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lean4-htt/tests/lean/run/omega.lean
Scott Morrison 88deb34ddb
chore: upstream omega (#3367) 
Co-authored-by: Joe Hendrix <joe@lean-fro.org>
2024-02-19 00:19:55 +00:00

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example : True := by
fail_if_success omega
trivial
-- set_option trace.omega true
example (_ : (1 : Int) < (0 : Int)) : False := by omega
example (_ : (0 : Int) < (0 : Int)) : False := by omega
example (_ : (0 : Int) < (1 : Int)) : True := by (fail_if_success omega); trivial
example {x : Int} (_ : 0 ≤ x) (_ : x ≤ 1) : True := by (fail_if_success omega); trivial
example {x : Int} (_ : 0 ≤ x) (_ : x ≤ -1) : False := by omega
example {x : Int} (_ : x % 2 < x - 2 * (x / 2)) : False := by omega
example {x : Int} (_ : x % 2 > 5) : False := by omega
example {x : Int} (_ : 2 * (x / 2) > x) : False := by omega
example {x : Int} (_ : 2 * (x / 2) ≤ x - 2) : False := by omega
example {x : Nat} : x / 0 = 0 := by omega
example {x : Int} : x / 0 = 0 := by omega
example {x : Int} : x / 2 + x / (-2) = 0 := by omega
example {x : Nat} (_ : x ≠ 0) : 0 < x := by omega
example {x y z : Nat} (_ : a ≤ c) (_ : b ≤ c) : a < Nat.succ c := by omega
example (_ : 7 < 3) : False := by omega
example (_ : 0 < 0) : False := by omega
example {x : Nat} (_ : x > 7) (_ : x < 3) : False := by omega
example {x : Nat} (_ : x ≥ 7) (_ : x ≤ 3) : False := by omega
example {x y : Nat} (_ : x + y > 10) (_ : x < 5) (_ : y < 5) : False := by omega
example {x y : Int} (_ : x + y > 10) (_ : 2 * x < 11) (_ : y < 5) : False := by omega
example {x y : Nat} (_ : x + y > 10) (_ : 2 * x < 11) (_ : y < 5) : False := by omega
example {x y : Int} (_ : 2 * x + 4 * y = 5) : False := by omega
example {x y : Nat} (_ : 2 * x + 4 * y = 5) : False := by omega
example {x y : Int} (_ : 6 * x + 7 * y = 5) : True := by (fail_if_success omega); trivial
example {x y : Nat} (_ : 6 * x + 7 * y = 5) : False := by omega
example {x y : Nat} (_ : x * 6 + y * 7 = 5) : False := by omega
example {x y : Nat} (_ : 2 * (3 * x) + y * 7 = 5) : False := by omega
example {x y : Nat} (_ : 2 * x * 3 + y * 7 = 5) : False := by omega
example {x y : Nat} (_ : 2 * 3 * x + y * 7 = 5) : False := by omega
example {x : Nat} (_ : x < 0) : False := by omega
example {x y z : Int} (_ : x + y > z) (_ : x < 0) (_ : y < 0) (_ : z > 0) : False := by omega
example {x y : Nat} (_ : x - y = 0) (_ : x > y) : False := by
fail_if_success omega (config := { splitNatSub := false })
omega
example {x y z : Int} (_ : x - y - z = 0) (_ : x > y + z) : False := by omega
example {x y z : Nat} (_ : x - y - z = 0) (_ : x > y + z) : False := by omega
example {a b c d e f : Nat} (_ : a - b - c - d - e - f = 0) (_ : a > b + c + d + e + f) :
False := by
omega
example {x y : Nat} (h₁ : x - y ≤ 0) (h₂ : y < x) : False := by omega
