name
stringlengths 5
12
| formal_statement
stringlengths 62
282
|
|---|---|
amgm_p1
|
theorem amgm_p1 (x y z: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x + y + z) / 3 ≥ (x * y * z) ^ (3⁻¹: ℝ ) := by
|
amgm_p2
|
theorem amgm_p2 (x y: ℝ) (hx : x > 0) (hy : y > 0) : (2 * x + y) / 3 ≥ (x * x * y) ^ (3⁻¹: ℝ ) := by
|
amgm_p3
|
theorem amgm_p3 (x y z w: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) : (x + y + z + w) / 4 ≥ (x * y * z * w) ^ (4⁻¹: ℝ ) := by
|
amgm_p4
|
theorem amgm_p4 (x y: ℝ ) (h : x > 0 ∧ y> 0): (2:ℝ) / 3 * x + (1:ℝ) / 3 * y ≥ x^((2:ℝ) / 3) * y^((1:ℝ) / 3) := by
|
amgm_p5
|
theorem amgm_p5 (x y: ℝ ) (h : x > 0 ∧ y> 0): (4:ℝ) / 7 * x + (3:ℝ) / 7 * y ≥ x^((4:ℝ) / 7) * y^((3:ℝ) / 7) := by
|
amgm_p6
|
theorem amgm_p6 (x y z: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (2:ℝ)/5 * x + (2:ℝ)/5 * y + (1:ℝ)/5 * z ≥ x ^ ((2:ℝ)/5) * y ^ ((2:ℝ)/5) * z ^ ((1:ℝ)/5) := by
|
amgm_p7
|
theorem amgm_p7 (x y z w: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) : (1:ℝ)/3 * x + (1:ℝ)/3 * y + (1:ℝ)/6 * z + (1:ℝ)/6 * w ≥ x ^ ((1:ℝ)/3) * y ^ ((1:ℝ)/3) * z ^ ((1:ℝ)/6) * w ^ ((1:ℝ)/6) := by
|
amgm_p8
|
theorem amgm_p8 (x y z: ℝ ) (h₁: x+ y + z = 3) (h₂ : x > 0 ∧ y> 0 ∧ z> 0): (x * y * z) ^ (3⁻¹: ℝ ) ≤ 1 := by
|
amgm_p9
|
theorem amgm_p9 (x y: ℝ ) (h₁: x+ 2 * y = 3) (h₂ : x > 0 ∧ y> 0): (x * y ^ 2) ^ (3⁻¹: ℝ ) ≤ 1 := by
|
amgm_p10
|
theorem amgm_p10 (x y: ℝ ) (h₁: x+ 2 * y = 3) (h₂ : x > 0 ∧ y> 0): x * y ^ 2 ≤ 1 := by
|
amgm_p11
|
theorem amgm_p11 (x y z: ℝ ) (h₁: x+ y + z = 3) (h₂ : x > 0 ∧ y> 0 ∧ z> 0): x * y * z ≤ 1 := by
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amgm_p12
|
theorem amgm_p12 (x y z: ℝ ) (h₁: x+ 2 * y + 2 * z = 10) (h₂ : x > 0 ∧ y> 0 ∧ z> 0): x * y ^ 2 * z ^ 2 ≤ 32 := by
|
amgm_p13
|
theorem amgm_p13 (x y: ℝ ) (h : x > 0 ∧ y> 0): (4:ℝ) / 5 * x ^ 5 + (1:ℝ) / 5 * y ^ 5 ≥ x^4 * y := by
|
amgm_p14
|
theorem amgm_p14 (x y: ℝ ) (h : x > 0 ∧ y> 0): (2:ℝ) / 3 * x ^ 6 + (1:ℝ) / 3 * y ^ 6 ≥ x^4 * y^2 := by
|
amgm_p15
|
theorem amgm_p15 (x y: ℝ ) (h : x > 0 ∧ y> 0): (4:ℝ) / 7 * x ^ 7 + (3:ℝ) / 7 * y ^ 7 ≥ x^4 * y^3 := by
|
amgm_p16
|
theorem amgm_p16 (x y: ℝ ) (h : x > 0 ∧ y> 0): (2:ℝ) / 3 * x ^ 3 + (1:ℝ) / 3 * y ^ 3 ≥ x^2 * y := by
|
amgm_p17
|
theorem amgm_p17 (x y z: ℝ ) (h : x > 0 ∧ y> 0 ∧ z> 0): (1:ℝ) / 2 * x ^ 4 + (1:ℝ) / 4 * y ^ 4 + (1:ℝ) / 4 * z ^ 4 ≥ x^2 * y * z := by
|
amgm_p18
|
theorem amgm_p18 (x y z: ℝ ) (h : x > 0 ∧ y> 0 ∧ z> 0): (2:ℝ) / 5 * x ^ 5 + (2:ℝ) / 5 * y ^ 5 + (1:ℝ) / 5 * z ^ 5 ≥ x^2 * y^2 * z := by
|
amgm_p19
|
theorem amgm_p19 (x y z: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (3:ℝ)/5 * x^5 + (1:ℝ)/5 * y^5 + (1:ℝ)/5 * z^5 ≥ x ^ 3 * y * z := by
|
amgm_p20
|
theorem amgm_p20 (x y z w: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) : (1:ℝ)/3 * x^6 + (1:ℝ)/3 * y^6 + (1:ℝ)/6 * z^6 + (1:ℝ)/6 * w^6 ≥ x^2 * y^2 * z * w := by
|
amgm_p21
|
theorem amgm_p21 (x y z: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (2:ℝ)/3 * x^2 + (1:ℝ)/6 * y^2 + (1:ℝ)/6 * z^2 ≥ x^((4:ℝ)/3) * y^((1:ℝ)/3) * z^((1:ℝ)/3) := by
