6970d77ae4
This PR add instances showing that the Grothendieck (i.e. additive) envelope of a semiring is an ordered ring if the original semiring is ordered (and satisfies ExistsAddOfLE), and in this case the embedding is monotone.
166 lines
5.4 KiB
Text
166 lines
5.4 KiB
Text
open Lean.Grind
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set_option grind.debug true
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b : α)
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: (2:Int)*a + b < b + a + a → False := by
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b : α)
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: (2:Int)*a + b < b + a + a → False := by
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grind
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b : α)
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: (2:Int)*a + b < b + a + a → False := by
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fail_if_success grind -linarith
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b : α)
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: (2:Int)*a + b ≥ b + a + a := by
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grind
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#guard_msgs (drop error) in
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b : α)
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: (2:Int)*a + b ≥ b + a + a := by
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fail_if_success grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: 2*a + b < b + a + a → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: 2 + 2*a + b + 1 < b + a + a + 3 → False := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b : α)
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: 2 + 2*a + b + 1 <= b + a + a + 3 := by
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grind
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b c : α)
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: a < b → b < c → c < a → False := by
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c : α)
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: a < b → b < c → a < c := by
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c : α)
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: a < b → b ≤ c → a < c := by
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c : α)
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: a ≤ b → b ≤ c → a ≤ c := by
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grind
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example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c d : α)
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: a < b → b < c + d → a - d < c := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d : α)
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: a < b → b < c + d → a - d < c := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b c : α)
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: a < b → 2*b < c → c < 2*a → False := by
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grind
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-- Test misconfigured instances
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/--
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trace: [grind.issues] type has a `Preorder` and is a `Semiring`, but is not an ordered ring, failed to synthesize
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OrderedRing α
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-/
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#guard_msgs (drop error, trace) in
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set_option trace.grind.issues true in
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example [CommRing α] [Preorder α] [OrderedAdd α] (a b c : α)
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: a < b → b + b < c → c < a → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a < 2 → b < a → b > 5 → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a < One.one + 4 → b < a → b ≥ 5 → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a < One.one + 5 → b < a → b ≥ 5 → False := by
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fail_if_success grind
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sorry
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b c d : α)
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: a < c → b = a + d → b - d > c → False := by
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grind
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example [IntModule α] [Preorder α] [OrderedAdd α] (a b c d : α)
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: a + d < c → b = a + (2:Int)*d → b - d > c → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a < 2 → b = a → b > 5 → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a < 2 → a = b + b → b > 5 → False := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b : α)
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: a < 2 → a = b + b → b < 1 := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b : α)
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: a ≤ 2 → a + b = 3*b → b ≤ 1 := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d e : α) :
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2*a + b ≥ 1 → b ≥ 0 → c ≥ 0 → d ≥ 0 → e ≥ 0
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→ a ≥ 3*c → c ≥ 6*e → d - e*5 ≥ 0
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→ a + b + 3*c + d + 2*e ≥ 0 := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b c d e : α) :
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2*a + b ≥ 1 → b ≥ 0 → c ≥ 0 → d ≥ 0 → e ≥ 0
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→ a ≥ 3*c → c ≥ 6*e → d - e*5 ≥ 0
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→ a + b + 3*c + d + 2*e < 0 → False := by
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grind
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example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
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: a = 0 → b = 1 → a + b > 2 → False := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c : α)
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: a = 0 → a + b > 2 → b = c → 1 = c → False := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b : α)
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: a = 0 → b = 1 → a + b ≤ 2 := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b : α)
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: a*b + b*a > 1 → a*b > 0 := by
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grind
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c : α)
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: a*b + c > 1 → c = b*a → a*b > 0 := by
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grind
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-- It must not internalize subterms `b + c + d` and `b + b + d`
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#guard_msgs (trace) in
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set_option trace.grind.linarith.internalize true
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example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d : α)
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: a < b + c + d → c = b → a < b + b + d := by
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grind
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/--
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trace: [grind.linarith.assert] -3 * y + 2 * x + One.one ≤ 0
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[grind.linarith.assert] 2 * z + -4 * x + One.one ≤ 0
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[grind.linarith.assert] -1 * z + 3 * y + One.one ≤ 0
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[grind.linarith.assert] 6 * y + -4 * x + 3 * One.one ≤ 0
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[grind.linarith.assert] 15 * One.one ≤ 0
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[grind.linarith.assert] Zero.zero < 0
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-/
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#guard_msgs (trace) in
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set_option trace.grind.cutsat.assert true in -- cutsat should **not** process the following constraints
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set_option trace.grind.linarith.assert true in -- linarith should take over
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set_option trace.grind.linarith.assert.store false in
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example (x y z : Int) (h1 : 2 * x < 3 * y) (h2 : -4 * x + 2 * z < 0) : ¬ 12*y - 4* z < 0 := by
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grind -cutsat
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