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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
induction_p1
|
theorem induction_p1 (n : ℕ) (h : n ≥ 4) : 2 ^ n ≥ n + 1 := by
|
induction_p2
|
theorem induction_p2 (x : ℝ) (n : ℕ) (h₀ : -1 < x) (h₁ : 0 < n) : 1 + ↑n * x ≤ (1 + x) ^ (n : ℕ) := by
|
induction_p3
|
theorem induction_p3 (n : ℕ) (h₀ : 4 ≤ n) : n ^ 2 ≤ n ! := by
|
induction_p4
|
theorem induction_p4 (n : ℕ) (h₀ : 3 ≤ n) : n ! < n ^ (n - 1) := by
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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