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4302c7830e25eb00
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2019_p21 (z : β„‚) (hβ‚€ : z = (1 + Complex.I) / Real.sqrt 2) : (βˆ‘ k ∈ Finset.Icc 1 12, z ^ k ^ 2) * βˆ‘ k ∈ Finset.Icc 1 12, 1 / z ^ k ^ 2 = (36) := by
z : β„‚ hβ‚€ : z = (1 + Complex.I) / β†‘βˆš2 ⊒ (βˆ‘ k ∈ Finset.Icc 1 12, z ^ k ^ 2) * βˆ‘ k ∈ Finset.Icc 1 12, 1 / z ^ k ^ 2 = 36
38a3b362ac4b77c8
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2015_p10 (x y : β„€) (hβ‚€ : 0 < y) (h₁ : y < x) (hβ‚‚ : x + y + x * y = 80) : x = (26) := by
x y : β„€ hβ‚€ : 0 < y h₁ : y < x hβ‚‚ : x + y + x * y = 80 ⊒ x = 26
f897047cc20bfb4d
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2008_p8 (x y : ℝ) (hβ‚€ : 0 < x ∧ 0 < y) (h₁ : y ^ 3 = 1) (hβ‚‚ : 6 * x ^ 2 = 2 * (6 * y ^ 2)) : x ^ 3 = (2 * Real.sqrt 2) := by
x y : ℝ hβ‚€ : 0 < x ∧ 0 < y h₁ : y ^ 3 = 1 hβ‚‚ : 6 * x ^ 2 = 2 * (6 * y ^ 2) ⊒ x ^ 3 = 2 * √2
9d8b27f7b8ae39f5
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_182 (y : β„‚) : 7 * (3 * y + 2) = (21 * y + 14) := by
y : β„‚ ⊒ 7 * (3 * y + 2) = 21 * y + 14
5e63c8dccd0f72c1
import Mathlib open scoped Nat Real Topology Polynomial theorem aime_1984_p5 (a b : ℝ) (hβ‚€ : 0 < a ∧ 0 < b) (h₁ : Real.logb 8 a + Real.logb 4 (b ^ 2) = 5) (hβ‚‚ : Real.logb 8 b + Real.logb 4 (a ^ 2) = 7) : a * b = (512) := by
a b : ℝ hβ‚€ : 0 < a ∧ 0 < b h₁ : Real.logb 8 a + Real.logb 4 (b ^ 2) = 5 hβ‚‚ : Real.logb 8 b + Real.logb 4 (a ^ 2) = 7 ⊒ a * b = 512
c26b3ff844891327
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_780 (m x : β„€) (hβ‚€ : 10 ≀ m) (h₁ : m ≀ 99) (hβ‚‚ : 6 * x % m = 1) (h₃ : (x - 6 ^ 2) % m = 0) : m = (43) := by
m x : β„€ hβ‚€ : 10 ≀ m h₁ : m ≀ 99 hβ‚‚ : 6 * x % m = 1 h₃ : (x - 6 ^ 2) % m = 0 ⊒ m = 43
f2b3cbe0e9377859
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_116 (k x : ℝ) (hβ‚€ : x = (13 - Real.sqrt 131) / 4) (h₁ : 2 * x ^ 2 - 13 * x + k = 0) : k = (19 / 4) := by
k x : ℝ hβ‚€ : x = (13 - √131) / 4 h₁ : 2 * x ^ 2 - 13 * x + k = 0 ⊒ k = 19 / 4
2701d78a75a79cc5
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_13 (u v : β„•) (S : Set β„•) (hβ‚€ : βˆ€ n : β„•, n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46) (h₁ : IsLeast S u) (hβ‚‚ : IsLeast (S \ {u}) v) : (u + v : β„š) / 2 = (64) := by
u v : β„• S : Set β„• hβ‚€ : βˆ€ (n : β„•), n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46 h₁ : IsLeast S u hβ‚‚ : IsLeast (S \ {u}) v ⊒ (↑u + ↑v) / 2 = 64
f92de45dcfb1fa02
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_169 : Nat.gcd 20! 200000 = (40000) := by
⊒ 20!.gcd 200000 = 40000
1aa15fdcb17604bc
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2009_p9 (a b c : ℝ) (f : ℝ β†’ ℝ) (hβ‚€ : βˆ€ x, f (x + 3) = 3 * x ^ 2 + 7 * x + 4) (h₁ : βˆ€ x, f x = a * x ^ 2 + b * x + c) : a + b + c = (2) := by
a b c : ℝ f : ℝ β†’ ℝ hβ‚€ : βˆ€ (x : ℝ), f (x + 3) = 3 * x ^ 2 + 7 * x + 4 h₁ : βˆ€ (x : ℝ), f x = a * x ^ 2 + b * x + c ⊒ a + b + c = 2
dd930016cd671a49
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2019_p9 (a : β„• β†’ β„š) (hβ‚€ : a 1 = 1) (h₁ : a 2 = 3 / 7) (hβ‚‚ : βˆ€ n β‰₯ 3, a n = a (n - 2 : β„•) * a (n - 1 : β„•) / (2 * a (n - 2 : β„•) - a (n - 1 : β„•))) : ↑(a 2019).den + (a 2019).num = (8078) := by
a : β„• β†’ β„š hβ‚€ : a 1 = 1 h₁ : a 2 = 3 / 7 hβ‚‚ : βˆ€ n β‰₯ 3, a n = a (n - 2) * a (n - 1) / (2 * a (n - 2) - a (n - 1)) ⊒ ↑(a 2019).den + (a 2019).num = 8078
e6daeda4bffdac89
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_13 (a b : ℝ) (hβ‚€ : βˆ€ x, x - 3 β‰  0 ∧ x - 5 β‰  0 β†’ 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)) : (a, b) = ((-6, 10)) := by
a b : ℝ hβ‚€ : βˆ€ (x : ℝ), x - 3 β‰  0 ∧ x - 5 β‰  0 β†’ 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5) ⊒ (a, b) = (-6, 10)
a6dc556812ddc873
