*x^2 + x + y + 1) (2*x + y) (3*y^2 + 2*y + 4) := by native_decide
|
|
example : check_simp' (z^4 + w^3 + x^2 + x + 1) (2*x + 1) (4*z^4 + 4*w^3 + 3) := by native_decide
|