uuid stringlengths 12 100 | data_source stringclasses 6
values | question stringlengths 573 25.4k | answer stringlengths 144 49.2k | prompt_planner stringlengths 574 25.4k | prompt_abstractor stringlengths 1.23k 58.6k |
|---|---|---|---|---|---|
Goedel-Pset-678601 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Theorem stating that among the elements in N₂O (dinitrogen monoxide),
nitrogen has a mass percentage closest to 63.64% -/
theorem nit... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem nitrogen_mass_percentage_in_n2o
(n_mass : ℝ := by
norm_num
<;> simp [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
<;> norm_num
<;> linarith
) :
Tr... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Theorem stating that among the elements in N₂O (dinitrogen monoxide),
nitrogen has a mass percentage closest to 63.64% -/
theorem nit... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-224873 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a regular hexagon with area 360, when randomly selecting three vertices,
the expected area of the intersection of the triangle... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem hexagon_intersection_area :
let hexagonArea : ℝ := 360
let triangleArea : ℝ := 12
let intersectionArea : ℝ := 12
-- The expected area of the intersectio... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a regular hexagon with area 360, when randomly selecting three vertices,
the expected area of the intersection of the triangle... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1567147 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equations \( f(x) + f(y) = f(x + y) \) and \( P(... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_52850 (f : ℝ → ℝ)
(h1 : ∀ x y : ℝ, f x + f y = f (x + y))
(h2 : ∀ P Q : Polynomial ℝ, P ≠ 0 → Q ≠ 0 → P.natDegree > 0 → Q.natDegree > 0 →
∀ ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equations \( f(x) + f(y) = f(x + y) \) and \( P(... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=109863 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_97f28b30_negation : ¬ ( ∀ (x y : ℝ) (hx : (0 : ℝ) < x) (hy : (0 : ℝ) < y) (h : x ^ (2 : ℕ) + y ^ (2 : ℕ) ≤ (1 : ℝ)) (h₃ : (0 :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_97f28b30_negation : ¬ ( ∀ (x y : ℝ) (hx : (0 : ℝ) < x) (hy : (0 : ℝ) < y) (h : x ^ (2 : ℕ) + y ^ (2 : ℕ) ≤ (1 : ℝ)) (h₃ : (0 : ℝ) < x) (h₄ : (0 : ℝ) < y) ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_97f28b30_negation : ¬ ( ∀ (x y : ℝ) (hx : (0 : ℝ) < x) (hy : (0 : ℝ) < y) (h : x ^ (2 : ℕ) + y ^ (2 : ℕ) ≤ (1 : ℝ)) (h₃ : (0 :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=348513 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_59277_17 (a b c k : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hk : 0 ≤ k) : (a^2 + k * b * c) * (b^2 + k * c * a) ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_59277_17 (a b c k : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hk : 0 ≤ k) : (a^2 + k * b * c) * (b^2 + k * c * a) * (c^2 + k * a * b) ≥ a * b *... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_59277_17 (a b c k : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hk : 0 ≤ k) : (a^2 + k * b * c) * (b^2 + k * c * a) ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1573448 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Ben starts each day with $50, spends $15 every day and saves the rest. After a few days, his mom doubled his total savings and his dad ga... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_7136 (d : ℕ)
(h₀ : 2 * (35 * d) + 10 = 500) :
d = 7 := by
have h₁ : 2 * (35 * d) + 10 = 500 := h₀
have h₂ : 2 * (35 * d) ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Ben starts each day with $50, spends $15 every day and saves the rest. After a few days, his mom doubled his total savings and his dad ga... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-723336 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Simplify the fraction $\dfrac{154}{10780}$.-/
theorem lean_workbook_plus_34089 :
(154 : ℝ) / 10780 = 1 / 70 := by sorry
```
