eccc472e8d
This PR removes the now unnecessary `set_option grind.warning false` statements, now that the warning is disabled by default.
34 lines
1.4 KiB
Text
34 lines
1.4 KiB
Text
axiom R : Type
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instance : Lean.Grind.CommRing R := sorry
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axiom cos : R → R
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axiom sin : R → R
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axiom trig_identity : ∀ x, (cos x)^2 + (sin x)^2 = 1
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grind_pattern trig_identity => cos x
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grind_pattern trig_identity => sin x
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-- Whenever `grind` sees `cos` or `sin`, it adds `(cos x)^2 + (sin x)^2 = 1` to the blackboard.
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-- That's a polynomial, so it is sent to the Grobner basis module.
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-- And we can prove equalities modulo that relation!
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example : (cos x + sin x)^2 = 2 * cos x * sin x + 1 := by
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grind
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-- `grind` notices that the two arguments of `f` are equal,
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-- and hence the function applications are too.
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example (f : R → Nat) : f ((cos x + sin x)^2) = f (2 * cos x * sin x + 1) := by
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grind
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-- After that, we can use basic modularity conditions:
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-- this reduces to `4 * x ≠ 2 + x` for some `x : ℕ`
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example (f : R → Nat) : 4 * f ((cos x + sin x)^2) ≠ 2 + f (2 * cos x * sin x + 1) := by
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grind
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-- A bit of case splitting is also fine.
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-- If `max = 3`, then `f _ = 0`, and we're done.
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-- Otherwise, the previous argument applies.
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example (f : R → Nat) : max 3 (4 * f ((cos x + sin x)^2)) ≠ 2 + f (2 * cos x * sin x + 1) := by
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grind
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-- See https://github.com/leanprover-community/mathlib4/blob/nightly-testing/MathlibTest/grind/trig.lean
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-- for the Mathlib version of this test, using the real `ℝ`, `cos`, `sin`, and `cos_sq_and_sin_sq` declarations.
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