name stringlengths 2 66 | header stringclasses 102
values | formal_statement stringlengths 29 4.73k |
|---|---|---|
vector_relation_14 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem vector_relation_14 (A B C M : β Γ β)
(hA : A = (-1, 0))
(hB : B = (3, 0))
(hC : C = (0, Real.sqrt 3))
(hM : B - M = 2 β’ (M - C)) :
β x y : β, x = 1/3 β§ y = 2/3 β§ M - A = x β’ (B - A) + y β’ (C - A) := by |
lean_workbook_plus_74438_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_74438_3 (s : β β β)
(hβ : β k in Finset.range 15, s k = 90 * 15)
(hβ : β k in Finset.range 15, s (k + 1) = 92 * 13)
(hβ : s 15 = 110) :
s 1 = 44 := by |
sequence_periodicity_25 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem sequence_periodicity_25 {n : β} (hn : n β₯ 3)
(a : β β β) -- sequence of real numbers
(h_rec : β i : β, i β₯ 1 β i β€ n β a i * a (i + 1) + 1 = a (i + 2)) -- recurrence relation
(h_period : β i : β, i β₯ 1 β i β€ n β a (i + 3) = a i) -- periodicity condition
: 3 β£ n := by |
periodic_function_solution_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem periodic_function_solution_3 (n : β€) (k m : β€)
(h1 : n + 1 β 0) (h2 : n - 2 β 0)
(h3 : 2/(n+1 : β) = 3/k)
(h4 : (n-2)/4 = 3/m) :
(n = 3 β¨ n = 1 β¨ n = 5 β¨ n = -1 β¨ n = 10 β¨ n = -6 β¨ n = 26 β¨ n = -22) := by |
angle_PMN_measure_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem angle_PMN_measure_2
(P Q R M N : β Γ β) -- Points as pairs of real numbers
(isIsosceles_PQR : abs (dist P R) = abs (dist R Q)) -- RP = RQ
(isIsosceles_PMN : abs (dist P M) = abs (dist P N)) -- PM = PN
(angle_PQR : Real.arccos ((P.1 - Q.1) * (R.1 - Q.1) + (P.2 - Q.2) * (R.2 - Q.2)) /
(Real.sqrt ... |
lean_workbook_plus_50998_7 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_50998_7 (a : β) :
(β x, a * x^2 + 2 * x + 1 = 0 β§ x < 0) β a β€ 1 := by |
lean_workbook_plus_46307_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_46307_3 (A B C a b c : β) : β S : Set β, β r : β, β x : β, x β S β x = A β§ x = B β§ x = C β§ r = a β§ r = b β§ r = c := by |
lean_workbook_12793_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_12793_4 (x : β)
(hβ : 0 < x)
(hβ : x < 1 / 8) :
β' n : β, β j in Finset.range (n + 1), ((3 * n + 2 - j).choose j * 2^j - (3 * n + 1 - j).choose (j - 1) * 2^(j - 1)) * x^n = 1 / (1 - 8 * x) := by |
euler_line_ratio | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem euler_line_ratio {A B C M O H : β Γ β}
(tri_ABC : A β B β§ B β C β§ C β A) -- Triangle exists
(is_M : M = (A + B + C) / 3) -- M is the centroid
(collinear : β (t : β), O = M + t β’ (H - M)) -- Points are collinear
: βH - Mβ / βM - Oβ = 2 := by |
range_of_f_inequality_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem range_of_f_inequality_4 (x : β) :
(Real.exp (x^2) + (x^2)^3 < Real.exp (3*x - 2) + (3*x - 2)^3) β
1 < x β§ x < 2 := by |
lean_workbook_plus_33872 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_33872 (a b c d e f r : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c β§ 0 < d β§ 0 < e β§ 0 < f)
(hβ : a β₯ b β§ b β₯ c)
(hβ : d β₯ e β§ e β₯ f)
(hβ : a + b β₯ c)
(hβ : d + e β₯ f)
(hβ
: a + c β₯ b)
(hβ : d + f β₯ e)
(hβ : a + b + c = d + e + f)
(hβ : r = a / d)
(hβ : r = b / e)
(hββ :... |
vector_magnitude_problem_225 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem vector_magnitude_problem_225
(a b : β Γ β) -- Two vectors in βΒ²
(ha : βaβ = 1) -- Magnitude of a is 1
