a2ceebe200
This PR adds propagation rules corresponding to the `Semiring` normalization rules introduced in #11628. The new rules apply only to non-commutative semirings, since support for them in `grind` is limited. The normalization rules introduced unexpected behavior in Mathlib because they neutralize parameters such as `one_mul`: any theorem instance associated with such a parameter is reduced to `True` by the normalizer.
77 lines
2 KiB
Text
77 lines
2 KiB
Text
section Mathlib.Data.Nat.Init
|
||
|
||
namespace Nat
|
||
|
||
class AtLeastTwo (n : Nat) : Prop where
|
||
prop : 2 ≤ n
|
||
|
||
instance (n : Nat) [NeZero n] : (n + 1).AtLeastTwo :=
|
||
⟨add_le_add (one_le_iff_ne_zero.mpr (NeZero.ne n)) (Nat.le_refl 1)⟩
|
||
|
||
end Nat
|
||
|
||
end Mathlib.Data.Nat.Init
|
||
|
||
section Mathlib.Data.Nat.Cast.Defs
|
||
|
||
instance {R : Type} {n : Nat} [NatCast R] [Nat.AtLeastTwo n] :
|
||
OfNat R n where
|
||
ofNat := n.cast
|
||
|
||
end Mathlib.Data.Nat.Cast.Defs
|
||
section Mathlib.Algebra.GroupWithZero.Defs
|
||
|
||
class MulZeroClass (α : Type) extends Mul α, Zero α where
|
||
mul_zero : ∀ a : α, a * 0 = 0
|
||
|
||
end Mathlib.Algebra.GroupWithZero.Defs
|
||
|
||
section Mathlib.Algebra.Ring.Defs
|
||
|
||
class Semiring (α : Type) extends
|
||
One α, NatCast α, Add α, Mul α, MulZeroClass α
|
||
|
||
end Mathlib.Algebra.Ring.Defs
|
||
|
||
section Mathlib.Algebra.Ring.GrindInstances
|
||
|
||
instance Semiring.toGrindSemiring (α : Type) [s : Semiring α] :
|
||
Lean.Grind.Semiring α :=
|
||
{ s with
|
||
nsmul := sorry
|
||
npow := sorry
|
||
ofNat | 0 | 1 | n + 2 => inferInstance
|
||
natCast := sorry
|
||
add_zero := sorry
|
||
mul_one := sorry
|
||
zero_mul := sorry
|
||
pow_zero := sorry
|
||
pow_succ := sorry
|
||
ofNat_eq_natCast := sorry
|
||
ofNat_succ := sorry
|
||
nsmul_eq_natCast_mul := sorry
|
||
add_comm := sorry
|
||
left_distrib := sorry
|
||
right_distrib := sorry
|
||
mul_zero := sorry
|
||
add_assoc := sorry
|
||
mul_assoc := sorry
|
||
one_mul := sorry }
|
||
|
||
end Mathlib.Algebra.Ring.GrindInstances
|
||
|
||
section Mathlib.Algebra.Polynomial.Coeff
|
||
|
||
theorem coeff_mul_X_pow {R : Type} [Semiring R] (p : R) (n d : Nat) :
|
||
∀ b, b.1 + b.2 = d + n → b ≠ (d, n) → p * (if n = b.2 then 1 else 0) = 0 := by
|
||
grind only [MulZeroClass.mul_zero]
|
||
|
||
theorem coeff_mul_X_pow' {R : Type} [Semiring R] (p : R) (n d : Nat) :
|
||
∀ b, b.1 + b.2 = d + n → b ≠ (d, n) → p * (if n = b.2 then 1 else 0) = 0 := by
|
||
grind only
|
||
|
||
example [Semiring α] (a b c : α) : b = 0 → a * b * c = 0 := by
|
||
grind only
|
||
|
||
example [Semiring α] (a b c : α) : c = 1 → a = 1 → a * b * c = b := by
|
||
grind only
|