|
amgm_p22
|
theorem amgm_p22 (x y z: ℝ ) (h : x > 0 ∧ y> 0 ∧ z> 0) (g : x * y * z = (1 : ℝ)) : (4:ℝ) / 7 * x^3 * y + (1:ℝ) / 7 * y^3 * z + (2:ℝ) / 7 * z^3 * x ≥ x := by
|
amgm_p23
|
theorem amgm_p23 (a b c d: ℝ) (ap : a > 0) (bp : b> 0) (cp : c> 0) ( dp : d> 0) (g : a * b * c * d = (1 : ℝ)) : (23:ℝ) / 51 * a^4 * b + (7:ℝ) / 51 * b^4 * c + (11:ℝ) / 51 * c^4 * d + (10:ℝ) / 51 * d^4 * a ≥ a := by
|
amgm_p24
|
theorem amgm_p24 (a b c : ℝ) (ap : a > 0) (bp : b> 0) (cp : c> 0) : a^3 + b^3 + c^3 ≥ a^2 * b + b^2 * c + c^2 * a := by
|
amgm_p25
|
theorem amgm_p25 (a b c : ℝ) (ap : a > 0) (bp : b> 0) (cp : c> 0) : a^7 + b^7 + c^7 ≥ a^4 * b^3 + b^4 * c^3 + c^4 * a^3 := by
|
cauchy_p1
|
theorem cauchy_p1 (x y : ℝ) (h₂ : x > 0 ∧ y > 0) : ( x + y ) * ( 1 / x + 1 / y ) ≥ 4 := by
|
cauchy_p2
|
theorem cauchy_p2 (x y z: ℝ) (h₂ : x > 0 ∧ y > 0 ∧ z > 0 ) : ( x + y + z ) * ( 1 / x + 1 / y + 1 / z ) ≥ 9 := by
|
cauchy_p3
|
theorem cauchy_p3 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hxy : x + y ≤ 1) : 4 * x^2 + 4 * y^2 + (1 - x - y)^2 ≥ 2 / 3 := by
|
cauchy_p4
|
theorem cauchy_p4 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hx1 : x ≤ 1) (hy1 : y ≤ 1) : x * √(1 - y^2) + y * √(1 - x^2) ≤ 1 := by
|
cauchy_p5
|
theorem cauchy_p5 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (g : x + y + z = 3) : 4 / x + 1 / y + 9 / z ≥ 12 := by
|
cauchy_p6
|
theorem cauchy_p6 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a / (b + c) + b / (c + a) + c / (a + b) ≥ 3 / 2 := by
|
cauchy_p7
|
theorem cauchy_p7 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (h : a + b + c + d = 1) : 1 / (b + c + d) + 1 / (c + d + a) + 1 / (a + b + d) + 1 / (a + b + c) ≥ 16 / 3 := by
|
cauchy_p8
|
theorem cauchy_p8 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (g : x * (x + y) + y * (y + z) + z * (z + x) = 1) : x / (x + y) + y / (y + z) + z / (z + x) ≥ (x + y + z) ^ 2 := by
|
cauchy_p9
|
theorem cauchy_p9 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) ( g : z * (x + y) + x * (y + z) + y * (z + x) = 1) : z / (x + y) + x / (y + z) + y / (z + x) ≥ (x + y + z) ^ 2 := by
|
cauchy_p10
|
theorem cauchy_p10 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : √(2 * x + 1) + √(2 * y + 3) = 4) : x + y ≥ 2 := by
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cauchy_p11
|
theorem cauchy_p11 (x y z: ℝ) (h : x^2 + 2 * y^2 + 4 * z^2 > 0) : (x + y + z)^2 / (x^2 + 2 * y^2 + 4 * z^2) ≤ 7 / 4 := by
|
cauchy_p12
|
theorem cauchy_p12 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : 1 / (2 * x + y) + 3 / (x + y) = 2) : 6 * x + 5 * y ≥ 13 / 2 + 2 * √3 := by
|
cauchy_p13
|
theorem cauchy_p13 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a + b + c = 1) : √(2 * a + 1) + √(2 * b + 1) + √(2 * c + 1) ≤ √15 := by
|
cauchy_p14
|
theorem cauchy_p14 (n : ℕ) (a b : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) : ∑ i, a i / b i ≥ (∑ i, a i)^2 / ∑ i, a i * b i := by