example {x y : Int} (_ : x / 2 - y / 3 < 1) (_ : 3 * x ≥ 2 * y + 6) : False := by omega
example {x y : Nat} (_ : x / 2 - y / 3 < 1) (_ : 3 * x ≥ 2 * y + 6) : False := by omega
example {x y : Nat} (_ : x / 2 - y / 3 < 1) (_ : 3 * x ≥ 2 * y + 4) : False := by omega
example {x y : Nat} (_ : x / 2 - y / 3 < x % 2) (_ : 3 * x ≥ 2 * y + 4) : False := by omega
example {x : Int} (h₁ : 5 ≤ x) (h₂ : x ≤ 4) : False := by omega
example {x : Nat} (h₁ : 5 ≤ x) (h₂ : x ≤ 4) : False := by omega
example {x : Nat} (h₁ : x / 3 ≥ 2) (h₂ : x < 6) : False := by omega
example {x : Int} {y : Nat} (_ : 0 < x) (_ : x + y ≤ 0) : False := by omega
example {a b c : Nat} (_ : a - (b - c) ≤ 5) (_ : b ≥ c + 3) (_ : a + c ≥ b + 6) : False := by omega
example {x : Nat} : 1 < (1 + ((x + 1 : Nat) : Int) + 2) / 2 := by omega
example {x : Nat} : (x + 4) / 2 ≤ x + 2 := by omega
example {x : Int} {m : Nat} (_ : 0 < m) (_ : ¬x % ↑m < (↑m + 1) / 2) : -↑m / 2 ≤ x % ↑m - ↑m := by
omega
example (h : (7 : Int) = 0) : False := by omega
example (h : (7 : Int) ≤ 0) : False := by omega
example (h : (-7 : Int) + 14 = 0) : False := by omega
example (h : (-7 : Int) + 14 ≤ 0) : False := by omega
example (h : (1 : Int) + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 0) : False := by
omega
example (h : (7 : Int) - 14 = 0) : False := by omega
example (h : (14 : Int) - 7 ≤ 0) : False := by omega
example (h : (1 : Int) - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 = 0) : False := by
omega
example (h : -(7 : Int) = 0) : False := by omega
example (h : -(-7 : Int) ≤ 0) : False := by omega
example (h : 2 * (7 : Int) = 0) : False := by omega
example (h : (7 : Int) < 0) : False := by omega
example {x : Int} (h : x + x + 1 = 0) : False := by omega
example {x : Int} (h : 2 * x + 1 = 0) : False := by omega
example {x y : Int} (h : x + x + y + y + 1 = 0) : False := by omega
example {x y : Int} (h : 2 * x + 2 * y + 1 = 0) : False := by omega
example {x : Int} (h₁ : 0 ≤ -7 + x) (h₂ : 0 ≤ 3 - x) : False := by omega
example {x : Int} (h₁ : 0 ≤ -7 + x) (h₂ : 0 < 4 - x) : False := by omega
example {x : Int} (h₁ : 0 ≤ 2 * x + 1) (h₂ : 2 * x + 1 ≤ 0) : False := by omega
example {x : Int} (h₁ : 0 < 2 * x + 2) (h₂ : 2 * x + 1 ≤ 0) : False := by omega
example {x y : Int} (h₁ : 0 ≤ 2 * x + 1) (h₂ : x = y) (h₃ : 2 * y + 1 ≤ 0) : False := by omega
example {x y z : Int} (h₁ : 0 ≤ 2 * x + 1) (h₂ : x = y) (h₃ : y = z) (h₄ : 2 * z + 1 ≤ 0) :
False := by omega
example {x1 x2 x3 x4 x5 x6 : Int} (h : 0 ≤ 2 * x1 + 1) (h : x1 = x2) (h : x2 = x3) (h : x3 = x4)
(h : x4 = x5) (h : x5 = x6) (h : 2 * x6 + 1 ≤ 0) : False := by omega
example {x : Int} (_ : 1 ≤ -3 * x) (_ : 1 ≤ 2 * x) : False := by omega
example {x y : Int} (_ : 2 * x + 3 * y = 0) (_ : 1 ≤ x) (_ : 1 ≤ y) : False := by omega
example {x y z : Int} (_ : 2 * x + 3 * y = 0) (_ : 3 * y + 4 * z = 0) (_ : 1 ≤ x) (_ : 1 ≤ -z) :
