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misc_type1.jsonl
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{"name": "jensen_p11", "
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{"name": "jensen_p12", "
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{"name": "jensen_p13", "
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{"name": "jensen_p14", "
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{"name": "jensen_p15", "
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{"name": "jensen_p16", "
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{"name": "jensen_p17", "
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{"name": "jensen_p18", "
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{"name": "jensen_p19", "
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{"name": "jensen_p20", "
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{"name": "induction_p6", "
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{"name": "induction_p7", "
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{"name": "induction_p8", "
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{"name": "induction_p9", "
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{"name": "induction_p10", "
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{"name": "schur_p6", "
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{"name": "schur_p7", "
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{"name": "schur_p8", "
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{"name": "schur_p9", "
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{"name": "schur_p10", "
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{"name": "sq_p6", "
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{"name": "sq_p7", "
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{"name": "sq_p8", "
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{"name": "sq_p9", "
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{"name": "sq_p10", "
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{"name": "jensen_p11", "formal_statement": "theorem jensen_p11 (x y z w : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) : ((1:ℝ)/3 * x + (2:ℝ)/3 * y) ^ 4 * ((1:ℝ)/3 * z + (2:ℝ)/3 * w) ^ 4 ≤ ((1:ℝ)/3 * x^4 + (2:ℝ)/3 * y ^ 4) * ((1:ℝ)/3 * z^4 + (2:ℝ)/3 * w ^ 4) := by\n"}
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{"name": "jensen_p12", "formal_statement": "theorem jensen_p12 (x y z w: ℝ) : Real.exp ((1:ℝ)/4 * x + (3:ℝ)/4 * y) * Real.exp ((1:ℝ)/4 * z + (3:ℝ)/4 * w) ≤ ((1:ℝ)/4 * Real.exp x + (3:ℝ)/4 * Real.exp y) * ((1:ℝ)/4 * Real.exp z + (3:ℝ)/4 * Real.exp w) := by\n"}
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{"name": "jensen_p13", "formal_statement": "theorem jensen_p13 (x y z w: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0): ((1:ℝ)/4 * x + (3:ℝ)/4 * y) * Real.log ((1:ℝ)/4 * x + (3:ℝ)/4 * y) + ((1:ℝ)/4 * z + (3:ℝ)/4 * w) * Real.log ((1:ℝ)/4 * z + (3:ℝ)/4 * w) ≤ ((1:ℝ)/4 * x * Real.log x + (3:ℝ)/4 * y * Real.log y) + ((1:ℝ)/4 * z * Real.log z + (3:ℝ)/4 * w * Real.log w) := by\n"}
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{"name": "jensen_p14", "formal_statement": "theorem jensen_p14 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : x + y + z = 3) (g : a + b + c = 3) : ((1:ℝ)/3 * x^6 + (1:ℝ)/3 * y ^ 6 + (1:ℝ)/3 * z ^ 6) * ((1:ℝ)/3 * a^6 + (1:ℝ)/3 * b ^ 6 + (1:ℝ)/3 * c ^ 6) ≥ 1 := by\n"}
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{"name": "jensen_p15", "formal_statement": "theorem jensen_p15 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 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 * a ^ ((1:ℝ)/3) + (3:ℝ)/8 * b ^ ((1:ℝ)/3) + (3:ℝ)/8 * c ^ ((1:ℝ)/3)) ≤ ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) ^ ((1:ℝ)/3) * ((1:ℝ)/4 * a + (3:ℝ)/8 * b + (3:ℝ)/8 * c) ^ ((1:ℝ)/3) := by\n"}
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{"name": "jensen_p16", "formal_statement": "theorem jensen_p16 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0): ((1:ℝ)/4 * Real.log x + (3:ℝ)/8 * Real.log y + (3:ℝ)/8 * Real.log z) + ((1:ℝ)/4 * Real.log a + (3:ℝ)/8 * Real.log b + (3:ℝ)/8 * Real.log c) ≤ (Real.log ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z)) + (Real.log ((1:ℝ)/4 * a + (3:ℝ)/8 * b + (3:ℝ)/8 * c)) := by\n"}
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{"name": "jensen_p17", "formal_statement": "theorem jensen_p17 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0): ((1:ℝ)/4 * Real.sqrt x + (3:ℝ)/8 * Real.sqrt y + (3:ℝ)/8 * Real.sqrt z) * ((1:ℝ)/4 * Real.sqrt a + (3:ℝ)/8 * Real.sqrt b + (3:ℝ)/8 * Real.sqrt c) ≤ (Real.sqrt ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z)) * (Real.sqrt ((1:ℝ)/4 * a + (3:ℝ)/8 * b + (3:ℝ)/8 * c)) := by\n"}
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{"name": "jensen_p18", "formal_statement": "theorem jensen_p18 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : x + y + z = (π:ℝ)) (g : a + b + c = (π:ℝ)): ((1:ℝ)/3 * Real.sin x + (1:ℝ)/3 * Real.sin y + (1:ℝ)/3 * Real.sin z) * ((1:ℝ)/3 * Real.sin a + (1:ℝ)/3 * Real.sin b + (1:ℝ)/3 * Real.sin c) ≤ (3:ℝ)/4 := by\n"}