(hb : βbβ = 1) -- Magnitude of b is 1
(h3 : β3 β’ a - 2 β’ bβ = Real.sqrt 7) : -- Given equation
-- Part I : Angle between vectors is Ο/3
(a β’ b = 1/2) β§
-- Part II : Value o... |
lean_workbook_22799_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_22799_3 (x : β)
(hβ : x = 8! )
(hβ : x = 2^i * 3^k * 5^m * 7^p)
(hβ : i + k + m + p = 11) :
k = 2 := by |
skating_speed_ratio_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem skating_speed_ratio_3
(vf vs : β) -- speeds of father and son
(h_pos_f : vf > 0) -- father's speed is positive
(h_pos_s : vs > 0) -- son's speed is positive
(h_f_faster : vf > vs) -- father is faster
(h_ratio : (vf + vs)/(vf - vs) = 5) -- relative speed ratio is 5
: vf/vs = 3/2 := by |
group_intersection | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem group_intersection (n : β) (students : Finset (Fin n))
(groups : Finset (Finset (Fin n)))
(h_size : students.card = 32)
(h_groups : groups.card = 33)
(h_size_groups : β g β groups, g.card = 3)
(h_sub : β g β groups, g β students)
(h_unique : β gβ gβ : Finset (Fin n), gβ β group... |
lean_workbook_plus_44423_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_44423_2 (f : β β β) (hf : β z, βzβ = 1 β βf zβ = 1) : β ΞΈ : β, β k : β, β z, f z = exp (ΞΈ * I) * z ^ k := by |
complex_equation_system_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem complex_equation_system_2
(p q r s t u : β)
(hp : p β 0) (hq : q β 0) (hr : r β 0)
(hs : s β 0) (ht : t β 0) (hu : u β 0)
(hs3 : s β 3) (ht3 : t β 3) (hu3 : u β 3)
(eq1 : p = (q + r)/(s - 3))
(eq2 : q = (p + r)/(t - 3))
(eq3 : r = (p + q)/(u - 3))
(sum_prod : s*t +... |
lean_workbook_plus_64732 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_64732 (a b c : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : c^2 = a^2 + b^2)
(hβ : Real.tan 30 = (b^2 / a) / (2 * c)) :
c / a = Real.sqrt 3 := by |
almost_all_perfect_squares | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem almost_all_perfect_squares
{n : β}
(hn : 2 β€ n)
(a : Fin n β β)
(ha : β i, 0 < a i)
(h_eq : β(β i, (a i : β)).sqrtβ = β i, β(a i : β).sqrtβ) :
β iβ, β i, i β iβ β β k, a i = k^2 := by |
lean_workbook_36611_6 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_36611_6 : β! (f : β β Finset (Fin 100 Γ Fin 100)), β n, (f n).card = 50 β§ β (a : Fin 100 Γ Fin 100), (a β f n β§ a β f (n+1)) β¨ (a β f n β§ a β f (n+1)) := by |
lean_workbook_plus_44720_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_44720_4 : β x : β, -2 * Real.sin (2 * x) = (β' k : β€, (-1 : β)^k * (Real.sin x - Real.cos x)^(2 * k + 2)) - 1 := by |
lean_workbook_plus_24332_6 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_24332_6 (k : β)
(hβ : 0 < k)
(hβ : β (n : β), n β Finset.Icc 1 (2 * k) β β! (p : β), p β Finset.Icc 1 k β§ n β£ p) :
k β€ 44 := by |
ineq_reciprocal_sum | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem ineq_reciprocal_sum {a b c : β}
(ha : a > 1) (hb : b > 1) (hc : c > 1)
(sum_eq : a + b + c = 4) :
1/(a-1) + 1/(b-1) + 1/(c-1) β₯ 8/(a+b) + 8/(b+c) + 8/(c+a) := by |
triangle_problem_184 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem triangle_problem_184
(a b c : β) -- sides of triangle
(h_pos_a : 0 < a) (h_pos_b : 0 < b) (h_pos_c : 0 < c) -- positive sides
(h_triangle : a < b + c β§ b < a + c β§ c < a + b) -- triangle inequality
(h_eq : (a + b + c) * (b + c - a) = 3 * b * c) -- given equation
(h_dot : b * c * cos ((pi : β) ... |