import Mathlib open scoped Nat Real Topology Polynomial theorem induction_sum2kp1npqsqm1 (n : β„•) : ↑(βˆ‘ k ∈ Finset.range n, (2 * k + 3)) = ↑(n + 1) ^ 2 - (1 : β„€) := by
n : β„• ⊒ ↑(βˆ‘ k ∈ Finset.range n, (2 * k + 3)) = ↑(n + 1) ^ 2 - 1
593155e948849d68
import Mathlib open scoped Nat Real Topology Polynomial theorem aime_1991_p6 (r : ℝ) (hβ‚€ : βˆ‘ k ∈ Finset.Icc (19 : β„•) 91, Int.floor (r + k / 100) = 546) : Int.floor (100 * r) = (743) := by
r : ℝ hβ‚€ : βˆ‘ k ∈ Finset.Icc 19 91, ⌊r + ↑k / 100βŒ‹ = 546 ⊒ ⌊100 * rβŒ‹ = 743
33406df7bf1b843f
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_149 : βˆ‘ k ∈ Finset.filter (fun x => x % 8 = 5 ∧ x % 6 = 3) (Finset.range 50), k = (66) := by
⊒ βˆ‘ k ∈ Finset.range 50 with k % 8 = 5 ∧ k % 6 = 3, k = 66
0f182193599dca48
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1984_p2 (a b : β„•) (hβ‚€ : 0 < a ∧ 0 < b) (h₁ : Β¬7 ∣ a) (hβ‚‚ : Β¬7 ∣ b) (h₃ : Β¬7 ∣ a + b) (hβ‚„ : 7 ^ 7 ∣ (a + b) ^ 7 - a ^ 7 - b ^ 7) : 19 ≀ a + b := by
a b : β„• hβ‚€ : 0 < a ∧ 0 < b h₁ : Β¬7 ∣ a hβ‚‚ : Β¬7 ∣ b h₃ : Β¬7 ∣ a + b hβ‚„ : 7 ^ 7 ∣ (a + b) ^ 7 - a ^ 7 - b ^ 7 ⊒ 19 ≀ a + b
ca716533245de685
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2008_p4 : ∏ k ∈ Finset.Icc (1 : β„•) 501, ((4 : ℝ) * k + 4) / (4 * k) = (502) := by
⊒ ∏ k ∈ Finset.Icc 1 501, (4 * ↑k + 4) / (4 * ↑k) = 502
732bf16fa733e606
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_2006_p3 : IsLeast {M : ℝ | βˆ€ a b c : ℝ, abs (a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)) ≀ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2} (9 * Real.sqrt 2 / 32) := by
⊒ IsLeast {M | βˆ€ (a b c : ℝ), |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≀ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2} (9 * √2 / 32)
996b38a1fc8ee61a
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_462 : ((1 : β„š) / 2 + 1 / 3) * (1 / 2 - 1 / 3) = (5 / 36) := by
⊒ (1 / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36
aaf43df039ab2ec3
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1964_p1_2 (n : β„•) : Β¬7 ∣ 2 ^ n + 1 := by
n : β„• ⊒ Β¬7 ∣ 2 ^ n + 1
95ccd12e5096d350
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_221 (S : Finset β„•) (hβ‚€ : βˆ€ x : β„•, x ∈ S ↔ 0 < x ∧ x < 1000 ∧ x.divisors.card = 3) : S.card = (11) := by
S : Finset β„• hβ‚€ : βˆ€ (x : β„•), x ∈ S ↔ 0 < x ∧ x < 1000 ∧ x.divisors.card = 3 ⊒ S.card = 11
095e7c1c8e3af948
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_64 : IsLeast {x : β„• | 30 * x ≑ 42 [MOD 47]} (39) := by
⊒ IsLeast {x | 30 * x ≑ 42 [MOD 47]} 39
e296ed472df217f2
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1987_p4 (f : β„• β†’ β„•) : βˆƒ n, f (f n) β‰  n + 1987 := by
f : β„• β†’ β„• ⊒ βˆƒ n, f (f n) β‰  n + 1987
2cbbc602d318190d
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_33 : βˆƒ (n : β„•), n < 398 ∧ n * 7 % 398 = 1 := by
⊒ βˆƒ n < 398, n * 7 % 398 = 1
95192a71852300c8
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12_2001_p9 (f : ℝ β†’ ℝ) (hβ‚€ : βˆ€ x > 0, βˆ€ y > 0, f (x * y) = f x / y) (h₁ : f 500 = 3) : f 600 = (5 / 2) := by
f : ℝ β†’ ℝ hβ‚€ : βˆ€ x > 0, βˆ€ y > 0, f (x * y) = f x / y h₁ : f 500 = 3 ⊒ f 600 = 5 / 2
140de00a6ace0b94
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1965_p1 : {x : ℝ | 0 ≀ x ∧ x ≀ 2 * Ο€ ∧ 2 * Real.cos x ≀ abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) ∧ abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) ≀ Real.sqrt 2} = (Set.Ic...
⊒ {x | 0 ≀ x ∧ x ≀ 2 * Ο€ ∧ 2 * Real.cos x ≀ |√(1 + Real.sin (2 * x)) - √(1 - Real.sin (2 * x))| ∧ |√(1 + Real.sin (2 * x)) - √(1 - Real.sin (2 * x))| ≀ √2} = Set.Icc (Ο€ / 4) (7 * Ο€ / 4)
6233383f12eabd6c
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_48 (b : β„•) (hβ‚€ : 0 < b) (h₁ : 3 * b ^ 2 + 2 * b + 1 = 57) : b = (4) := by