False := by omega
example {x y z : Int} (_ : 2 * x + 3 * y + 4 * z = 0) (_ : 1 ≤ x + y) (_ : 1 ≤ y + z)
(_ : 1 ≤ x + z) : False := by omega
example {x y : Int} (_ : 1 ≤ 3 * x) (_ : y ≤ 2) (_ : 6 * x - 2 ≤ y) : False := by omega
example {x y : Int} (_ : y = x) (_ : 0 ≤ x - 2 * y) (_ : x - 2 * y ≤ 1) (_ : 1 ≤ x) : False := by
omega
example {x y : Int} (_ : y = x) (_ : 0 ≤ x - 2 * y) (_ : x - 2 * y ≤ 1) (_ : x ≥ 1) : False := by
omega
example {x y : Int} (_ : y = x) (_ : 0 ≤ x - 2 * y) (_ : x - 2 * y ≤ 1) (_ : 0 < x) : False := by
omega
example {x y : Int} (_ : y = x) (_ : 0 ≤ x - 2 * y) (_ : x - 2 * y ≤ 1) (_ : x > 0) : False := by
omega
example {x : Nat} (_ : 10 ∣ x) (_ : ¬ 5 ∣ x) : False := by omega
example {x y : Nat} (_ : 5 ∣ x) (_ : ¬ 10 ∣ x) (_ : y = 7) (_ : x - y ≤ 2) (_ : x ≥ 6) : False := by
omega
example (x : Nat) : x % 4 - x % 8 = 0 := by omega
example {n : Nat} (_ : n > 0) : (2*n - 1) % 2 = 1 := by omega
example (x : Int) (_ : x > 0 ∧ x < -1) : False := by omega
example (x : Int) (_ : x > 7) : x < 0 ∨ x > 3 := by omega
example (_ : ∃ n : Nat, n < 0) : False := by omega
example (_ : { x : Int // x < 0 ∧ x > 0 }) : False := by omega
example {x y : Int} (_ : x < y) (z : { z : Int // y ≤ z ∧ z ≤ x }) : False := by omega
example (a b c d e : Int)
(ha : 2 * a + b + c + d + e = 4)
(hb : a + 2 * b + c + d + e = 5)
(hc : a + b + 2 * c + d + e = 6)
(hd : a + b + c + 2 * d + e = 7)
(he : a + b + c + d + 2 * e = 8) : e = 3 := by omega
example (a b c d e : Int)
(ha : 2 * a + b + c + d + e = 4)
(hb : a + 2 * b + c + d + e = 5)
(hc : a + b + 2 * c + d + e = 6)
(hd : a + b + c + 2 * d + e = 7)
(he : a + b + c + d + 2 * e = 8 ∨ e = 3) : e = 3 := by
fail_if_success omega (config := { splitDisjunctions := false })
omega
example {x : Int} (h : x = 7) : x.natAbs = 7 := by
fail_if_success omega (config := { splitNatAbs := false })
fail_if_success omega (config := { splitDisjunctions := false })
omega
example {x y : Int} (_ : (x - y).natAbs < 3) (_ : x < 5) (_ : y > 15) : False := by
omega
example {a b : Int} (h : a < b) (w : b < a) : False := by omega
example (_e b c a v0 v1 : Int) (_h1 : v0 = 5 * a) (_h2 : v1 = 3 * b) (h3 : v0 + v1 + c = 10) :
v0 + 5 + (v1 - 3) + (c - 2) = 10 := by omega
example (h : (1 : Int) < 0) (_ : ¬ (37 : Int) < 42) (_ : True) (_ : (-7 : Int) < 5) :
(3 : Int) < 7 := by omega
example (A B : Int) (h : 0 < A * B) : 0 < 8 * (A * B) := by omega
example (A B : Nat) (h : 7 < A * B) : 0 < A*B/8 := by omega
example (A B : Int) (h : 7 < A * B) : 0 < A*B/8 := by omega
example (ε : Int) (h1 : ε > 0) : ε / 2 + ε / 3 + ε / 7 < ε := by omega
example (x y z : Int) (h1 : 2*x < 3*y) (h2 : -4*x + z/2 < 0) (h3 : 12*y - z < 0) : False := by omega
example (ε : Int) (h1 : ε > 0) : ε / 2 < ε := by omega
example (ε : Int) (_ : ε > 0) : ε - 2 ≤ ε / 3 + ε / 3 + ε / 3 := by omega