lean_workbook_plus_75075_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_75075_4 (a b c A : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : a + b > c)
(hβ : a + c > b)
(hβ : b + c > a)
(hβ : 0 < A β§ 0 < B β§ 0 < C)
(hβ
: A + B + C = 180)
(hβ : a / Real.cos A = c / (2 - Real.cos C)) :
b = 4 β a = 2 := by |
hyperbola_parabola_intersection_eccentricity_9 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem hyperbola_parabola_intersection_eccentricity_9
(a b c : β)
(ha : 0 < a)
(hb : 0 < b)
(hc : 0 < c)
(h_e : c^2 = a^2 + b^2) -- relation for hyperbola eccentricity
(x y : β)
(h_hyp : x^2/a^2 - y^2/b^2 = 1) -- point on hyperbola
(h_par : y^2 = 4*c*x) -- point on parabola
(h_perp : x = c)... |
fraction_maximum_value | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem fraction_maximum_value {A B : β}
(hA : A < 1000) (hB : B < 1000) -- A and B are less than 1000
(hne : A β B) -- A and B are different
(hA_pos : A > 0) (hB_pos : B > 0) -- A and B are positive
: (A - B : β)/(A + B) β€ 499/500 := by |
box_cut_max_diagonal | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem box_cut_max_diagonal :
let d1 := space_diagonal_squared 1 1 16
let d2 := space_diagonal_squared 1 2 8
d1 = 258 β§ d2 = 69 β§ d1 > d2 := by |
lean_workbook_plus_79435_7 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_79435_7 (a b : β€)
(hβ : 1996 * a + b / 96 = a + b) :
a / b = 1 / 2016 β¨ b / a = 2016 := by |
problem_solution_26 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem problem_solution_26 (n : β) (a : β) (b : β)
(h_a : a = 30)
(h_pos : 0 < a)
(h_square : n * a = b * b)
(h_min : β k, 0 < k β k < a β Β¬β m, n * k = m * m) : n = 30 := by |
lean_workbook_plus_25916_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_25916_2 (a b c d m : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c β§ 0 < d)
(hβ : a + b = c + d)
(hβ : a * b = c * d)
(hβ : b + c = m)
(hβ : b * c = n) :
(m - a) * (d - b) = (c - a) * (m - b) := by |
polynomial_identity_8 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem polynomial_identity_8 (aβ aβ aβ aβ aβ : β)
(h : β x : β, (x + Real.sqrt 2)^4 = aβ + aβ*x + aβ*x^2 + aβ*x^3 + aβ*x^4) :
(aβ + aβ + aβ)^2 - (aβ + aβ)^2 = 1 := by |
lean_workbook_41047_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_41047_2 (f : β β β)
(hβ : β x, x < 3 β f x = 12 * x + 21)
(hβ : β x, 3 β€ x β f x = 3 * x - 27)
(hβ : f x = 0) :
x = -7 / 4 β¨ x = 9 := by |
sequence_formula_561 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem sequence_formula_561 (a : β β β) (h_pos : β n, a n > 0)
(h_a1 : a 1 = 3)
(h_rec : β n, a n * (a n ^ 2 + 1 - 1) = 2 * a (n + 1) * (a n ^ 2 - 1)) :
let b : β β β := fun n β¦ a n - 1/(a n)
β n, b n = 8/3 * 2^(n-1) := by |
lean_workbook_plus_72079_6 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_72079_6 (a b : β)
(f : β β β)
(hβ : β x, f x = x * exp x - a * exp x - b * x)
(hβ : β x, f x = x - 1) :
a = 1 β§ b = -1 := by |
lean_workbook_plus_27454 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_27454 (b : β)
(a : β β β)
(hβ : 2 β€ b)
(hβ : β n, 0 β€ a n β§ a n < b)
(hβ : a 0 β 0)
(hβ : β T, 0 < T β§ β n, a (n + T) = a n)
(hβ : β n, a n β 0)
(hβ
: S = {n | n β£ (a 0 * b^n + β i in Finset.range n, a i * b^i)}) :
S.Infinite β β p, p.Prime β§ β n, p β£ (a 0 * b^n + β ... |
lean_workbook_plus_21516_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_21516_4 (a b : β)
(hβ : a / (1 - Complex.I) + b / (2 - Complex.I) = 1 / (3 - Complex.I)) :