|
cauchy_p15
|
theorem cauchy_p15 (n : ℕ) (a b : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) : ∑ i, a i / (b i)^2 ≥ (∑ i, a i / b i)^2 / ∑ i, a i := by
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cauchy_p16
|
theorem cauchy_p16 (x y a b: ℝ) (hy : y ≠ 0) (hb : b ≠ 0) (hxy : x^2 + 1 / y^2 = 1) (hab : a^2 + 1 / b^2 = 4) : |a / y + x / b| ≤ 2 := by
|
cauchy_p17
|
theorem cauchy_p17 (a b c d e : ℝ) (h : (a - b)^2 + (b - c)^2 + (c - d)^2 + (d - e)^2 = 1) : a - 2 * b - c + 2 * e ≤ √10 := by
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cauchy_p18
|
theorem cauchy_p18 (n : ℕ) (hn : n > 2) (a : Fin n → ℝ) (ha1 : ∀ i, a i < 1) (ha2 : ∀ i, a i ≥ 0) (hs : ∑ i : Fin n, a i = n - 2) : ∑ i : Fin n, ((a i)^2 / (1 - a i)) ≥ ((n : ℝ) - 2)^2 / 2 := by
|
cauchy_p19
|
theorem cauchy_p19 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : 1 / (1 + x^2) + 1 / (1 + y^2) + 1 / (1 + z^2) = 2) : x^2 + y^2 + z^2 + 3 ≥ (x + y + z)^2 := by
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cauchy_p20
|
theorem cauchy_p20 (a b c : ℝ) (ha : a > 1) (hb : b > 1) (hc : c > 1) (h : (a^2-1)/2 + (b^2-1)/2 + (c^2-1)/3 = 1) : a + b + c ≤ 7 * √3 / 3 := by
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cauchy_p21
|
theorem cauchy_p21 (n : ℕ) (a b : Fin n → ℝ) (hn : n > 0) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (sum_eq : ∑ i, a i = ∑ i , b i): ∑ i, (a i) ^ 2 / (a i + b i) ≥ (∑ i, a i) / 2 := by
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cauchy_p22
|
theorem cauchy_p22 (a b c d e s : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (he : e > 0) (hs : s = a + b + c + d + e) : a^2 / (a^2 + b * (s - b)) + b^2 / (b^2 + c * (s - c)) + c^2 / (c^2 + d * (s - d)) + d^2 / (d^2 + e * (s - e)) + e^2 / (e^2 + a * (s - a)) ≥ 1 := by
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cauchy_p23
|
theorem cauchy_p23 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : x^2 + y^2 / 2 = 1) : x + √(2 + 3 * y^2) ≤ 2 * √21 / 3 := by
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cauchy_p24
|
theorem cauchy_p24 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (hxy : 2 * x - y^2 / x > 0) (hyz : 2 * y - z^2 / y > 0) (hzx : 2 * z - x^2 / z > 0) : x^3 / (2 * x - y^2 / x) + y^3 / (2 * y - z^2 / y) + z^3 / (2 * z - x^2 / z) ≥ x^2 + y^2 + z^2 := by
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cauchy_p25
|
theorem cauchy_p25 (n : ℕ) (x : Fin n → ℝ) (s : ℝ) (hn : n > 2) (hs : s = ∑ i, x i) (hx : ∀ i, x i < s - x i) : ∑ i, (x i)^2 / (s - 2 * x i) ≥ s / (n - 2) := by
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jensen_p1
|
theorem jensen_p1 (x y : ℝ) (h : x > 0) (g : y > 0) : ((1:ℝ)/3 * x + (2:ℝ)/3 * y) ^ 4 ≤ (1:ℝ)/3 * x^4 + (2:ℝ)/3 * y ^ 4 := by
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jensen_p2
|
theorem jensen_p2 (x y : ℝ) : Real.exp ((1:ℝ)/4 * x + (3:ℝ)/4 * y) ≤ (1:ℝ)/4 * Real.exp x + (3:ℝ)/4 * Real.exp y := by
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jensen_p3