Replace e... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_34089 :
(154 : ℝ) / 10780 = 1 / 70 := by
/-
To simplify the fraction \(\dfrac{154}{10780}\), we start by finding the greatest comm... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Simplify the fraction $\dfrac{154}{10780}$.-/
theorem lean_workbook_plus_34089 :
(154 : ℝ) / 10780 = 1 / 70 := by sorry
```
Replace e... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1031945 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_394844_0_gp2 : ∀ n : ℕ, (∃ k : ℤ, (2^n : ℤ) + 7 = k^2) ↔ n = 1 := by sorry
```
Replace every sorry statement with an app... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_394844_0_gp2 : ∀ n : ℕ, (∃ k : ℤ, (2^n : ℤ) + 7 = k^2) ↔ n = 1 := by
intro n
have h_main : (∃ k : ℤ, (2^n : ℤ) + 7 = k^2) ↔ n = 1 := by
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_394844_0_gp2 : ∀ n : ℕ, (∃ k : ℤ, (2^n : ℤ) + 7 = k^2) ↔ n = 1 := by sorry
```
Replace every sorry statement with an app... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=296121 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem soln_better1 : (2 - Real.sqrt 2) * Real.sqrt (a ^ 2 + b ^ 2) ≥ (Real.sqrt 2 - 1) * (a + b) := by sorry
```
Replace every sorry stateme... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem soln_better1 : (2 - Real.sqrt 2) * Real.sqrt (a ^ 2 + b ^ 2) ≥ (Real.sqrt 2 - 1) * (a + b) := by
have h_main : (2 - Real.sqrt 2) * Real.sqrt (a ^ 2 + b ^ 2) ≥ ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem soln_better1 : (2 - Real.sqrt 2) * Real.sqrt (a ^ 2 + b ^ 2) ≥ (Real.sqrt 2 - 1) * (a + b) := by sorry
```
Replace every sorry stateme... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-767591 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_30429 (a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : a + b + c = 1) :
27 ≤ 1 / a^2 + 1 / b^2 + 1 / c^2 := by so... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_30429 (a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : a + b + c = 1) :
27 ≤ 1 / a^2 + 1 / b^2 + 1 / c^2 := by
have h_main : 27 ≤ 1 / a^2 +... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_30429 (a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : a + b + c = 1) :
27 ≤ 1 / a^2 + 1 / b^2 + 1 / c^2 := by so... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=125482 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_49393 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 3) : 3 / ((1 + a * b) ^ 2 + (1 + b * c) ^ 2 +... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_49393 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 3) : 3 / ((1 + a * b) ^ 2 + (1 + b * c) ^ 2 + (1 + c * a) ^ 2) ≥ 3 / 27 / ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_49393 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 3) : 3 / ((1 + a * b) ^ 2 + (1 + b * c) ^ 2 +... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=77877 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_11273 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (habc : a + b + c = 3) : a * (2 * a + b) * b * c / (3 / c + c) ≤... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_11273 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (habc : a + b + c = 3) : a * (2 * a + b) * b * c / (3 / c + c) ≤ 27 := by
have h_main : ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_11273 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (habc : a + b + c = 3) : a * (2 * a + b) * b * c / (3 / c + c) ≤... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=803508 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2940 (n k : ℕ) (h₁ : 0 < k) (h₂ : k ≤ n) : n.choose k * k ! * (n - k)! ≤ (n + 1)! * (n + 1 - 1) ! := by sorry
```
Repl... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2940 (n k : ℕ) (h₁ : 0 < k) (h₂ : k ≤ n) : n.choose k * k ! * (n - k)! ≤ (n + 1)! * (n + 1 - 1) ! := by
have h₃ : n.choose k * (k ! * (n - k)!)... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2940 (n k : ℕ) (h₁ : 0 < k) (h₂ : k ≤ n) : n.choose k * k ! * (n - k)! ≤ (n + 1)! * (n + 1 - 1) ! := by sorry
```