a = -1 / 5 β§ b = 1 := by |
vector_n_solution | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem vector_n_solution (m n : β Γ β)
(h_m : m = (1, 1))
(h_dot : m.1 * n.1 + m.2 * n.2 = -1)
(h_angle : (m.1 * n.1 + m.2 * n.2) /
(Real.sqrt ((m.1)^2 + (m.2)^2) * Real.sqrt ((n.1)^2 + (n.2)^2)) = -Real.sqrt 2/2) :
n = (-1, 0) β¨ n = (0, -1) := by |
lean_workbook_plus_38296_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_38296_3 (m : β)
(A B : Set β)
(hβ : A = {(-1), 3, m})
(hβ : B = {3, 4})
(hβ : B β© A = B) :
m = 4 := by |
parabola_chord_length_46 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem parabola_chord_length_46
(A B : β Γ β) -- Points A and B on the parabola
(F : β Γ β) -- Focus point
(hF : F = (3/2, 0)) -- Focus coordinates
(hA : A.2 ^ 2 = 6 * A.1) -- A is on parabola
(hB : B.2 ^ 2 = 6 * B.1) -- B is on parabola
(hline : β t : β, t β Set.Icc 0 1 β
((... |
lean_workbook_plus_25154 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_25154 (k : β)
(hβ : -x^2 - (k + 12) * x - 8 = -(x - 2) * (x - 4)) :
k = -18 := by |
particle_trajectory_equation | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem particle_trajectory_equation
(b m g : β) -- parameters
(h_pos_b : b > 0)
(h_pos_m : m > 0)
(h_pos_g : g > 0)
(t : β) : -- time variable
let x := (b * t) / m -- horizontal position
let y := (b/2) * Real.exp (t/(m/b)) - (b/2) * Real.exp (-t/(m/b)) - (g/2) * b * t^2 -- vertical position
let... |
geometric_sequence_max_ratio | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem geometric_sequence_max_ratio (a : β β β) (b : β β β) (S : β β β)
(h_geom : β n, a (n + 1) = a n * q) -- geometric sequence condition
(h_q_bounds : 0 < q β§ q < 1) -- ratio bounds
(h_sum : a 3 + a 5 = 5) -- first given equation
(h_prod : a 2 * a 6 = 4) -- second given equation
(h_b_def ... |
lean_workbook_plus_58053_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_58053_3 (a : β β NNReal)
(n : β)
(hβ : 0 < n)
(hβ : β n, 0 < a n)
(hβ : β n, (β i in Finset.range (n + 1), (a i)^3) = (β i in Finset.range (n + 1), a i)^2) :
a n = n := by |
centroid_distance_inequality_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem centroid_distance_inequality_5 {A B C G : β Γ β} (tβ tβ : β)
(h_tri : A β B β§ B β C β§ C β A) -- Triangle exists and is non-degenerate
(h_centroid : G = ((A.1 + B.1 + C.1)/3, (A.2 + B.2 + C.2)/3)) -- G is the centroid
(h_tβ : tβ = dist G A + dist G B + dist G C) -- Definition of tβ
(h_tβ ... |
lean_workbook_40541_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_40541_2 (a b c d : β)
(hβ : a = 1 / 2)
(hβ : b = -Real.pi)
(hβ : c = -0.7)
(hβ : d = -3 / 3) :
c < 0 β§ β n : β€, c = n / 3 := by |
sum_of_two_from_different_sets | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem sum_of_two_from_different_sets (n : β) (A B C : Set β)
(h_pos : n > 0)
(h_partition : A βͺ B βͺ C = Finset.range (3 * n + 1) \ {0})
(h_disjoint : A β© B = β
β§ B β© C = β
β§ A β© C = β
)
(h_equal : A.ncard = n β§ B.ncard = n β§ C.ncard = n) :
β (a b c : β), a β b β§ b β c β§ a β c β§
a β A ... |
telescoping_sum_equals_1008014 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem telescoping_sum_equals_1008014 :
let S := β i in Finset.range 1003,
Real.sqrt (1 + 1/(i + 1)^2 + 1/(i + 2)^2)
βS^2β = 1008014 := by |
factorial_div_990 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem factorial_div_990 :