b : β„• hβ‚€ : 0 < b h₁ : 3 * b ^ 2 + 2 * b + 1 = 57 ⊒ b = 4
65b56dd5b158b479
import Mathlib open scoped Nat Real Topology Polynomial theorem numbertheory_sqmod4in01d (a : β„€) : a ^ 2 % 4 = 0 ∨ a ^ 2 % 4 = 1 := by
a : β„€ ⊒ a ^ 2 % 4 = 0 ∨ a ^ 2 % 4 = 1
ad9eb02ddb681cc1
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_466 : (βˆ‘ k ∈ Finset.range 11, k) % 9 = (1) := by
⊒ (βˆ‘ k ∈ Finset.range 11, k) % 9 = 1
11b5722d8a500b1d
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_48 (q e : β„‚) (hβ‚€ : q = 9 - 4 * Complex.I) (h₁ : e = -3 - 4 * Complex.I) : q - e = (12) := by
q e : β„‚ hβ‚€ : q = 9 - 4 * Complex.I h₁ : e = -3 - 4 * Complex.I ⊒ q - e = 12
59c1ada943500f1a
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12_2000_p15 (f : β„‚ β†’ β„‚) (hβ‚€ : βˆ€ x, f (x / 3) = x ^ 2 + x + 1) (h₁ : Fintype (f ⁻¹' {7})) : βˆ‘ y ∈ (f ⁻¹' {7}).toFinset, y / 3 = (-1 / 9) := by
f : β„‚ β†’ β„‚ hβ‚€ : βˆ€ (x : β„‚), f (x / 3) = x ^ 2 + x + 1 h₁ : Fintype ↑(f ⁻¹' {7}) ⊒ βˆ‘ y ∈ (f ⁻¹' {7}).toFinset, y / 3 = -1 / 9
a096805797f5dfd1
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_132 : 2004 % 12 = (0) := by
⊒ 2004 % 12 = 0
7a42500799290a6b
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2009_p5 (x : ℝ) (hβ‚€ : x ^ 3 - (x + 1) * (x - 1) * x = 5) : x ^ 3 = (125) := by
x : ℝ hβ‚€ : x ^ 3 - (x + 1) * (x - 1) * x = 5 ⊒ x ^ 3 = 125
4fb65db037d80b12
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_188 : Nat.gcd 180 168 = (12) := by
⊒ Nat.gcd 180 168 = 12
1eb8300c0a409f42
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_224 (S : Finset β„•) (hβ‚€ : βˆ€ n : β„•, n ∈ S ↔ Real.sqrt n < 7 / 2 ∧ 2 < Real.sqrt n) : S.card = (8) := by
S : Finset β„• hβ‚€ : βˆ€ (n : β„•), n ∈ S ↔ βˆšβ†‘n < 7 / 2 ∧ 2 < βˆšβ†‘n ⊒ S.card = 8
e7cb9b8380d9c339
import Mathlib open scoped Nat Real Topology Polynomial theorem induction_divisibility_3divnto3m2n (n : β„•) : 3 ∣ n ^ 3 + 2 * n := by
n : β„• ⊒ 3 ∣ n ^ 3 + 2 * n
0324044868d977ad
import Mathlib open scoped Nat Real Topology Polynomial theorem induction_sum_1oktkp1 (n : β„•) : βˆ‘ k ∈ Finset.range n, (1 : ℝ) / ((k + 1) * (k + 2)) = n / (n + 1) := by
n : β„• ⊒ βˆ‘ k ∈ Finset.range n, 1 / ((↑k + 1) * (↑k + 2)) = ↑n / (↑n + 1)
2d4c2fcd5b1dc617
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_32 (S : Finset β„•) (hβ‚€ : βˆ€ n : β„•, n ∈ S ↔ n ∣ 36) : βˆ‘ k ∈ S, k = (91) := by
S : Finset β„• hβ‚€ : βˆ€ (n : β„•), n ∈ S ↔ n ∣ 36 ⊒ βˆ‘ k ∈ S, k = 91
33cf4cd6541974a0
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_422 (x : ℝ) (Οƒ : Equiv ℝ ℝ) (hβ‚€ : βˆ€ x, Οƒ.1 x = 5 * x - 12) (h₁ : Οƒ.1 (x + 1) = Οƒ.2 x) : x = (47 / 24) := by
x : ℝ Οƒ : ℝ ≃ ℝ hβ‚€ : βˆ€ (x : ℝ), Οƒ.toFun x = 5 * x - 12 h₁ : Οƒ.toFun (x + 1) = Οƒ.invFun x ⊒ x = 47 / 24
1b709f98a18c5f97
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12b_2002_p11 (a b : β„•) (hβ‚€ : Nat.Prime a) (h₁ : Nat.Prime b) (hβ‚‚ : Nat.Prime (a + b)) (h₃ : Nat.Prime (a - b : β„•)) : letI P : β„• β†’ Prop := (Nat.Prime) P (a + b + ((a - b : β„•) + (a + b))) := by
a b : β„• hβ‚€ : Nat.Prime a h₁ : Nat.Prime b hβ‚‚ : Nat.Prime (a + b) h₃ : Nat.Prime (a - b) ⊒ Nat.Prime (a + b + (a - b + (a + b)))
ec83b4154c4c24fd
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_73 (p q r x : β„‚) (hβ‚€ : (x - p) * (x - q) = (r - p) * (r - q)) (h₁ : x β‰  r) : x = (p + q - r) := by
p q r x : β„‚ hβ‚€ : (x - p) * (x - q) = (r - p) * (r - q) h₁ : x β‰  r ⊒ x = p + q - r
59640eba40e7b6a3
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_109 (v : β„• β†’ β„•) (hβ‚€ : βˆ€ n, v n = (2 * n - 1 : β„•)) : (βˆ‘ k ∈ Finset.Icc 1 100, v k) % 7 = (4) := by
v : β„• β†’ β„• hβ‚€ : βˆ€ (n : β„•), v n = 2 * n - 1 ⊒ (βˆ‘ k ∈ Finset.Icc 1 100, v k) % 7 = 4
85ee06a42648499e
