f40d72ea47
This PR adds draft typeclasses for `grind` to process facts about ordered modules. These interfaces will evolve as the implementation develops.
72 lines
3 KiB
Text
72 lines
3 KiB
Text
/-
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Copyright (c) 2025 Lean FRO, LLC. or its affiliates. All Rights Reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Kim Morrison
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-/
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module
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prelude
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import Init.Data.Int.Order
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namespace Lean.Grind
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class NatModule (M : Type u) extends Zero M, Add M, HSMul Nat M M where
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add_zero : ∀ a : M, a + 0 = a
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zero_add : ∀ a : M, 0 + a = a
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add_comm : ∀ a b : M, a + b = b + a
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add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
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zero_smul : ∀ a : M, 0 • a = 0
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one_smul : ∀ a : M, 1 • a = a
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add_smul : ∀ n m : Nat, ∀ a : M, (n + m) • a = n • a + m • a
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smul_zero : ∀ n : Nat, n • (0 : M) = 0
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smul_add : ∀ n : Nat, ∀ a b : M, n • (a + b) = n • a + n • b
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mul_smul : ∀ n m : Nat, ∀ a : M, (n * m) • a = n • (m • a)
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class IntModule (M : Type u) extends Zero M, Add M, Neg M, Sub M, HSMul Int M M where
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add_zero : ∀ a : M, a + 0 = a
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zero_add : ∀ a : M, 0 + a = a
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add_comm : ∀ a b : M, a + b = b + a
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add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
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zero_smul : ∀ a : M, (0 : Int) • a = 0
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one_smul : ∀ a : M, (1 : Int) • a = a
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add_smul : ∀ n m : Int, ∀ a : M, (n + m) • a = n • a + m • a
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neg_smul : ∀ n : Int, ∀ a : M, (-n) • a = - (n • a)
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smul_zero : ∀ n : Int, n • (0 : M) = 0
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smul_add : ∀ n : Int, ∀ a b : M, n • (a + b) = n • a + n • b
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mul_smul : ∀ n m : Int, ∀ a : M, (n * m) • a = n • (m • a)
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neg_add_cancel : ∀ a : M, -a + a = 0
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sub_eq_add_neg : ∀ a b : M, a - b = a + -b
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instance IntModule.toNatModule (M : Type u) [i : IntModule M] : NatModule M :=
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{ i with
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hSMul a x := (a : Int) • x
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smul_zero := by simp [IntModule.smul_zero]
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add_smul := by simp [IntModule.add_smul]
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smul_add := by simp [IntModule.smul_add]
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mul_smul := by simp [IntModule.mul_smul] }
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/--
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We keep track of rational linear combinations as integer linear combinations,
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but with the assurance that we can cancel the GCD of the coefficients.
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-/
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class RatModule (M : Type u) extends IntModule M where
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no_int_zero_divisors : ∀ (k : Int) (a : M), k ≠ 0 → k • a = 0 → a = 0
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/-- A preorder is a reflexive, transitive relation `≤` with `a < b` defined in the obvious way. -/
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class Preorder (α : Type u) extends LE α, LT α where
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le_refl : ∀ a : α, a ≤ a
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le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c
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lt := fun a b => a ≤ b ∧ ¬b ≤ a
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lt_iff_le_not_le : ∀ a b : α, a < b ↔ a ≤ b ∧ ¬b ≤ a := by intros; rfl
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class IntModule.IsOrdered (M : Type u) [Preorder M] [IntModule M] where
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neg_le_iff : ∀ a b : M, -a ≤ b ↔ -b ≤ a
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neg_lt_iff : ∀ a b : M, -a < b ↔ -b < a
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add_lt_left : ∀ a b c : M, a < b → a + c < b + c
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add_lt_right : ∀ a b c : M, a < b → c + a < c + b
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smul_pos : ∀ (k : Int) (a : M), 0 < a → (0 < k ↔ 0 < k • a)
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smul_neg : ∀ (k : Int) (a : M), a < 0 → (0 < k ↔ k • a < 0)
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smul_nonneg : ∀ (k : Int) (a : M), 0 ≤ a → 0 ≤ k → 0 ≤ k • a
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smul_nonpos : ∀ (k : Int) (a : M), a ≤ 0 → 0 ≤ k → k • a ≤ 0
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end Lean.Grind
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