example (ε : Int) (_ : ε > 0) : ε / 3 + ε / 3 + ε / 3 ≤ ε := by omega
example (ε : Int) (_ : ε > 0) : ε - 2 ≤ ε / 3 + ε / 3 + ε / 3 ∧ ε / 3 + ε / 3 + ε / 3 ≤ ε := by
omega
example (x : Int) (h : 0 < x) : 0 < x / 1 := by omega
example (x : Int) (h : 5 < x) : 0 < x/2/3 := by omega
example (_a b _c : Nat) (h2 : b + 2 > 3 + b) : False := by omega
example (_a b _c : Int) (h2 : b + 2 > 3 + b) : False := by omega
example (g v V c h : Int) (_ : h = 0) (_ : v = V) (_ : V > 0) (_ : g > 0)
(_ : 0 ≤ c) (_ : c < 1) : v ≤ V := by omega
example (x y z : Int) (h1 : 2 * x < 3 * y) (h2 : -4 * x + 2 * z < 0) (h3 : 12 * y - 4 * z < 0) :
False := by
omega
example (x y z : Int) (h1 : 2 * x < 3 * y) (h2 : -4 * x + 2 * z < 0) (_h3 : x * y < 5)
(h3 : 12 * y - 4 * z < 0) : False := by omega
example (a b c : Int) (h1 : a > 0) (h2 : b > 5) (h3 : c < -10) (h4 : a + b - c < 3) : False := by
omega
example (_ b _ : Int) (h2 : b > 0) (h3 : ¬ b ≥ 0) : False := by
omega
example (x y z : Int) (hx : x ≤ 3 * y) (h2 : y ≤ 2 * z) (h3 : x ≥ 6 * z) : x = 3 * y := by
omega
example (x y z : Int) (h1 : 2 * x < 3 * y) (h2 : -4 * x + 2 * z < 0) (_h3 : x * y < 5) :
¬ 12 * y - 4 * z < 0 := by
omega
example (x y z : Int) (hx : ¬ x > 3 * y) (h2 : ¬ y > 2 * z) (h3 : x ≥ 6 * z) : x = 3 * y := by
omega
example (x y : Int) (h : 6 + ((x + 4) * x + (6 + 3 * y) * y) = 3) (h' : (x + 4) * x ≥ 0)
(h'' : (6 + 3 * y) * y ≥ 0) : False := by omega
example (a : Int) (ha : 0 ≤ a) : 0 * 0 ≤ 2 * a := by omega
example (x y : Int) (h : x < y) : x ≠ y := by omega
example (x y : Int) (h : x < y) : ¬ x = y := by omega
example (x : Int) : id x ≥ x := by omega
example (prime : Nat → Prop) (x y z : Int) (h1 : 2 * x + ((-3) * y) < 0) (h2 : (-4) * x + 2* z < 0)
(h3 : 12 * y + (-4) * z < 0) (_ : prime 7) : False := by omega
example (i n : Nat) (h : (2 : Int) ^ i ≤ 2 ^ n) : (0 : Int) ≤ 2 ^ n - 2 ^ i := by omega
-- Check we use `exfalso` on non-comparison goals.
example (prime : Nat → Prop) (_ b _ : Nat) (h2 : b > 0) (h3 : b < 0) : prime 10 := by
omega
example (a b c : Nat) (h2 : (2 : Nat) > 3) : a + b - c ≥ 3 := by omega
-- Verify that we split conjunctions in hypotheses.
example (x y : Int)
(h : 6 + ((x + 4) * x + (6 + 3 * y) * y) = 3 ∧ (x + 4) * x ≥ 0 ∧ (6 + 3 * y) * y ≥ 0) :
False := by omega
example (mess : Nat → Nat) (S n : Nat) :
mess S + (n * mess S + n * 2 + 1) < n * mess S + mess S + (n * 2 + 2) := by omega
example (p n p' n' : Nat) (h : p + n' = p' + n) : n + p' = n' + p := by
omega
example (a b c : Int) (h1 : 32 / a < b) (h2 : b < c) : 32 / a < c := by omega
-- Check that `autoParam` wrappers do not get in the way of using hypotheses.
example (i n : Nat) (hi : i ≤ n := by omega) : i < n + 1 := by
omega
-- Test that we consume expression metadata when necessary.
example : 0 = 0 := by
have : 0 = 0 := by omega
omega -- This used to fail.