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : β„€) (hβ‚€ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) : x + y + z + 6 ∣ x ^ 3 + y ^ 3 + z ^ 3 := by
x y z : β„€ hβ‚€ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z ⊒ x + y + z + 6 ∣ x ^ 3 + y ^ 3 + z ^ 3
815cda9b1ebe631c
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1962_p4 (S : Set ℝ) (hβ‚€ : S = {x : ℝ | Real.cos x ^ 2 + Real.cos (2 * x) ^ 2 + Real.cos (3 * x) ^ 2 = 1}) : S = ({x : ℝ | βˆƒ m : β„€, x = Ο€ / 2 + m * Ο€ ∨ x = Ο€ / 4 + m * Ο€ / 2 ∨ x = Ο€ / 6 + m * Ο€ ∨ x = 5 * Ο€ / 6 + m * Ο€}) := by
S : Set ℝ hβ‚€ : S = {x | Real.cos x ^ 2 + Real.cos (2 * x) ^ 2 + Real.cos (3 * x) ^ 2 = 1} ⊒ S = {x | βˆƒ m, x = Ο€ / 2 + ↑m * Ο€ ∨ x = Ο€ / 4 + ↑m * Ο€ / 2 ∨ x = Ο€ / 6 + ↑m * Ο€ ∨ x = 5 * Ο€ / 6 + ↑m * Ο€}
1b575acce73e9c03
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_236 : 1999 ^ 2000 % 5 = (1) := by
⊒ 1999 ^ 2000 % 5 = 1
d7ddead4e423d260
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_24 : (βˆ‘ k ∈ Finset.Icc 1 9, 11 ^ k) % 100 = (59) := by
⊒ (βˆ‘ k ∈ Finset.Icc 1 9, 11 ^ k) % 100 = 59
2cfd271b26af4e28
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_amgm_prod1toneq1_sum1tongeqn (a : β„• β†’ NNReal) (n : β„•) (hβ‚€ : ∏ i ∈ Finset.range n, a i = 1) : βˆ‘ i ∈ Finset.range n, a i β‰₯ n := by
a : β„• β†’ NNReal n : β„• hβ‚€ : ∏ i ∈ Finset.range n, a i = 1 ⊒ βˆ‘ i ∈ Finset.range n, a i β‰₯ ↑n
40aed4f1ed212d67
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_101 : {x : ℝ | x ^ 2 - 5 * x - 4 ≀ 10} = (Set.Icc (-2) 7) := by
⊒ {x | x ^ 2 - 5 * x - 4 ≀ 10} = Set.Icc (-2) 7
7faae039f80262a7
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_257 (x : β„•) (hβ‚€ : 1 ≀ x ∧ x ≀ 100) (h₁ : 77 ∣ (βˆ‘ k ∈ Finset.range 101, k - x : β„•)) : x = (45) := by
x : β„• hβ‚€ : 1 ≀ x ∧ x ≀ 100 h₁ : 77 ∣ βˆ‘ k ∈ Finset.range 101, k - x ⊒ x = 45
04c0486b1a87b44e
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12_2000_p5 (x p : ℝ) (hβ‚€ : x < 2) (h₁ : abs (x - 2) = p) : x - p = (2 - 2 * p) := by
x p : ℝ hβ‚€ : x < 2 h₁ : |x - 2| = p ⊒ x - p = 2 - 2 * p
233baeacd990da62
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_547 (x y : ℝ) (hβ‚€ : x = 5) (h₁ : y = 2) : Real.sqrt (x ^ 3 - 2 ^ y) = (11) := by
x y : ℝ hβ‚€ : x = 5 h₁ : y = 2 ⊒ √(x ^ 3 - 2 ^ y) = 11
30f95da79bfd3a93
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_200 : 139 % 11 = (7) := by
⊒ 139 % 11 = 7
3612dde9f76b7c3f
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_510 (x y : ℝ) (hβ‚€ : x + y = 13) (h₁ : x * y = 24) : Real.sqrt (x ^ 2 + y ^ 2) = (11) := by
x y : ℝ hβ‚€ : x + y = 13 h₁ : x * y = 24 ⊒ √(x ^ 2 + y ^ 2) = 11
4eae6ed0f748e988
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_140 (a b c : ℝ) (hβ‚€ : 0 < a ∧ 0 < b ∧ 0 < c) (h₁ : βˆ€ x, 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c)) : a * b - 3 * c = (-9) := by
a b c : ℝ hβ‚€ : 0 < a ∧ 0 < b ∧ 0 < c h₁ : βˆ€ (x : ℝ), 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c) ⊒ a * b - 3 * c = -9
fa89a7da02c5e78f
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_455 (x : ℝ) (hβ‚€ : 2 * (2 * (2 * (2 * x))) = 48) : x = (3) := by
x : ℝ hβ‚€ : 2 * (2 * (2 * (2 * x))) = 48 ⊒ x = 3
2f1d523900798a1d
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = (23) := by
⊒ Nat.gcd 6432 132 + 11 = 23
72ebb485ab03d1db
import Mathlib open scoped Nat Real Topology Polynomial theorem aime_1994_p4 (n : β„•) (hβ‚€ : n > 0) (h₁ : βˆ‘ k ∈ Finset.Icc 1 n, Int.floor (Real.logb 2 k) = 1994) : n = (312) := by
n : β„• hβ‚€ : n > 0 h₁ : βˆ‘ k ∈ Finset.Icc 1 n, ⌊Real.logb 2 ↑kβŒ‹ = 1994 ⊒ n = 312
9b7530a5896da2fd
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_739 : 9! % 10 = (0) := by
⊒ 9! % 10 = 0
2fda6a1b55630219