Repl... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-216054 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The probability of picking exactly three red marbles in six trials when randomly selecting
from eight red marbles and seven blue marb... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem marble_probability :
let total_marbles : ℕ := 8 + 7
let red_marbles : ℕ := 8
let blue_marbles : ℕ := 7
let probability : ℚ := 6881280 / 38107875
proba... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The probability of picking exactly three red marbles in six trials when randomly selecting
from eight red marbles and seven blue marb... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1008585 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_44043 (c : ℝ)
(h₀ : 0 < c)
(h₁ : 30 / (4 + c) = 18 / (4 - c)) :
c = 1 := by sorry
```
Replace every sorr... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_44043 (c : ℝ)
(h₀ : 0 < c)
(h₁ : 30 / (4 + c) = 18 / (4 - c)) :
c = 1 := by
have h_main : c = 1 := by
have h₂ : 4 - c ≠ 0 ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_44043 (c : ℝ)
(h₀ : 0 < c)
(h₁ : 30 / (4 + c) = 18 / (4 - c)) :
c = 1 := by sorry
```
Replace every sorr... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1119789 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given:
- Original price of t-shirt is $51
- Discount per t-shirt is $8
- Number of t-shirts sold is 130
Prove that the total money made ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem tshirt_sales_total :
let original_price : ℕ := 51
let discount_per_tshirt : ℕ := 8
let num_tshirts : ℕ := 130
let total_money : ℕ := 5590
total_money ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given:
- Original price of t-shirt is $51
- Discount per t-shirt is $8
- Number of t-shirts sold is 130
Prove that the total money made ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-781208 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given that the average of the numbers $x_1, x_2, \ldots, x_n$ is $\overline{x}$ and the standard deviation is $s$, find the average and s... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_41775 (x : ℕ → ℝ)
(n : ℕ)
(h₀ : 0 < n)
(h₁ : ∑ k in Finset.range n, x k = ∑ k in Finset.range n, (2 * x k + 1))
(h₂ : ∑ k in Finse... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given that the average of the numbers $x_1, x_2, \ldots, x_n$ is $\overline{x}$ and the standard deviation is $s$, find the average and s... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-822324 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Sara got fast food for lunch. She spent some amount on a hotdog and $5.1 on a salad. The total of the lunch bill was $10.46. How much did... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_45210 (h₁ : 10.46 - 5.10 = 5.36) : 10.46 - 5.10 = 5.36 := by
/-
Given that Sara spent a total of $10.46 on a hotdog and $5.10 on a salad, ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Sara got fast food for lunch. She spent some amount on a hotdog and $5.1 on a salad. The total of the lunch bill was $10.46. How much did... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-512608 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- For any real number x, x(x+3) ≥ 0 if and only if x ≥ 0 or x ≤ -3 -/
theorem quadratic_inequality_solution (x : ℝ) :
x * (x + 3) ≥ 0 ↔ ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem quadratic_inequality_solution (x : ℝ) :
x * (x + 3) ≥ 0 ↔ x ≥ 0 ∨ x ≤ -3 := by
constructor
intro h
apply or_iff_not_imp_right.mpr
intro h... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- For any real number x, x(x+3) ≥ 0 if and only if x ≥ 0 or x ≤ -3 -/
theorem quadratic_inequality_solution (x : ℝ) :
x * (x + 3) ≥ 0 ↔ ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1116627 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_340954 (k : ℝ) (P : ℝ → ℝ)
(h_constraint : ∀ V, P V * V + (P V)^2 = k)
(h_diff : DifferentiableAt ℝ P (2*k))
(h_co... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_340954 (k : ℝ) (P : ℝ → ℝ)
