93e0ebf25c
This PR is initially motivated by noticing `Lean.Grind.Preorder.toLE` appearing in long Mathlib typeclass searches; this change will prevent these searches. These changes are also helpful preparation for potentially dropping the custom `Lean.Grind.*` typeclasses, and unifying with the new typeclasses introduced in #9729.
93 lines
3.2 KiB
Text
93 lines
3.2 KiB
Text
open Lean Grind
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set_option grind.debug true
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example [Field α] [IsCharP α 0] (a b c : α) : a/3 = b → c = a/3 → a/2 + a/2 = b + 2*c := by
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grind
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example [Field α] (a b : α) : b = 0 → (a + a) / 0 = b := by
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grind
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example [Field α] [IsCharP α 3] (a b : α) : a/3 = b → b = 0 := by
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grind
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example [Field α] [IsCharP α 7] (a b c : α) : a/3 = b → c = a/3 → a/2 + a/2 = b + 2*c + 7 := by
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grind
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example [Field R] [IsCharP R 0] (x : R) (cos : R → R) :
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(cos x ^ 2 + (2 * cos x ^ 2 - 1) ^ 2 + (4 * cos x ^ 3 - 3 * cos x) ^ 2 - 1) / 4 =
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cos x * (cos x ^ 2 - 1 / 2) * (4 * cos x ^ 3 - 3 * cos x) := by
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grind
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example [Field α] (a : α) : (1 / 2) * a = a / 2 := by grind
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example [Field α] (a : α) : 2⁻¹ * a = a / 2 := by grind
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example [Field α] (a : α) : a⁻¹⁻¹ = a := by grind
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example [Field α] [IsCharP α 0] (a : α) : a / 2 + a / 3 = 5 * a / 6 := by
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grind
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example [Field α] (a b : α) : a ≠ 0 → b ≠ 0 → a / (a / b) = b := by
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grind
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example [Field α] (a b : α) : a ≠ 0 → a / (a / b) = b := by
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grind
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example [Field α] [IsCharP α 0] (x : α)
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: x ≠ 0 → (4 / x)⁻¹ * ((3 * x^3) / x)^2 * ((1 / (2 * x))⁻¹)^3 = 18 * x^8 := by
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grind
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example [Field α] (a : α) : 2 * a ≠ 0 → 1 / a + 1 / (2 * a) = 3 / (2 * a) := by
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grind
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example [Field α] [IsCharP α 0] (a : α) : 1 / a + 1 / (2 * a) = 3 / (2 * a) := by
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grind
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example [Field α] [IsCharP α 0] (a b : α) : 2*b - a = a + b → 1 / a + 1 / (2 * a) = 3 / b := by
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grind
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example [Field α] [NoNatZeroDivisors α] (a : α) : 1 / a + 1 / (2 * a) = 3 / (2 * a) := by
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grind
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example [Field α] {x y z w : α} : x / y = z / w → y ≠ 0 → w ≠ 0 → x * w = z * y := by
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grind
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example [Field α] (a : α) : a = 0 → a ≠ 1 := by
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grind
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example [Field α] (a : α) : a = 0 → a ≠ 1 - a := by
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grind
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example [Field α] {sqrtTwo a b c : α} :
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sqrtTwo / 32 * ((a - b) ^ 2 + (b - c) ^ 2 + (c - a) ^ 2 + (-(a + b + c)) ^ 2) ^ 2 =
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9 * sqrtTwo / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by
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grind
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-- The following example should not split on `2 = 0` because a linear ordered field has
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-- characteristic zero.
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#guard_msgs (trace) in
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set_option trace.grind.split true in
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example [Field α] [LE α] [LT α] [LinearOrder α] [OrderedRing α] (x y z : α)
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: x > 0 → y > 0 → z > 0 → x * y * z ≥ 1 →
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(x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +
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(z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) =
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1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2) := by
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grind
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example [Field α] (a : α) : a^2 = 0 → a = 0 := by
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grind
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example [Field α] (a : α) : a^3 = 0 → a = 0 := by
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grind
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/-- trace: [grind.debug.ring.rabinowitsch] (b + a - (c - b + b)) * (b + a - (c - b + b))⁻¹ -/
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#guard_msgs (trace) in
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set_option trace.grind.debug.ring.rabinowitsch true in
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example [Field α] (a b c : α) : a^2 = 0 → c = b → b + a = c - b + b := by
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grind
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/-- trace: [grind.debug.ring.rabinowitsch] (b + a - c) * (b + a - c)⁻¹ -/
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#guard_msgs (trace) in
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set_option trace.grind.debug.ring.rabinowitsch true in
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example [Field α] (a b c : α) : a^2 = 0 → c = b → b + a - c = 0 := by
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grind
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