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_245 (x : ℝ) (hβ‚€ : x β‰  0) : (4 / x)⁻¹ * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))⁻¹ ^ 3 = (18 * x ^ 8) := by
x : ℝ hβ‚€ : x β‰  0 ⊒ (4 / x)⁻¹ * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))⁻¹ ^ 3 = 18 * x ^ 8
a84cfa87cb3bded8
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_apb4leq8ta4pb4 (a b : ℝ) (hβ‚€ : 0 < a ∧ 0 < b) : (a + b) ^ 4 ≀ 8 * (a ^ 4 + b ^ 4) := by
a b : ℝ hβ‚€ : 0 < a ∧ 0 < b ⊒ (a + b) ^ 4 ≀ 8 * (a ^ 4 + b ^ 4)
504a03bd394a9b78
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_28 : IsGreatest {c : ℝ | βˆƒ x : ℝ, 2 * x ^ 2 + 5 * x + c = 0} (25 / 8) := by
⊒ IsGreatest {c | βˆƒ x, 2 * x ^ 2 + 5 * x + c = 0} (25 / 8)
c1abccbe022c0cba
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_543 : (βˆ‘ k ∈ Nat.divisors (30 ^ 4), 1 - 2 : β„•) = (123) := by
⊒ βˆ‘ k ∈ (30 ^ 4).divisors, 1 - 2 = 123
bc349c831538aca6
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_480 (f : ℝ β†’ ℝ) (hβ‚€ : βˆ€ x < 0, f x = -x ^ 2 - 1) (h₁ : βˆ€ x, 0 ≀ x ∧ x < 4 β†’ f x = 2) (hβ‚‚ : βˆ€ x β‰₯ 4, f x = Real.sqrt x) : f Ο€ = (2) := by
f : ℝ β†’ ℝ hβ‚€ : βˆ€ x < 0, f x = -x ^ 2 - 1 h₁ : βˆ€ (x : ℝ), 0 ≀ x ∧ x < 4 β†’ f x = 2 hβ‚‚ : βˆ€ x β‰₯ 4, f x = √x ⊒ f Ο€ = 2
97442ef8a2ddc6eb
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_69 (rows seats : β„•) (hβ‚€ : rows * seats = 450) (h₁ : (rows + 5) * (seats - 3) = 450) : rows = (25) := by
rows seats : β„• hβ‚€ : rows * seats = 450 h₁ : (rows + 5) * (seats - 3) = 450 ⊒ rows = 25
4558ece0f712e4ac
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_433 (f : ℝ β†’ ℝ) (hβ‚€ : βˆ€ x, f x = 3 * Real.sqrt (2 * x - 7) - 8) : f 8 = (1) := by
f : ℝ β†’ ℝ hβ‚€ : βˆ€ (x : ℝ), f x = 3 * √(2 * x - 7) - 8 ⊒ f 8 = 1
d12378aec6268835
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_126 (x y : ℝ) (hβ‚€ : 2 * 3 = x - 9) (h₁ : 2 * -5 = y + 1) : (x, y) = ((15, -11)) := by
x y : ℝ hβ‚€ : 2 * 3 = x - 9 h₁ : 2 * -5 = y + 1 ⊒ (x, y) = (15, -11)
a7743ff9a3443664
import Mathlib open scoped Nat Real Topology Polynomial theorem aimeII_2020_p6 (t : β„• β†’ β„š) (hβ‚€ : t 1 = 20) (h₁ : t 2 = 21) (hβ‚‚ : βˆ€ n β‰₯ 3, t n = (5 * t (n - 1 : β„•) + 1) / (25 * t (n - 2 : β„•))) : ↑(t 2020).den + (t 2020).num = (626) := by
t : β„• β†’ β„š hβ‚€ : t 1 = 20 h₁ : t 2 = 21 hβ‚‚ : βˆ€ n β‰₯ 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2)) ⊒ ↑(t 2020).den + (t 2020).num = 626
fb6543226eecc643
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2008_p2 (x : ℝ) (hβ‚€ : x * (1 / 2 + 2 / 3) = 1) : x = (6 / 7) := by
x : ℝ hβ‚€ : x * (1 / 2 + 2 / 3) = 1 ⊒ x = 6 / 7
a0aa41ee445d7147
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_35 (p q : ℝ β†’ ℝ) (hβ‚€ : βˆ€ x, p x = 2 - x ^ 2) (h₁ : βˆ€ x β‰  0, q x = 6 / x) : p (q 2) = (-7) := by
p q : ℝ β†’ ℝ hβ‚€ : βˆ€ (x : ℝ), p x = 2 - x ^ 2 h₁ : βˆ€ (x : ℝ), x β‰  0 β†’ q x = 6 / x ⊒ p (q 2) = -7
62ee38aba94d61f1
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x : βˆ€ x > 0, 2 - Real.sqrt 2 β‰₯ 2 - x - 1 / (2 * x) := by
⊒ βˆ€ x > 0, 2 - √2 β‰₯ 2 - x - 1 / (2 * x)
b5a3aa23e805351a
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_335 (n : β„•) (hβ‚€ : n % 7 = 5) : 5 * n % 7 = (4) := by
n : β„• hβ‚€ : n % 7 = 5 ⊒ 5 * n % 7 = 4
88b9ca94495082c1
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_35 : βˆ‘ k ∈ Nat.divisors (Nat.sqrt 196), k = (24) := by
⊒ βˆ‘ k ∈ (Nat.sqrt 196).divisors, k = 24
4fc20dec3c416f85
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2021_p7 : IsLeast {v : ℝ | βˆƒ x y : ℝ, (x * y - 1) ^ 2 + (x + y) ^ 2 = v} (1) := by
⊒ IsLeast {v | βˆƒ x y, (x * y - 1) ^ 2 + (x + y) ^ 2 = v} 1
155b4681a29a4cf4
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_327 : {a : ℝ | 1 / 5 * abs (9 + 2 * a) < 1} = (Set.Icc (-7) (-2)) := by
⊒ {a | 1 / 5 * |9 + 2 * a| < 1} = Set.Icc (-7) (-2)
627a62ca87713166