(h_constraint : ∀ V, P V * V + (P V)^2 = k)
(h_diff : DifferentiableAt ℝ P (2*k))
(h_condition : P (2*k + P (2*k)) = k... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_340954 (k : ℝ) (P : ℝ → ℝ)
(h_constraint : ∀ V, P V * V + (P V)^2 = k)
(h_diff : DifferentiableAt ℝ P (2*k))
(h_co... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-676364 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Expand and simplify the expression: $\frac{3}{7}\left(\frac{7}{x^2} - 5x^3 + 2\right)$.-/
theorem lean_workbook_plus_30121 (x : ℝ) : (3 ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_30121 (x : ℝ) : (3 / 7) * (7 / x^2 - 5 * x^3 + 2) = 3 / x^2 - 15 * x^3 / 7 + 6 / 7 := by
/-
To prove the given theorem, we start by ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Expand and simplify the expression: $\frac{3}{7}\left(\frac{7}{x^2} - 5x^3 + 2\right)$.-/
theorem lean_workbook_plus_30121 (x : ℝ) : (3 ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1192067 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_74576315 (x : ℝ) (h₀ : (-1 : ℝ) < x ∧ x ≤ (2 : ℝ))
(h₁ :
√((2 : ℝ) - x) + √((2 : ℝ) + (2 : ℝ) * x) =
√((x ^ (4 : ℕ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_74576315 (x : ℝ) (h₀ : (-1 : ℝ) < x ∧ x ≤ (2 : ℝ))
(h₁ :
√((2 : ℝ) - x) + √((2 : ℝ) + (2 : ℝ) * x) =
√((x ^ (4 : ℕ) + (1 : ℝ)) / (x ^ (2 : ℕ) ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_74576315 (x : ℝ) (h₀ : (-1 : ℝ) < x ∧ x ≤ (2 : ℝ))
(h₁ :
√((2 : ℝ) - x) + √((2 : ℝ) + (2 : ℝ) * x) =
√((x ^ (4 : ℕ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=387128 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_43817' (a b c : ℝ) (h : a + b + c = 0) : (a * b) ^ 2 / (a ^ 2 + b ^ 2 + c ^ 2) + (b * c) ^ 2 / (b ^ 2 + a ^ 2 + c ^ 2) + ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_43817' (a b c : ℝ) (h : a + b + c = 0) : (a * b) ^ 2 / (a ^ 2 + b ^ 2 + c ^ 2) + (b * c) ^ 2 / (b ^ 2 + a ^ 2 + c ^ 2) + (c * a) ^ 2 / (c ^ 2 + a ^ 2 ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_43817' (a b c : ℝ) (h : a + b + c = 0) : (a * b) ^ 2 / (a ^ 2 + b ^ 2 + c ^ 2) + (b * c) ^ 2 / (b ^ 2 + a ^ 2 + c ^ 2) + ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-935558 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Ethan and Arianna are running a km long race. Ethan finished the race when Arianna was some distance from the start line. They were 816 m... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_25688 (d : ℝ)
(h₀ : d = 1000 - 816) :
d = 184 := by
rw [h₀]
ring
<;> linarith
```
<|endoftext|> | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Ethan and Arianna are running a km long race. Ethan finished the race when Arianna was some distance from the start line. They were 816 m... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1023926 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The units digit of 2^n follows a pattern of [2,4,8,6], and 2^2015 has the same units digit as 2^3 -/
theorem units_digit_power_two_2015 :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem units_digit_power_two_2015 : 2^2015 % 10 = 8 := by
norm_num [Nat.mod_eq_of_lt (by norm_num : 2 < 10)]
<;> rfl
<;> simp [Nat.pow_mod, Nat.mod_eq_of_lt (... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The units digit of 2^n follows a pattern of [2,4,8,6], and 2^2015 has the same units digit as 2^3 -/
theorem units_digit_power_two_2015 :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=425694 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2188 : ∀ (a b c d : ℝ), a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧ a * b * c * d = 1 → 1 / (a + b + 2) + 1 / (c + d + 2) ≤ 1 / 2 * ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2188 : ∀ (a b c d : ℝ), a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧ a * b * c * d = 1 → 1 / (a + b + 2) + 1 / (c + d + 2) ≤ 1 / 2 * √ (50 / 3) := by