(β n : β, n < 11 β Β¬(990 β£ n!)) β§ (990 β£ 11!) := by |
subset_size_bound_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem subset_size_bound_3 {n : β} (X : Finset (Fin n)) (S : Finset (Finset (Fin n)))
(h_size : X.card = n)
(h_S_sub : β s β S, s β X)
(h_S_size : β s β S, s.card = 3)
(h_S_intersect : β sβ sβ : Finset (Fin n), sβ β S β sβ β S β sβ β sβ β
(sβ β© sβ).card β€ 1) :
β A : Finset (Fin n), ... |
handshake_parity_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem handshake_parity_3 {n : β} (a : Fin n β β) (h_sum_even : Even (β i, a i)) :
Even (Fintype.card {i : Fin n | Odd (a i)}) := by |
figure_perimeter | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem figure_perimeter
(BD BC CD DE AE AB : β) -- Side lengths
(angle_DBC angle_BCD : β) -- Angles in degrees
(h1 : BD = BC) -- Isosceles condition for BCD
(h2 : angle_DBC = angle_BCD) -- Equal angles in BCD
(h3 : angle_DBC = 60) -- Angle value in degrees
(h4 : AB = AE) -- Is... |
lean_workbook_plus_39823_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_39823_4 (x y t : β)
(hβ : 0 < x β§ 0 < y)
(hβ : y^2 = 8 * x)
(hβ : -4 * (-4, t) = (x - 2, y)) :
x + 2 = 20 := by |
lean_workbook_plus_29675_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_29675_3 (x : β) : (x = 7 ^ 2011) β (Nat.digits 10 x) = [1, 8, 8, 1, 4, 5] := by |
lean_workbook_plus_69303 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_69303 (x y : β β β)
(n : β)
(c : β)
(hβ : 0 < n)
(hβ : β i, 1 β€ i β§ i β€ n β y i = x i + c)
(hβ : c β 0) :
(n : β)β»ΒΉ * β i in Finset.Icc 1 n, y i = (n : β)β»ΒΉ * β i in Finset.Icc 1 n, x i + c := by |
lean_workbook_plus_71398_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_71398_5 (a b c : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : a + b > c)
(hβ : a + c > b)
(hβ : b + c > a)
(hβ
: 60 = Real.arccos ((b^2 + c^2 - a^2) / (2 * b * c))) :
(Real.arccos ((b^2 + c^2 - a^2) / (2 * b * c)) = Real.arccos ((c^2 + a^2 - b^2) / (2 * c * a))) := by |
right_triangle_projections | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem right_triangle_projections
(A B C U D E F G : β Γ β) -- Points in the plane
(right_angle : (C.1 - A.1) * (C.1 - B.1) + (C.2 - A.2) * (C.2 - B.2) = 0) -- C is right angle
(on_AC : D.2 = A.2 + (C.2 - A.2)/(C.1 - A.1) * (D.1 - A.1)) -- D lies on AC
(on_BC : E.2 = B.2 + (C.2 - B.2)/(C.1 - B.1) * (E.1... |
poly_value_bound_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem poly_value_bound_2
{k n : β}
{x : Fin k β β€}
{y : Fin n β β€}
{P : β€ β β€}
(hk : k > 0)
(hn : n > 0)
(hx_inj : Function.Injective x)
(hy_inj : Function.Injective y)
(hP_k : β i : Fin k, P (x i) = 54)
(hP_n : β j : Fin n, P (y j) = 2013) :
k * n β€ 6 :=... |
lean_workbook_plus_23122_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_23122_3 (a b : β) (hβ : a + b * Complex.I = (1 + Complex.I) * (2 - Complex.I)) : a + b = 4 := by |
lean_workbook_plus_14899_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_14899_5 (f : β β β) (hf : f (3*x + 2*y) = f x * f y) : (β x y : β, f (3*x + 2*y) = f x * f y) β β a :β, β x : β, f x = a ^ x := by |
sin_cos_period_odd | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem sin_cos_period_odd : β x : β,
(sin x * cos x = -sin (-x) * cos (-x)) β§ -- odd function property
(sin x * cos x = sin (x + Real.pi) * cos (x + Real.pi)) -- period Ο := by |
lean_workbook_plus_33960_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_33960_2 (n : β)