/-! ### `Prod.Lex` -/
-- This example comes from the termination proof
-- for `permutationsAux.rec` in `Mathlib.Data.List.Defs`.
example {x y : Nat} : Prod.Lex (· < ·) (· < ·) (x, x) (Nat.succ y + x, Nat.succ y) := by omega
-- We test the termination proof in-situ:
def List.permutationsAux.rec' {C : List α → List α → Sort v} (H0 : ∀ is, C [] is)
(H1 : ∀ t ts is, C ts (t :: is) → C is [] → C (t :: ts) is) : ∀ l₁ l₂, C l₁ l₂
| [], is => H0 is
| t :: ts, is =>
H1 t ts is (permutationsAux.rec' H0 H1 ts (t :: is)) (permutationsAux.rec' H0 H1 is [])
termination_by ts is => (length ts + length is, length ts)
decreasing_by all_goals simp_wf; omega
example {x y w z : Nat} (h : Prod.Lex (· < ·) (· < ·) (x + 1, y + 1) (w, z)) :
Prod.Lex (· < ·) (· < ·) (x, y) (w, z) := by omega
-- Verify that we can handle `iff` statements in hypotheses:
example (a b : Int) (h : a < 0 ↔ b < 0) (w : b > 3) : a ≥ 0 := by omega
-- Verify that we can prove `iff` goals:
example (a b : Int) (h : a > 7) (w : b > 2) : a > 0 ↔ b > 0 := by omega
-- Verify that we can prove implications:
example (a : Int) : a > 0 → a > -1 := by omega
-- Verify that we can introduce multiple arguments:
example (x y : Int) : x + 1 ≤ y → ¬ y + 1 ≤ x := by omega
-- Verify that we can handle double negation:
example (x y : Int) (_ : x < y) (_ : ¬ ¬ y < x) : False := by omega
-- Verify that we don't treat function goals as implications.
example (a : Nat) (h : a < 0) : Nat → Nat := by omega
-- Example from Cedar:
example {a₁ a₂ p₁ p₂ : Nat}
(h₁ : a₁ = a₂ → ¬p₁ = p₂) :
(a₁ < a₂ ∨ a₁ = a₂ ∧ p₁ < p₂) ∨ a₂ < a₁ ∨ a₂ = a₁ ∧ p₂ < p₁ := by omega
-- From https://github.com/leanprover/std4/issues/562
example {i : Nat} (h1 : i < 330) (_h2 : 7 ∣ (660 + i) * (1319 - i)) : 1319 - i < 1979 := by
omega
example {a : Int} (_ : a < min a b) : False := by omega (config := { splitMinMax := false })
example {a : Int} (_ : max a b < b) : False := by omega (config := { splitMinMax := false })
example {a : Nat} (_ : a < min a b) : False := by omega (config := { splitMinMax := false })
example {a : Nat} (_ : max a b < b) : False := by omega (config := { splitMinMax := false })
example {a b : Nat} (_ : a = 7) (_ : b = 3) : min a b = 3 := by
fail_if_success omega (config := { splitMinMax := false })
omega
example {a b : Nat} (_ : a + b = 9) : (min a b) % 2 + (max a b) % 2 = 1 := by
fail_if_success omega (config := { splitMinMax := false })
omega
example {a : Int} (_ : a < if a ≤ b then a else b) : False := by omega
example {a b : Int} : (if a < b then a else b - 1) ≤ b := by omega
-- Check that we use local values.
example (i j : Nat) (p : i ≥ j) : True := by
let l := j - 1
have _ : i ≥ l := by omega
trivial
example (i j : Nat) (p : i ≥ j) : True := by
let l := j - 1
let k := l
have _ : i ≥ k := by omega
trivial
example (i : Fin 7) : (i : Nat) < 8 := by omega
example (x y z i : Nat) (hz : z ≤ 1) : x % 2 ^ i + y % 2 ^ i + z < 2 * 2^ i := by omega
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