568a1b1a81
This PR makes the following modifications to the new comm ring procedure in `grind` 1. Adds data-structures for representing equations (and their justifications), basis, and queue of equations to be processed. 2. Adds `RingM` helper monad. 3. Adds equation simplification main loop
64 lines
2.4 KiB
Text
64 lines
2.4 KiB
Text
import Lean.Meta.Tactic.Grind.Arith.CommRing.Poly
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open Lean.Grind.CommRing
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def w : Expr := .var 0
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def x : Expr := .var 1
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def y : Expr := .var 2
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def z : Expr := .var 3
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instance : Add Expr where
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add a b := .add a b
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instance : Sub Expr where
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sub a b := .sub a b
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instance : Neg Expr where
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neg a := .neg a
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instance : Mul Expr where
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mul a b := .mul a b
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instance : HPow Expr Nat Expr where
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hPow a k := .pow a k
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instance : OfNat Expr n where
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ofNat := .num n
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def spol' (p₁ p₂ : Poly) : Poly :=
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p₁.spol p₂ |>.spol
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def check_spoly (e₁ e₂ r : Expr) : Bool :=
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let p₁ := e₁.toPoly
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let p₂ := e₂.toPoly
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let r := r.toPoly
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let s := p₁.spol p₂
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spol' p₁ p₂ == r &&
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spol' p₂ p₁ == r.mulConst (-1) &&
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s.spol == r &&
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r == (p₁.mulMon s.c₁ s.m₁).combine (p₂.mulMon s.c₂ s.m₂)
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example : check_spoly (y^2 - x + 1) (x*y - 1 + y) (-x^2 + y + x - y^2) := by native_decide
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example : check_spoly (y - z + 1) (x*y - 1) (-x*z + 1 + x) := by native_decide
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example : check_spoly (z^3 - x*y) (z*y - 1) (z^2 - x*y^2) := by native_decide
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example : check_spoly (x + 1) (z + 1) (z - x) := by native_decide
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example : check_spoly (w^2*x - y) (w*x^2 - z) (-y*x + z*w) := by native_decide
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example : check_spoly (2*z^3 - x*y) (3*z*y - 1) (2*z^2 - 3*x*y^2) := by native_decide
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example : check_spoly (2*x + 3) (3*z + 1) (9*z - 2*x) := by native_decide
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example : check_spoly (2*y^2 - x + 1) (2*x*y - 1 + y) (-x^2 + y + x - y^2) := by native_decide
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example : check_spoly (2*y^2 - x + 1) (4*x*y - 1 + y) (-2*x^2 + y + 2*x - y^2) := by native_decide
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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
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def simp? (p₁ p₂ : Poly) : Option Poly :=
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(·.p) <$> p₁.simp? p₂
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partial def simp' (p₁ p₂ : Poly) : Poly :=
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if let some r := p₁.simp? p₂ then
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assert! r.p == (p₁.mulMon r.k₁ r.m).combine (p₂.mulConst r.k₂)
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simp' p₁ r.p
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else
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p₂
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def check_simp' (e₁ e₂ r : Expr) : Bool :=
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r.toPoly == simp' e₁.toPoly e₂.toPoly
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example : check_simp' (x*y - y) (x^2*y - 1) (y - 1) := by native_decide
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example : check_simp' (2*x + 1) (x^2 + x + 1) 3 := by native_decide
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example : check_simp' (2*x + 1) (3*x^2 + x + y + 1) (4*y + 5) := by native_decide
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example : check_simp' (2*x + y) (3*x^2 + x + y + 1) (3*y^2 + 2*y + 4) := by native_decide
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example : check_simp' (2*x + 1) (z^4 + w^3 + x^2 + x + 1) (4*z^4 + 4*w^3 + 3) := by native_decide
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