import Mathlib open scoped Nat Real Topology Polynomial theorem aime_1984_p15 (x y z w : ℝ) (hβ‚€ : x^2 / (2^2 - 1) + y^2 / (2^2 - 3^2) + z^2 / (2^2 - 5^2) + w^2 / (2^2 - 7^2) = 1) (h₁ : x^2 / (4^2 - 1) + y^2 / (4^2 - 3^2) + z^2 / (4^2 - 5^2) + w^2 / (4^2 - 7^2) = 1) (hβ‚‚ : x^2 / (6^2 - 1) + y^2 / (6^2 - 3^2...
x y z w : ℝ hβ‚€ : x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) + w ^ 2 / (2 ^ 2 - 7 ^ 2) = 1 h₁ : x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) + w ^ 2 / (4 ^ 2 - 7 ^ 2) = 1 hβ‚‚ : x ^ 2 / (6 ^ 2 - 1) + y ^ 2 / (6 ^ 2 - 3 ^ 2) + z ^ 2 / (6 ^ 2 - 5 ^ 2) + w ^ 2 / (6 ^ 2 ...
207b0fffd3200c67
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4 (x y : ℝ) (hβ‚€ : 0 < x ∧ 0 < y) (h₁ : y ≀ x) (hβ‚‚ : Real.sqrt (x * y) * (x - y) = x + y) : x + y β‰₯ 4 := by
x y : ℝ hβ‚€ : 0 < x ∧ 0 < y h₁ : y ≀ x hβ‚‚ : √(x * y) * (x - y) = x + y ⊒ x + y β‰₯ 4
6c80baad2cbf1131
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2002_p21 (u : β„• β†’ β„•) (hβ‚€ : u 0 = 4) (h₁ : u 1 = 7) (hβ‚‚ : βˆ€ n, u (n + 2) = (u n + u (n + 1)) % 10) : IsLeast {n : β„• | βˆ‘ k ∈ Finset.range n, u k > 10000} (1999) := by
u : β„• β†’ β„• hβ‚€ : u 0 = 4 h₁ : u 1 = 7 hβ‚‚ : βˆ€ (n : β„•), u (n + 2) = (u n + u (n + 1)) % 10 ⊒ IsLeast {n | βˆ‘ k ∈ Finset.range n, u k > 10000} 1999
184afa75b49c6932
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_192 (q e d : β„‚) (hβ‚€ : q = 11 - 5 * Complex.I) (h₁ : e = 11 + 5 * Complex.I) (hβ‚‚ : d = 2 * Complex.I) : q * e * d = (292 * Complex.I) := by
q e d : β„‚ hβ‚€ : q = 11 - 5 * Complex.I h₁ : e = 11 + 5 * Complex.I hβ‚‚ : d = 2 * Complex.I ⊒ q * e * d = 292 * Complex.I
00829ff60f18023a
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12b_2002_p6 (a b : ℝ) (hβ‚€ : a β‰  0 ∧ b β‰  0) (h₁ : βˆ€ x, x ^ 2 + a * x + b = (x - a) * (x - b)) : (a, b) = ((1, -2)) := by
a b : ℝ hβ‚€ : a β‰  0 ∧ b β‰  0 h₁ : βˆ€ (x : ℝ), x ^ 2 + a * x + b = (x - a) * (x - b) ⊒ (a, b) = (1, -2)
89a8631a4c5bbd4b
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_102 : 2 ^ 8 % 5 = (1) := by
⊒ 2 ^ 8 % 5 = 1
863e5008caa08fca
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2010_p22 : IsLeast { βˆ‘ k ∈ Finset.Icc 1 119, abs ((k : β„•) * x - 1) | (x : ℝ)} (49) := by
⊒ IsLeast {x | βˆƒ x_1, βˆ‘ k ∈ Finset.Icc 1 119, |↑k * x_1 - 1| = x} 49
f2ef7aee87fe1002
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_81 : 71 % 3 = (2) := by
⊒ 71 % 3 = 2
ed97b12e80195484
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_155 : Finset.card (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)) = (48) := by
⊒ {x ∈ Finset.Icc 100 999 | x % 19 = 7}.card = 48
4f8666e60b5d71a8
import Mathlib open scoped Nat Real Topology Polynomial theorem imo_1978_p5 (n : β„•) (a : β„• β†’ β„•) (hβ‚€ : Function.Injective a) (h₁ : a 0 = 0) (hβ‚‚ : 0 < n) : βˆ‘ k ∈ Finset.Icc 1 n, (1 : ℝ) / k ≀ βˆ‘ k ∈ Finset.Icc 1 n, (a k / k ^ 2 : ℝ) := by
n : β„• a : β„• β†’ β„• hβ‚€ : Function.Injective a h₁ : a 0 = 0 hβ‚‚ : 0 < n ⊒ βˆ‘ k ∈ Finset.Icc 1 n, 1 / ↑k ≀ βˆ‘ k ∈ Finset.Icc 1 n, ↑(a k) / ↑k ^ 2
f829d6e5cd474673
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2017_p7 (f : β„• β†’ ℝ) (hβ‚€ : f 1 = 2) (h₁ : βˆ€ n, 1 < n ∧ Even n β†’ f n = f (n - 1 : β„•) + 1) (hβ‚‚ : βˆ€ n, 1 < n ∧ Odd n β†’ f n = f (n - 2 : β„•) + 2) : f 2017 = (2018) := by
f : β„• β†’ ℝ hβ‚€ : f 1 = 2 h₁ : βˆ€ (n : β„•), 1 < n ∧ Even n β†’ f n = f (n - 1) + 1 hβ‚‚ : βˆ€ (n : β„•), 1 < n ∧ Odd n β†’ f n = f (n - 2) + 2 ⊒ f 2017 = 2018
b1c4f4388d8bc36a
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_42 (S : Set β„•) (u v : β„•) (hβ‚€ : βˆ€ a : β„•, a ∈ S ↔ 0 < a ∧ 27 * a % 40 = 17) (h₁ : IsLeast S u) (hβ‚‚ : IsLeast (S \ {u}) v) : u + v = (62) := by
S : Set β„• u v : β„• hβ‚€ : βˆ€ (a : β„•), a ∈ S ↔ 0 < a ∧ 27 * a % 40 = 17 h₁ : IsLeast S u hβ‚‚ : IsLeast (S \ {u}) v ⊒ u + v = 62
