d2eb1bc9f5
This PR reviews the expected-to-fail-right-now tests for `grind`, moving some (now passing) tests to the main test suite, updating some tests, and adding some tests about normalisation of exponents.
48 lines
1.9 KiB
Text
48 lines
1.9 KiB
Text
-- Remark: we need some support for nonlinear in `cutsat`
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example {a b c d : Nat} (h : a = b + c * d) (w : 1 ≤ d) : a ≥ c := by
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-- grind -- fails, but would be lovely
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subst h
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apply Nat.le_add_left_of_le
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apply Nat.le_mul_of_pos_right
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assumption
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example {a b c d : Int} (h : a = b + c * d) (hb : 0 ≤ b) (hc : 0 ≤ c) (w : 1 ≤ d) : a ≥ c := by
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-- grind -- fails also
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subst h
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conv => rhs; rw [← Int.zero_add c]
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apply Int.add_le_add
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· assumption
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· have : 0 ≤ c * (d - 1) := Int.mul_nonneg (by omega) (by omega)
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rw [Int.mul_sub, Int.mul_one, Int.sub_nonneg] at this
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exact this
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-- Note: if we can automate the `Int` version here, we can also automate the `Nat` version just by embedding in `Int`.
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open Std Lean Grind
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example {α : Type} [IntModule α] [LE α] [IsPreorder α] [OrderedAdd α] {a b c : α} {d : Int}
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(wb : 0 ≤ b) (wc : 0 ≤ c)
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(h : a = b + d • c) (w : 1 ≤ d) : a ≥ c := by
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subst h
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conv => rhs; rw [← AddCommMonoid.zero_add c]
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apply OrderedAdd.add_le_add
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· exact wb
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· have := OrderedAdd.zsmul_le_zsmul_of_le_of_le_of_nonneg_of_nonneg w (le_refl c) (by decide) wc
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rwa [IntModule.one_zsmul] at this
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-- We can prove this directly in an ordered NatModule, from the axioms. (But shouldn't, see below.)
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example {α : Type} [NatModule α] [LE α] [IsPreorder α] [OrderedAdd α] {a b c : α} {d : Nat}
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(wb : 0 ≤ b) (wc : 0 ≤ c)
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(h : a = b + d • c) (w : 1 ≤ d) : a ≥ c := by
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subst h
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conv => rhs; rw [← AddCommMonoid.zero_add c]
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apply OrderedAdd.add_le_add
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· exact wb
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· have := OrderedAdd.nsmul_le_nsmul_of_le_of_le_of_nonneg w (le_refl c) wc
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rwa [NatModule.one_nsmul] at this
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-- The correct proof is to embed a NatModule in its IntModule envelope.
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-- From Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization
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theorem HomogeneousLocalization.Away.isLocalization_mul.extracted_1 (d e : Nat) : (d + e) • (d + e) = e • e + (2 * e + d) • d := by grind
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