196d899c02
This PR adds a monolithic `CommRing` class, for internal use by `grind`, and includes instances for `Int`/`BitVec`/`IntX`/`UIntX`.
62 lines
1.6 KiB
Text
62 lines
1.6 KiB
Text
set_option autoImplicit true
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section Mathlib.Algebra.Group.Defs
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class MulOneClass (M : Type) extends One M, Mul M where
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one_mul : ∀ a : M, 1 * a = a
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export MulOneClass (one_mul)
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end Mathlib.Algebra.Group.Defs
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section Mathlib.Algebra.Ring.Defs
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class Distrib (R : Type) extends Mul R, Add R where
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right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c
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class RightDistribClass (R : Type) [Mul R] [Add R] : Prop where
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right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c
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instance Distrib.rightDistribClass (R : Type) [Distrib R] : RightDistribClass R :=
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⟨Distrib.right_distrib⟩
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theorem add_mul [Mul R] [Add R] [RightDistribClass R] (a b c : R) :
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(a + b) * c = a * c + b * c :=
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RightDistribClass.right_distrib a b c
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theorem add_one_mul [Add α] [MulOneClass α] [RightDistribClass α] (a b : α) :
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(a + 1) * b = a * b + b := by
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rw [add_mul, one_mul]
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class Semiring (R : Type) extends Distrib R, MulOneClass R
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end Mathlib.Algebra.Ring.Defs
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section Mathlib.Data.Nat.Basic
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instance : Semiring Nat where
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add := Nat.add
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mul := Nat.mul
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one := Nat.succ Nat.zero
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one_mul := sorry
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right_distrib := sorry
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end Mathlib.Data.Nat.Basic
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#synth MulOneClass Nat -- works
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#synth RightDistribClass Nat -- works
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theorem ex1 [Add α] [MulOneClass α] [RightDistribClass α] (a b : α) :
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(a + 1) * b = a * b + b := by
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sorry
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#check (ex1) -- should work
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#check (add_one_mul) -- should work
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#check @add_one_mul
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example {a b : Nat} : (a + 1) * b = a * b + b := by
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have := add_one_mul a b -- works
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rw [add_one_mul] -- should work
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example {a b : Nat} : (a + 1) * b = a * b + b := by
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rw [add_one_mul] -- should work
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