795db81db6b47511
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_110 (q e : β„‚) (hβ‚€ : q = 2 - 2 * Complex.I) (h₁ : e = 5 + 5 * Complex.I) : q * e = (20) := by
q e : β„‚ hβ‚€ : q = 2 - 2 * Complex.I h₁ : e = 5 + 5 * Complex.I ⊒ q * e = 20
4f1fbbfbd584c505
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12b_2021_p21 (S : Finset ℝ) (hβ‚€ : βˆ€ x : ℝ, x ∈ S ↔ 0 < x ∧ x ^ (2 : ℝ) ^ Real.sqrt 2 = Real.sqrt 2 ^ (2 : ℝ) ^ x) : βˆ‘ k ∈ S, k ∈ (Set.Icc (2 : ℝ) 6) := by
S : Finset ℝ hβ‚€ : βˆ€ (x : ℝ), x ∈ S ↔ 0 < x ∧ x ^ 2 ^ √2 = √2 ^ 2 ^ x ⊒ βˆ‘ k ∈ S, k ∈ Set.Icc 2 6
b85784e202806953
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_405 (S : Finset β„•) (hβ‚€ : βˆ€ x, x ∈ S ↔ 0 < x ∧ x ^ 2 + 4 * x + 4 < 20) : S.card = (2) := by
S : Finset β„• hβ‚€ : βˆ€ (x : β„•), x ∈ S ↔ 0 < x ∧ x ^ 2 + 4 * x + 4 < 20 ⊒ S.card = 2
f2d146ec4391bcfa
import Mathlib open scoped Nat Real Topology Polynomial theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β„•) (hβ‚€ : 0 < n) : βˆ‘ k ∈ Finset.Icc 1 n, k * Nat.choose n k = (n * 2 ^ (n - 1)) := by
n : β„• hβ‚€ : 0 < n ⊒ βˆ‘ k ∈ Finset.Icc 1 n, k * n.choose k = n * 2 ^ (n - 1)
fed4f7a6c17dc0aa
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_393 (Οƒ : Equiv ℝ ℝ) (hβ‚€ : βˆ€ x, Οƒ.1 x = 4 * x ^ 3 + 1) : Οƒ.2 33 = (2) := by
Οƒ : ℝ ≃ ℝ hβ‚€ : βˆ€ (x : ℝ), Οƒ.toFun x = 4 * x ^ 3 + 1 ⊒ Οƒ.invFun 33 = 2
ed6b96fa632ab0ec
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12b_2004_p3 (x y : β„•) (hβ‚€ : 2 ^ x * 3 ^ y = 1296) : x + y = (8) := by
x y : β„• hβ‚€ : 2 ^ x * 3 ^ y = 1296 ⊒ x + y = 8
343bffafae02e014
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_303 (S : Finset β„•) (hβ‚€ : βˆ€ n : β„•, n ∈ S ↔ 2 ≀ n ∧ 171 ≑ 80 [MOD n] ∧ 468 ≑ 13 [MOD n]) : βˆ‘ k ∈ S, k = (111) := by
S : Finset β„• hβ‚€ : βˆ€ (n : β„•), n ∈ S ↔ 2 ≀ n ∧ 171 ≑ 80 [MOD n] ∧ 468 ≑ 13 [MOD n] ⊒ βˆ‘ k ∈ S, k = 111
06269bdcf5a5fcc9
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_151 : Int.ceil (Real.sqrt 27) - Int.floor (Real.sqrt 26) = (1) := by
⊒ ⌈√27βŒ‰ - ⌊√26βŒ‹ = 1
624a7a5c058e8b9b
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12a_2011_p18 : IsGreatest { x^2 - 6 * x + y^2 | (x : ℝ) (y : ℝ) (hβ‚€ : abs (x + y) + abs (x - y) = 2)} (8) := by
⊒ IsGreatest {x | βˆƒ x_1 y, βˆƒ (_ : |x_1 + y| + |x_1 - y| = 2), x_1 ^ 2 - 6 * x_1 + y ^ 2 = x} 8
066b42e19278f1b2
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_algebra_15 (s : β„• β†’ β„• β†’ β„•) (hβ‚€ : βˆ€ a b, 0 < a ∧ 0 < b β†’ s a b = a ^ (b : β„•) + b ^ (a : β„•)) : s 2 6 = (100) := by
s : β„• β†’ β„• β†’ β„• hβ‚€ : βˆ€ (a b : β„•), 0 < a ∧ 0 < b β†’ s a b = a ^ b + b ^ a ⊒ s 2 6 = 100
320e35f6e72afb74
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_211 : Finset.card (Finset.filter (fun n => 6 ∣ 4 * ↑n - (2 : β„€)) (Finset.range 60)) = (20) := by
⊒ {n ∈ Finset.range 60 | 6 ∣ 4 * ↑n - 2}.card = 20
df2e017834b83107
import Mathlib open scoped Nat Real Topology Polynomial theorem mathd_numbertheory_640 : (91145 + 91146 + 91147 + 91148) % 4 = (2) := by
⊒ (91145 + 91146 + 91147 + 91148) % 4 = 2
af8261d93339fd91
import Mathlib open scoped Nat Real Topology Polynomial theorem amc12b_2003_p6 (a r : ℝ) (u : β„• β†’ ℝ) (hβ‚€ : βˆ€ k, u k = a * r ^ k) (h₁ : u 1 = 2) (hβ‚‚ : u 3 = 6) : let a := (2 / Real.sqrt 3); u 0 = a ∨ u 0 = -a := by
a r : ℝ u : β„• β†’ ℝ hβ‚€ : βˆ€ (k : β„•), u k = a * r ^ k h₁ : u 1 = 2 hβ‚‚ : u 3 = 6 ⊒ let a := 2 / √3; u 0 = a ∨ u 0 = -a
c5cb845a6c1c9dd6
import Mathlib open scoped Nat Real Topology Polynomial theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β„‚) : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
a : β„‚ ⊒ (a - 10) * (a + 11) = a ^ 2 + a - 110
End of preview. Expand in Data Studio