(hβ : 0 < n)
(hβ : (2 * n) * (n - 1) = 7 * (2 * n)) :
n = 8 := by |
lean_workbook_plus_12878_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_12878_4 :
0.6 + 0.4 * Real.log 0.4 = 0.2335 := by |
min_distance_between_circles_17 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem min_distance_between_circles_17 :
let Cβ := {P : β Γ β | (P.1 - 4)^2 + (P.2 - 2)^2 = 9}
let Cβ := {Q : β Γ β | (Q.1 + 2)^2 + (Q.2 + 1)^2 = 4}
β P β Cβ, β Q β Cβ, βP - Qβ β₯ 3 * Real.sqrt 5 - 5 β§
β Pβ β Cβ, β Qβ β Cβ, βPβ - Qββ = 3 * Real.sqrt 5 - 5 := by |
lean_workbook_plus_29296_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_29296_3 :
IsGreatest {y : β | β x : β, y = (4 * x^2 + 8 * x + 19) / (4 * x^2 + 8 * x + 5)} 15 := by |
midpoint_segment_diff | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem midpoint_segment_diff
(A B C D M P N Q : β Γ β) -- Points in 2D plane
(hM : M = (A + B) / 2) -- M is midpoint of AB
(hP : P = (B + C) / 2) -- P is midpoint of BC
(hN : N = (C + D) / 2) -- N is midpoint of CD
(hQ : Q = (D + A) / 2) -- Q is midpoint of DA
(hAB : βB - Aβ = 5) -... |
lean_workbook_plus_1680_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_1680_2 (f : β β β)
(hβ : β k, f k β€ k^2 β f (k + 1) β€ (k + 1)^2)
(hβ : f 7 = 50) :
β k β€ 7, f k > k^2 := by |
functional_equation_unique_and_sum | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem functional_equation_unique_and_sum :
β! f : β β β,
(β n : β, n > 0 β f n > 0) β§
IsFunctionalSolution f β§
(β i in Finset.range 19, f (i + 1) = 1995) := by |
lean_workbook_plus_43814_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_43814_2 (f : β β β)
(hβ : β x, f (-x) = -f x)
(hβ : β x < 0, f x = 3 * Real.sin x + 4 * Real.cos x + 1) :
β x > 0, f x = 3 * Real.sin x - 4 * Real.cos x - 1 := by |
inscribed_triangle_perimeter_bound | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem inscribed_triangle_perimeter_bound :
β (t : β), t > 4 β β (n : β), β (s : Set β),
(β x β s, x = 4 + 2 * Ξ΅ β§ Ξ΅ > 0) β n * (Real.sqrt ((2 + Ξ΅) * Ξ΅^3)) > Real.pi := by |
quadratic_function_range_46 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem quadratic_function_range_46 (a : β) :
(β x : β, β y : β, y = (a^2 - 2*a - 3)*x^2 + (a - 3)*x + 1) β§
(β y : β, β x : β, y = (a^2 - 2*a - 3)*x^2 + (a - 3)*x + 1) β
a = -1 := by |
lean_workbook_39902_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_39902_4 (x : β)
(hβ : 1614 - x = 1360) :
1614 % x = 90 := by |
lean_workbook_26089_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_26089_5 (x y : β) :
3 β€ 4 + x^2 * y^4 + x^4 * y^2 - 3 * x^2 * y^2 := by |
sequence_sum_squares_22 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem sequence_sum_squares_22 (a b : β β β)
(h_init : a 0 = 2 β§ b 0 = 2)
(h_rec_a : β n, a (n + 1) = a n * Real.sqrt (1 + a n ^ 2 + b n ^ 2) - b n)
(h_rec_b : β n, b (n + 1) = b n * Real.sqrt (1 + a n ^ 2 + b n ^ 2) + a n)
(n : β) : a n ^ 2 + b n ^ 2 = 3 ^ (2 ^ n) := by |
lean_workbook_plus_54777 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_54777 (n : β)
(ΞΈ : β)
(a : β β β)
(hβ : 0 < ΞΈ β§ ΞΈ < Real.pi / 2)
(hβ : a 1 = 2 * Real.cos ΞΈ)
(hβ : β n, a (n + 1) = Real.sqrt (2 + a n)) :
a n = 2 * Real.cos (ΞΈ / 2^(n - 1)) := by |
point_on_line_7 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem point_on_line_7 (m : β) :
let pβ : β Γ β := (7, 10)
let pβ : β Γ β := (-3, m)
let pβ : β Γ β := (-11, 5)