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{"name": "jensen_p19", "formal_statement": "theorem jensen_p19 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : x + y + z = (π:ℝ)) (g : a + b + c = (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)/3 * Real.sin (a/2) + (1:ℝ)/3 * Real.sin (b/2) + (1:ℝ)/3 * Real.sin (c/2)) ≤ (1:ℝ) / 4 := by\n"}
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{"name": "jensen_p20", "formal_statement": "theorem jensen_p20 (x y z a b c: ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : x + y + z = (π:ℝ)) (g : a + b + c = (π:ℝ)): ((1:ℝ)/3 * Real.cos (x/2) + (1:ℝ)/3 * Real.cos (y/2) + (1:ℝ)/3 * Real.cos (z/2)) * ((1:ℝ)/3 * Real.cos (a/2) + (1:ℝ)/3 * Real.cos (b/2) + (1:ℝ)/3 * Real.cos (c/2)) ≤ (3:ℝ)/4 := by\n"}
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{"name": "induction_p6", "formal_statement": "theorem induction_p6 (n m: ℕ) (h : n ≥ 4) (g : m ≥ 4) : 2 ^ n * 2 ^ m ≥ (n + 1) * (m + 1) := by\n"}
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{"name": "induction_p7", "formal_statement": "theorem induction_p7 (x y: ℝ) (n m: ℕ) (h₀ : 0 < x) (h₁ : 0 < n) (g₀ : 0 < y) (g₁ : 0 < m): (1 + ↑n * x) * (1 + ↑m * y) ≤ (1 + x) ^ (n : ℕ) * (1 + y) ^ (m : ℕ) := by\n"}
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{"name": "induction_p8", "formal_statement": "theorem induction_p8 (n m: ℕ) (h : 4 ≤ n) (g : 4 ≤ m) : n ^ 2 * m ^ 2 ≤ (n !) * (m !) := by\n"}
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{"name": "induction_p9", "formal_statement": "theorem induction_p9 (n m: ℕ) (h : 3 ≤ n) (g : 3 ≤ m) : (n !) * (m !) < n ^ (n - 1) * m ^ (m - 1) := by\n"}
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{"name": "induction_p10", "formal_statement": "theorem induction_p10 (n m: ℕ) (h : 0 < n) (g : 0 < m) : (∏ k in Finset.Icc 1 n, (1 + (1 : ℝ) / k ^ 3)) * (∏ k in Finset.Icc 1 m, (1 + (1 : ℝ) / k ^ 3)) ≤ ((3 : ℝ) - 1 / ↑n) * ((3 : ℝ) - 1 / ↑m) := by\n"}
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{"name": "schur_p6", "formal_statement": "theorem schur_p6 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : a * b * c = 1) (g : x * y * z = 1) : (a - 1 + 1 / b) * (b - 1 + 1 / c) * (c - 1 + 1 / a) + (x - 1 + 1 / y) * (y - 1 + 1 / z) * (z - 1 + 1 / x) ≤ 2 := by\n"}
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{"name": "schur_p7", "formal_statement": "theorem schur_p7 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : a * b * c = 1) (g : x * y * z = 1) : (3 + a / b + b / c + c / a) * (3 + x / y + y / z + z / x) ≥ (a + b + c + 1 / a + 1 / b + 1 / c) * (x + y + z + 1 / x + 1 / y + 1 / z) := by\n"}
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{"name": "schur_p8", "formal_statement": "theorem schur_p8 (a b c x y z t s: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hab : a ≥ b)(hbc : b ≥ c) (hxy : x ≥ y)(hyz : y ≥ z) (ht : t > 0) (hs : s > 0) : (a^t * (a - b) * (a - c) + b^t * (b - c) * (b - a) + c^t * (c - a) * (c - b)) * (x^s * (x - y) * (x - z) + y^s * (y - z) * (y - x) + z^s * (z - x) * (z - y)) ≥ 0 := by\n"}
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{"name": "schur_p9", "formal_statement": "theorem schur_p9 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : a + b + c = 1) (g : x + y + z = 1): 25 * (a^2 + b^2 + c^2) * (x^2 + y^2 + z^2) ≤ (6 * (a^3 + b^3 + c^3) + 1) * (6 * (x^3 + y^3 + z^3) + 1) := by\n"}
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{"name": "schur_p10", "formal_statement": "theorem schur_p10 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hab : a + b > c) (hbc : b + c > a) (hca : c + a > b) (hxy : x + y > z) (hyz : y + z > x) (hzx : z + x > y) : (2 * a^2 * (b + c) + 2 * b^2 * (c + a) + 2 * c^2 * (a + b)) * (2 * x^2 * (y + z) + 2 * y^2 * (z + x) + 2 * z^2 * (x + y)) ≥ (a^3 + b^3 + c^3 + 9 * a * b * c) * (x^3 + y^3 + z^3 + 9 * x * y * z) := by\n"}
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{"name": "sq_p6", "formal_statement": "theorem sq_p6 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (a+b) * (b+c) * (c+a) * (x+y) * (y+z) * (z+x) ≥ 64 * a * b * c * x * y * z := by\n"}
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{"name": "sq_p7", "formal_statement": "theorem sq_p7 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (a^2 * b^2 + b^2 * c^2 + c^2 * a^2) * (x^2 * y^2 + y^2 * z^2 + z^2 * x^2) ≥ a * b * c * (a + b + c) * x * y * z * (x + y + z) := by\n"}
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{"name": "sq_p8", "formal_statement": "theorem sq_p8 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (a^4 + b^4 + c^4) * (x^4 + y^4 + z^4) ≥ a * b * c * (a + b + c) * x * y * z * (x + y + z) := by\n"}
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{"name": "sq_p9", "formal_statement": "theorem sq_p9 (a b c x y z: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (a+b+c)^3 * (x+y+z)^3 ≥ 27 * a * b * c * 27 * x * y * z := by\n"}
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{"name": "sq_p10", "formal_statement": "theorem sq_p10 (a b c d x y z w: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) : (a^2 + b^2 + c^2 + d^2) * (x^2 + y^2 + z^2 + w^2) ≥ (a*b + b*c + c*d + d*a) * (x*y + y*z + z*w + w*x) := by\n"}
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