|
theorem jensen_p3 (x y : ℝ) (h : x > 0) (g : y > 0): ((1:ℝ)/4 * x + (3:ℝ)/4 * y) * Real.log ((1:ℝ)/4 * x + (3:ℝ)/4 * y) ≤ (1:ℝ)/4 * x * Real.log x + (3:ℝ)/4 * y * Real.log y := by
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jensen_p4
|
theorem jensen_p4 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = 3) : (1:ℝ)/3 * x^6 + (1:ℝ)/3 * y ^ 6 + (1:ℝ)/3 * z ^ 6 ≥ 1 := by
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jensen_p5
|
theorem jensen_p5 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * x ^ ((1:ℝ)/3) + (3:ℝ)/8 * y ^ ((1:ℝ)/3) + (3:ℝ)/8 * z ^ ((1:ℝ)/3) ≤ ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) ^ ((1:ℝ)/3) := by
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jensen_p6
|
theorem jensen_p6 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * Real.log x + (3:ℝ)/8 * Real.log y + (3:ℝ)/8 * Real.log z ≤ Real.log ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) := by
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jensen_p7
|
theorem jensen_p7 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * Real.sqrt x + (3:ℝ)/8 * Real.sqrt y + (3:ℝ)/8 * Real.sqrt z ≤ Real.sqrt ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) := by
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jensen_p8
|
theorem jensen_p8 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.sin x + (1:ℝ)/3 * Real.sin y + (1:ℝ)/3 * Real.sin z ≤ √3 / 2 := by
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jensen_p9
|
theorem jensen_p9 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.sin (x/2) + (1:ℝ)/3 * Real.sin (y/2) + (1:ℝ)/3 * Real.sin (z/2) ≤ (1:ℝ) / 2 := by
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jensen_p10
|
theorem jensen_p10 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.cos (x/2) + (1:ℝ)/3 * Real.cos (y/2) + (1:ℝ)/3 * Real.cos (z/2) ≤ √3 / 2 := by
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induction_p1
|
theorem induction_p1 (n : ℕ) (h : n ≥ 4) : 2 ^ n ≥ n + 1 := by
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induction_p2
|
theorem induction_p2 (x : ℝ) (n : ℕ) (h₀ : -1 < x) (h₁ : 0 < n) : 1 + ↑n * x ≤ (1 + x) ^ (n : ℕ) := by
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induction_p3
|
theorem induction_p3 (n : ℕ) (h₀ : 4 ≤ n) : n ^ 2 ≤ n ! := by
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induction_p4
|
theorem induction_p4 (n : ℕ) (h₀ : 3 ≤ n) : n ! < n ^ (n - 1) := by
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induction_p5
|
theorem induction_p5 (n : ℕ) (h₀ : 0 < n) : (∏ k in Finset.Icc 1 n, (1 + (1 : ℝ) / k ^ 3)) ≤ (3 : ℝ) - 1 / ↑n := by
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schur_p1
|
theorem schur_p1 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a * b * c = 1) : (a - 1 + 1 / b) * (b - 1 + 1 / c) * (c - 1 + 1 / a) ≤ 1 := by
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schur_p2
|