(pβ.2 - pβ.2)/(pβ.1 - pβ.1) = (pβ.2 - pβ.2)/(pβ.1 - pβ.1) β m = 65/9 := by |
triangle_length_problem_20 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem triangle_length_problem_20
(A B C M H D : β Γ β) -- Points in RΒ²
(right_angle_C : (B.1 - C.1) * (A.1 - C.1) + (B.2 - C.2) * (A.2 - C.2) = 0) -- Right angle at C
(M_on_AC : β t : β, 0 β€ t β§ t β€ 1 β§ M = (1 - t) β’ A + t β’ C) -- M lies on AC
(AM_length : Real.sqrt ((M.1 - A.1)^2 + (M.2 - A.2)^2) = ... |
imaginary_part_is_minus_four | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Complex
| theorem imaginary_part_is_minus_four
(z : β) -- The complex number z
(h : (1 + I)/(3*I + z) = I) -- The given equation
: z.im = -4 := by |
cos_sq_sum_half_degrees | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem cos_sq_sum_half_degrees :
let angles : Fin 181 β β := fun n => (n : β) * (1/2 : β)
let cos_sq_sum := β n in Finset.range 181, (cos (angles n))^2
cos_sq_sum = 90.5 := by |
simplify_expression_15 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem simplify_expression_15 : (((64 :β)^(1/4) - Real.sqrt (37/4))^2) = 69/4 - 2 * Real.sqrt 74 := by |
lean_workbook_plus_78348_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_78348_2 (Ξ± Ξ² : β)
(hβ : 0 < Ξ± β§ Ξ± < Real.pi / 3)
(hβ : Real.pi / 6 < Ξ² β§ Ξ² < Real.pi / 2)
(hβ : 5 * Real.sqrt 3 * Real.sin Ξ± + 5 * Real.cos Ξ± = 8)
(hβ : Real.sqrt 2 * Real.sin Ξ² + Real.sqrt 6 * Real.cos Ξ² = 2) :
Real.cos (Ξ± + Ξ²) = -Real.sqrt 2 / 10 := by |
lean_workbook_plus_28107_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_28107_3 (b c : β)
(hβ : 0 < b β§ 0 < c)
(hβ : b + c = 100)
(hβ : b = c - 90) :
b / c = 1 / 10 := by |
lean_workbook_plus_43881 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_43881 (a b c : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : a + b > c)
(hβ : a + c > b)
(hβ : b + c > a)
: β k : β, k > 0 β§ k * (b * c + c * a + a * b) = a^3 + b^3 + c^3 := by |
parabola_slope_ratio | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem parabola_slope_ratio
(xβ yβ xβ yβ xβ yβ xβ yβ : β) -- Points P, Q, M, N
(h_paraβ : yβ^2 = -4*xβ) -- P lies on parabola
(h_paraβ : yβ^2 = -4*xβ) -- Q lies on parabola
(h_paraβ : yβ^2 = -4*xβ) -- M lies on parabola
(h_paraβ : yβ^2 = -4*xβ) -- N lies on parabola
(h... |
lean_workbook_plus_60408_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_60408_5 : Β¬ (β a : β β β, (β i : β, i < 19 β a i < a (i + 1)) β§ (β i in Finset.range 19, a i = 1999) β§ (β i : β, i < 19 β (Nat.digits 10 (a i)).sum = (Nat.digits 10 (a (i + 1))).sum)) := by |
pyramid_angles_sides | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem pyramid_angles_sides (Ξ± Ξ² : β) (k : β) (n : β)
(h_k : k = 2)
(h_angles : tan Ξ± = k * tan Ξ²)
(h_regular : n β₯ 3)
(h_cos : cos (180 / n) = 1 / k) : n = 3 := by |
lean_workbook_plus_34389_2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_34389_2 (h : β β β)
(h_def : β x, (x < 0 β h x = 4 * x + 4) β§ (0 β€ x β h x = 3 * x - 18))
(h_eq : h x = 0) :
x = -1 β¨ x = 6 := by |
lean_workbook_29097 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_29097 (q e : β)
(hβ : q = β' n : β, (3^(1 / (4^n)))) :
q = (81 :β)^(1 / 9) := by |
min_sum_vector_squares | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem min_sum_vector_squares {n : β} (a b c d : Fin n β β)
(h_nonzero : a β 0 β¨ b β 0) :
let f (t : β) := (β i, (t * a i + c i)^2) + (β i, (t * b i + d i)^2)
