196d899c02
This PR adds a monolithic `CommRing` class, for internal use by `grind`, and includes instances for `Int`/`BitVec`/`IntX`/`UIntX`.
44 lines
1.6 KiB
Text
44 lines
1.6 KiB
Text
class OfNatSound (α : Type u) [Add α] [(n : Nat) → OfNat α n] : Prop where
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ofNat_add (n m : Nat) : (OfNat.ofNat n : α) + OfNat.ofNat m = OfNat.ofNat (n+m)
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export OfNatSound (ofNat_add)
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theorem ex1 {α : Type u} [Add α] [(n : Nat) → OfNat α n] [OfNatSound α] : (10000000 : α) + 10000000 = 20000000 :=
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ofNat_add ..
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-- Some example structure
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class S (α : Type u) extends Add α, Mul α, Zero α, One α where
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add_assoc (a b c : α) : a + b + c = a + (b + c)
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add_zero (a : α) : a + 0 = a
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zero_add (a : α) : 0 + a = a
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mul_zero (a : α) : a * 0 = 0
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mul_one (a : α) : a * 1 = a
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left_distrib (a b c : α) : a * (b + c) = a * b + a * c
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-- Very simply default `ofNat` for `S`
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protected def S.ofNat (α : Type u) [S α] : Nat → α
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| 0 => 0
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| n+1 => S.ofNat α n + 1
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instance [S α] : OfNat α n where
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ofNat := S.ofNat α n
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instance [S α] : OfNatSound α where
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ofNat_add n m := by
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induction m with
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| zero => simp [S.ofNat]; erw [S.add_zero]; done
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| succ m ih => simp [OfNat.ofNat, S.ofNat] at *; erw [← ih]; rw [S.add_assoc]
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theorem S.ofNat_mul [S α] (n m : Nat) : (OfNat.ofNat n : α) * OfNat.ofNat m = OfNat.ofNat (n * m) := by
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induction m with
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| zero => rw [S.mul_zero, Nat.mul_zero]
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| succ m ih =>
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show OfNat.ofNat (α := α) n * OfNat.ofNat (m + 1) = OfNat.ofNat (n * m.succ)
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rw [Nat.mul_succ, ← ofNat_add, ← ofNat_add, ← ih, left_distrib]
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simp [OfNat.ofNat, S.ofNat]
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erw [S.zero_add, S.mul_one]
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theorem ex2 [S α] : (100000000000000000 : α) * 20000000000000000 = 2000000000000000000000000000000000 :=
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S.ofNat_mul ..
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#print ex2
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