d64ae32965
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 ```
40 lines
1.5 KiB
Text
40 lines
1.5 KiB
Text
import Lean
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open Lean.Grind
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open Lean.Grind.CommRing
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-- Convenient RArray literals
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elab tk:"#R[" ts:term,* "]" : term => do
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let ts : Array Lean.Syntax := ts
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let es ← ts.mapM fun stx => Lean.Elab.Term.elabTerm stx none
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if h : 0 < es.size then
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Lean.RArray.toExpr (← Lean.Meta.inferType es[0]!) id (Lean.RArray.ofArray es h)
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else
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throwErrorAt tk "RArray cannot be empty"
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example (x y : Int) : (x + y) * (x + y + 1) = x * (1 + y + x) + (y + 1 + x) * y :=
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let ctx := #R[x, y]
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let lhs : Expr := .mul (.add (.var 0) (.var 1)) (.add (.add (.var 0) (.var 1)) (.num 1))
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let rhs : Expr := .add (.mul (.var 0) (.add (.add (.num 1) (.var 1)) (.var 0)))
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(.mul (.add (.add (.var 1) (.num 1)) (.var 0)) (.var 1))
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Expr.eq_of_toPoly_eq ctx lhs rhs (Eq.refl true)
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example (x y : UInt8) : (128 * x + y) * 2 = y + y :=
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let ctx := #R[x, y]
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let lhs : Expr := .mul (.add (.mul (.num 128) (.var 0)) (.var 1)) (.num 2)
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let rhs : Expr := .add (.var 1) (.var 1)
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Expr.eq_of_toPolyC_eq ctx lhs rhs (Eq.refl true)
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def q₁ : Poly := Expr.toPoly (.var 1)
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def lhs₁ : Expr := .var 0
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def rhs₁ : Expr := .var 1
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def q₂ : Poly := Expr.toPoly (.num 1)
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def lhs₂ : Expr := .add (.sub (.var 1) (.mul (.var 0) (.var 1))) (.pow (.var 1) 2)
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def rhs₂ : Expr := .num 1
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def lhs : Expr := .var 1
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def rhs : Expr := .num 1
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def nc : NullCert := .add q₁ lhs₁ rhs₁ (.add q₂ lhs₂ rhs₂ .empty)
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example (x y : Int) : x = y → y - x*y + y^2 = 1 → y = 1 :=
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let ctx := #R[x, y]
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nc.eq ctx (lhs := lhs) (rhs := rhs) (Eq.refl true)
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