08eb78a5b2
This PR sets up the new integrated test/bench suite. It then migrates all benchmarks and some related tests to the new suite. There's also some documentation and some linting. For now, a lot of the old tests are left alone so this PR doesn't become even larger than it already is. Eventually, all tests should be migrated to the new suite though so there isn't a confusing mix of two systems.
96 lines
2.4 KiB
Text
96 lines
2.4 KiB
Text
import Lean
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universe u
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structure Magma where
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α : Type u
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mul : α → α → α
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instance : CoeSort Magma (Type u) where
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coe s := s.α
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abbrev mul {M : Magma} (a b : M) : M :=
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M.mul a b
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set_option pp.all true
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infixl:70 (priority := high) "*" => mul
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structure Semigroup extends Magma where
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mul_assoc (a b c : α) : a * b * c = a * (b * c)
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instance : CoeSort Semigroup (Type u) where
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coe s := s.α
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structure CommSemigroup extends Semigroup where
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mul_comm (a b : α) : a * b = b * a
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structure Monoid extends Semigroup where
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one : α
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one_mul (a : α) : one * a = a
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mul_one (a : α) : a * one = a
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instance : CoeSort Monoid (Type u) where
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coe s := s.α
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structure CommMonoid extends Monoid where
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mul_comm (a b : α) : a * b = b * a
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instance : Coe CommMonoid CommSemigroup where
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coe s := {
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α := s.α
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mul := s.mul
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mul_assoc := s.mul_assoc
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mul_comm := s.mul_comm
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}
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structure Group extends Monoid where
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inv : α → α
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mul_left_inv (a : α) : (inv a) * a = one
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instance : CoeSort Group (Type u) where
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coe s := s.α
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abbrev inv {G : Group} (a : G) : G :=
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G.inv a
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postfix:max "⁻¹" => inv
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instance (G : Group) : OfNat (CoeSort.coe G.toMagma) (nat_lit 1) where
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ofNat := G.one
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instance (G : Group) : OfNat (G.toMagma.α) (nat_lit 1) where
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ofNat := G.one
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structure CommGroup extends Group where
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mul_comm (a b : α) : a * b = b * a
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instance : CoeSort CommGroup (Type u) where
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coe s := s.α
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theorem inv_mul_cancel_left {G : Group} (a b : G) : a⁻¹ * (a * b) = b := by
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rw [← G.mul_assoc, G.mul_left_inv, G.one_mul]
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theorem toMonoidOneEq {G : Group} : G.toMonoid.one = 1 :=
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rfl
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theorem inv_eq_of_mul_eq_one {G : Group} {a b : G} (h : a * b = 1) : a⁻¹ = b := by
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rw [← G.mul_one a⁻¹, toMonoidOneEq, ←h, ← G.mul_assoc, G.mul_left_inv, G.one_mul]
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theorem inv_inv {G : Group} (a : G) : (a⁻¹)⁻¹ = a :=
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inv_eq_of_mul_eq_one (G.mul_left_inv a)
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theorem mul_right_inv {G : Group} (a : G) : a * a⁻¹ = 1 := by
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have : a⁻¹⁻¹ * a⁻¹ = 1 := by rw [G.mul_left_inv]; rfl
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rw [inv_inv] at this
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assumption
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unif_hint (G : Group) where
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|- G.toMonoid.one =?= 1
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theorem mul_inv_rev {G : Group} (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by
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apply inv_eq_of_mul_eq_one
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rw [G.mul_assoc, ← G.mul_assoc b, mul_right_inv, G.one_mul, mul_right_inv]
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theorem mul_inv {G : CommGroup} (a b : G) : (a * b)⁻¹ = a⁻¹ * b⁻¹ := by
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rw [mul_inv_rev, G.mul_comm]
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