minif2f-satp-alphaproof

The canonical 488-base-problem view of the miniF2F benchmark used by AlphaProof, ported to Lean 4.26.0 and packaged with initial proof goal_state values generated for SATP v2.

This dataset is intended as the held-out evaluation (test) and hyperparameter-tuning (validation) benchmark for SATP / sketch-and-prove pipelines running in the same Lean 4.26 environment.

The source is Google DeepMind miniF2F@f0a20e1. Its README identifies this as the benchmark version used to evaluate AlphaProof. There is no upstream alphaproof branch/tag and no official Lean 4.26 snapshot; this dataset is therefore a reproducible Lean 4.26 port of that exact commit.

Related SATP datasets:

Splits 

from datasets import load_dataset

ds = load_dataset("ChristianZ97/minif2f-satp-alphaproof")
ds["test"]        # 244 rows β€” held-out evaluation
ds["validation"]  # 244 rows β€” hyperparameter tuning

The upstream repository contains 500 theorem declarations: 244 in Test.lean and 256 in Valid.lean. Twelve validation declarations have .variants. in their names and are alternative formalizations. This default config publishes the traditional base benchmark: 244 validation + 244 test = 488 rows. The excluded declaration names are recorded in manifest.json.

Split Upstream declarations Excluded .variants. Published rows v2 meta-extracted goals
test 244 0 244 244
validation 256 12 244 244
Total 500 12 488 488

Goal-state provenance (read this first)

Every published goal_state was generated fresh under Lean 4.26 after Lean elaborated the corresponding theorem declaration. No goal string was copied from DSP-Plus or from the older ChristianZ97/minif2f-satp dataset.

Extraction uses the exact collectGoalState implementation from the ChristianZ97/LeanSATP v2 branch, pinned at dfb0c2558997bc23da0b004ba5eacc4c8369dbae:

private def collectGoalState : TacticM String := withMainContext do
  return (← ppGoal (← getMainGoal)).pretty

This is Lean 4.26's standard Meta.ppGoal rendering. Same-type binders are therefore merged into one line, for example:

b h v : ℝ
hβ‚€ : 0 < b ∧ 0 < h ∧ 0 < v
...
⊒ v = 65

This exact rendering matters because satp-policy-v2 consumes byte-sensitive goal-state text. Raw leanprover/repl sorries[].goal output is not used: for some declarations it expands notation differently from Meta.ppGoal (for example, 20! versus (Nat.factorial 20)).

The AlphaProof source form and the standalone Mathlib-only Hub form were both elaborated independently. The v2 meta renderer produced byte-identical initial goals for 488/488 paired declarations.

The isolated meta-programming renderer is included as port/SATPGoal.lean, and the full extraction pipeline is included as port/build_dataset.py.

Schema

Column Type Notes
uuid string sha256(canonical(goal_state))[:16]
formal_statement string Self-contained Lean 4.26 code ending in := by
goal_state string Exact LeanSATP v2 Meta.ppGoal output at proof entry

Canonicalization

canonical(s) removes lines whose trimmed contents begin with Lean -- comments, collapses whitespace runs to one space, and strips leading/trailing whitespace. UUIDs identify elaborated proof obligations, not theorem text.

Formal-statement/header contract

Each row is independently executable in the SATP Lean 4.26 environment:

  • begins with import Mathlib;
  • retains open scoped Nat Real Topology Polynomial, required by AlphaProof statements using notation such as postfix factorial;
  • contains the theorem declaration with its complete proposition;
  • ends in := by, ready for tactic insertion;
  • contains no proof sorry and no upstream proof body.

All 488 published rows were re-elaborated from this stored form under Lean 4.26 with 0 errors.

Source

The theorem declarations come from:

  • repository: google-deepmind/miniF2F
  • commit: f0a20e14c1eeccd859d51bb4c2b3ee487889c303
  • test source: MiniF2F/Test.lean
  • validation source: MiniF2F/Valid.lean

Google DeepMind's README states that this is the miniF2F version on which AlphaProof is evaluated. The upstream repository currently targets Lean 4.27; the package-only 4.26 port pins the official Formal Conjectures v4.26.0 tag. No benchmark proposition needed a source patch.

Lean environment

Component Pin
Lean v4.26.0 β€” d8204c9fd894f91bbb2cdfec5912ec8196fd8562
Mathlib v4.26.0 β€” 2df2f0150c275ad53cb3c90f7c98ec15a56a1a67
Formal Conjectures v4.26.0 β€” 84450b941eee69d63d9e5f27e626fc58de767d75
AlphaProof miniF2F source f0a20e14c1eeccd859d51bb4c2b3ee487889c303

With Formal Conjectures pinned to its official v4.26.0 tag, the three upstream miniF2F source modules compile unchanged on Lean 4.26. Merely changing lean-toolchain is insufficient because an unpinned Formal Conjectures dependency resolves back to Lean 4.27.

Hub runtime form

AlphaProof's validation source uses the informational answer(term) elaborator and one Real.nthRoot definition supplied by Formal Conjectures. SATP verifies standalone rows in a Mathlib-only environment, so the published rows apply two audited compatibility transforms:

  1. 193 answer(term) markers are changed to parenthesized term; the elaborator is an annotation and elaborates to that same term.
  2. The exact Formal Conjectures v4.26 Real.nthRoot definition is vendored in the single affected row, mathd_algebra_282.

Every upstream proposition and its Mathlib-only Hub form was elaborated independently under Lean 4.26. Their SATP-rendered initial goals are byte-for-byte identical for all 488 rows. The source files themselves need no proposition patch.

Reproducibility and provenance

  • manifest.json contains source hashes, dependency commits, split counts, Parquet hashes, and excluded variants.
  • port/build_dataset.py extracts declarations, compiles support modules, performs both elaborations, enforces goal equality, computes UUIDs, and writes the Parquet shards.
  • port/PORTING.md documents the upstream Lean 4.26 project pin and the exact build command.

Validation gates enforced by the builder include exact source/dependency pins, 244 rows per split, unique/disjoint theorem names, exact three-column schema, 488/488 paired v2-meta goal equality, UUID recomputation, and Parquet round-trip checks. Exact source hashes and output shard hashes are in manifest.json.

Citation 

@inproceedings{zheng2022minif2f,
  title     = {miniF2F: a Cross-System Benchmark for Formal Olympiad-Level Mathematics},
  author    = {Zheng, Kunhao and Han, Jesse Michael and Polu, Stanislas},
  booktitle = {International Conference on Learning Representations},
  year      = {2022}
}

License

The DeepMind miniF2F repository is Apache-2.0 and retains the original OpenAI copyright notices in its source. The vendored Formal Conjectures compatibility definition is also Apache-2.0. See LICENSE and the exact source links above.
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