let tβ := -(β i, (a i * c i + b i * d i))/(β i, (a i^2 + b i^2))
β t, f tβ β€ f t := by |
limit_diff_quotient_at_three | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem limit_diff_quotient_at_three :
β Ξ΅ > 0, β Ξ΄ > 0, β h : β,
0 < |h| β§ |h| < Ξ΄ β |((3 + h)^2 - 3^2)/h - 6| < Ξ΅ := by |
geometric_distance_calculation_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem geometric_distance_calculation_3
(AB : β) (DC : β) (AD : β) (BD : β) (BC : β) (AE : β) (AC : β)
(h_AB : AB = 12)
(h_DC : DC = 15)
(h_AD : AD = 9)
(h_BD : BD ^ 2 = AB ^ 2 - AD ^ 2)
(h_BC : BC ^ 2 = DC ^ 2 - BD ^ 2)
(h_AE : AE = AD + BC) :
abs (AC - Real.sqrt 585) < 0.1 := by |
lean_workbook_plus_49601_4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_49601_4 (a b c A : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : a + b > c)
(hβ : a + c > b)
(hβ : b + c > a)
(hβ : b = Real.sqrt 7)
(hβ
: A = Real.arccos (1 / 2))
(hβ : Real.sin A = 3 * Real.sqrt 3 / 2) :
a + c = 5 := by |
lean_workbook_plus_68562_5 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_68562_5 (a b : β)
(hβ : 0 < a β§ 0 < b)
(hβ : a < b)
(hβ : abs (Real.log a) = abs (Real.log b)) :
2 < a + b := by |
cubic_collinear_roots_k | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Complex
| theorem cubic_collinear_roots_k : β k : β€,
(let p (x : β) := x^3 - 15*x^2 + k*x - 1105
let zβ : β := 5 + 14*I
let zβ : β := 5
let zβ : β := 5 - 14*I
-- The points are roots of the polynomial
(p zβ = 0 β§ p zβ = 0 β§ p zβ = 0) β§
-- The points are distinct
(zβ β zβ β§ zβ β zβ β§ zβ β zβ) β§
... |
intersection_line_tangent_circle | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem intersection_line_tangent_circle :
β (a : β), a > 0 β§
(β (l : Set (β Γ β)),
-- l is the line through P(1,1) perpendicular to x-y+1=0
(l = {p : β Γ β | p.1 + p.2 = 2}) β§
-- l is tangent to the circle (x-a)Β²+yΒ²=8
(|a - 2|/Real.sqrt 2 = 2*Real.sqrt 2)) := by |
lean_workbook_plus_41366 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem lean_workbook_plus_41366 (a : β) : β b, b β a β§ (b β£ a β¨ b β£ a + 1) := by |
savings_equality_3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem savings_equality_3 (weekly_savings1 : β := 7) (weekly_savings2 : β := 5)
(weeks : β := 25) (initial2 : β := 210)
(h_equal : β (initial1 : β), initial1 + weekly_savings1 * weeks = initial2 + weekly_savings2 * weeks β initial1 = 160) :
160 + weekly_savings1 * weeks = initial2 + weekly_sa... |
disinfectant_sales_problem | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem disinfectant_sales_problem
(x : β) -- selling price
(h_range : 8 β€ x β§ x β€ 15) -- price range constraint
: let y := -5*x + 150 -- sales quantity
let w := (x - 8)*y -- profit function
-- Part 1 : Linear relationship is unique
(y = -5*x + 150) β§
-- Part 2 : 425 yuan profit occurs at x... |
not_singleton_set | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem not_singleton_set (A : Set β)
(h1 : (1 : β) β A)
(h2 : β x β A, (1/(1-x)) β A) :
Β¬ β (a : β), A = {a} := by |
min_tan_angle_BAD | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem min_tan_angle_BAD
(A B C D : β Γ β) -- Points in RΒ²
(right_angle : (C.1 - A.1) * (C.1 - B.1) + (C.2 - A.2) * (C.2 - B.2) = 0) -- β C = 90Β°
(BC_length : Real.sqrt ((C.1 - B.1)^2 + (C.2 - B.2)^2) = 6) -- BC = 6
(D_on_BC : β t : β, 0 β€ t β§ t β€ 1 β§
D = (B.1 + t * (C.1 - B.1), B.2 + t ... |
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