theorem schur_p2 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a * b * c = 1) : 3 + a / b + b / c + c / a ≥ a + b + c + 1 / a + 1 / b + 1 / c := by
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schur_p3
|
theorem schur_p3 (a b c t: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a ≥ b)(hbc : b ≥ c) (ht : t > 0) : a^t * (a - b) * (a - c) + b^t * (b - c) * (b - a) + c^t * (c - a) * (c - b) ≥ 0 := by
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schur_p4
|
theorem schur_p4 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a + b + c = 1): 5 * (a^2 + b^2 + c^2) ≤ 6 * (a^3 + b^3 + c^3) + 1 := by
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schur_p5
|
theorem schur_p5 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a + b > c) (hbc : b + c > a) (hca : c + a > b) : 2 * a^2 * (b + c) + 2 * b^2 * (c + a) + 2 * c^2 * (a + b) ≥ a^3 + b^3 + c^3 + 9 * a * b * c := by
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sq_p1
|
theorem sq_p1 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a+b) * (b+c) * (c+a) ≥ 8 * a * b * c := by
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sq_p2
|
theorem sq_p2 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a^2 * b^2 + b^2 * c^2 + c^2 * a^2 ≥ a * b * c * (a + b + c) := by
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sq_p3
|
theorem sq_p3 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a ^ 4 + b^4 + c^4 ≥ a * b * c * (a + b + c) := by
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sq_p4
|
theorem sq_p4 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a+b+c)^3 ≥ 27 * a * b * c := by
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sq_p5
|
theorem sq_p5 (a b c d: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) : a^2 + b^2 + c^2 + d^2 ≥ a*b + b*c + c*d + d*a := by
|
Introduction
We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers—including Goedel, STP, and Kimina-7B—struggle significantly. DeepSeek-Prover-V2 shows relative robustness—possibly because it is trained to decompose the problems into sub-problems—but still suffers a 20% performance drop (pass@32). Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition.
Quick Start
The proof of the seed problems and the evaluation scripts can be found at https://github.com/haoyuzhao123/LeanIneqComp
Ineq-Comp Benchmark
We provide 5 splits: seed, type1, type2, mix, and real. For seed, type1, and type2 splits, each contains 75 problems. mix split contains 100 problems generated by Ineq-Mix, and real split contains 50 real-world inequality problems. Please refer to the github repo for more fine-grained splits.
Citation
@article{zhao2025ineq,
title={Ineq-Comp: Benchmarking Human-Intuitive Compositional Reasoning in Automated Theorem Proving on Inequalities},
author={Zhao, Haoyu and Geng, Yihan and Tang, Shange and Lin, Yong and Lyu, Bohan and Lin, Hongzhou and Jin, Chi and Arora, Sanjeev},
journal={arXiv preprint arXiv:2505.12680},
year={2025}
}
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