intro... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_2188 : ∀ (a b c d : ℝ), a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧ a * b * c * d = 1 → 1 / (a + b + 2) + 1 / (c + d + 2) ≤ 1 / 2 * ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=849120 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_49284 (x y z : ℝ) (hx : x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0) (h : x + y + z = 1) : x ^ 3 * y ^ 3 * z ^ 3 + 4 * (x * y * z) ≤ (x +... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_49284 (x y z : ℝ) (hx : x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0) (h : x + y + z = 1) : x ^ 3 * y ^ 3 * z ^ 3 + 4 * (x * y * z) ≤ (x + y + z) ^ 3 := by
have h... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_49284 (x y z : ℝ) (hx : x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0) (h : x + y + z = 1) : x ^ 3 * y ^ 3 * z ^ 3 + 4 * (x * y * z) ≤ (x +... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=69723 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_1105 (x : ℝ) : 1 / 12 ≤ (1 / 4) * (sin x ^ 4 + cos x ^ 4) - (1 / 6) * (sin x ^ 6 + cos x ^ 6) ∧ (1 - sin (2*x) ^ 2) ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_1105 (x : ℝ) : 1 / 12 ≤ (1 / 4) * (sin x ^ 4 + cos x ^ 4) - (1 / 6) * (sin x ^ 6 + cos x ^ 6) ∧ (1 - sin (2*x) ^ 2) * cos x ≤ 1 := by
have h... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_1105 (x : ℝ) : 1 / 12 ≤ (1 / 4) * (sin x ^ 4 + cos x ^ 4) - (1 / 6) * (sin x ^ 6 + cos x ^ 6) ∧ (1 - sin (2*x) ^ 2) ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-120270 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given a grocer's transaction where:
- Buying rate is 3 pounds per $0.50
- Selling rate is 4 pounds per $1.00
- Total profit is $8.00
The... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem grocer_banana_problem :
let buy_rate : Rat := 3
let sell_rate : Rat := 4
let profit_per : Rat := 2
let profit : Rat := 8
let total_quantity : Rat := 9... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given a grocer's transaction where:
- Buying rate is 3 pounds per $0.50
- Selling rate is 4 pounds per $1.00
- Total profit is $8.00
The... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1046749 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that if $ a $ and $ b $ are integers such that $ ab $ is a prime number, then one of $ a $ or $ b $ must be $ \pm 1 $, and the other... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_87144_0_gs1 (a b : ℤ) (h : Nat.Prime (Int.natAbs (a * b))) :
((a = 1 ∨ a = -1) ∧ Nat.Prime (Int.natAbs b)) ∨
((b = 1 ∨ b = -1) ∧ Nat.Prime (Int.nat... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that if $ a $ and $ b $ are integers such that $ ab $ is a prime number, then one of $ a $ or $ b $ must be $ \pm 1 $, and the other... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1394908 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_207705_0_gs2_f0_gs4 (a b c : ℕ)
(ha : a > 0) (hb : b > 0) (hc : c > 0)
(d : ℕ) (hd_pos : d > 1) (hd_gcd : Nat.gcd (N... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_207705_0_gs2_f0_gs4 (a b c : ℕ)
(ha : a > 0) (hb : b > 0) (hc : c > 0)
(d : ℕ) (hd_pos : d > 1) (hd_gcd : Nat.gcd (Nat.gcd a b) c = d) :
Set.Infi... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_207705_0_gs2_f0_gs4 (a b c : ℕ)
(ha : a > 0) (hb : b > 0) (hc : c > 0)
(d : ℕ) (hd_pos : d > 1) (hd_gcd : Nat.gcd (N... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-913701 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given two bikers with speeds 18 mph and 15 mph respectively biking for 6 hours,
prove that the difference in their traveled distances... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem bike_distance_difference
(alberto_speed bjorn_speed : ℝ)
(time : ℝ)
(h_alberto : alberto_speed = 18)
(h_bjorn : bjorn_speed = 15)
(h_time : time = 6) :... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given two bikers with speeds 18 mph and 15 mph respectively biking for 6 hours,
prove that the difference in their traveled distances... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1523648 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- There are no real solutions to the equation (2x - 6)² + 4 = -(x - 3) -/
theorem no_solutions_quadratic : ∀ x : ℝ, (2*x - 6)^2 + 4 ≠ -(x -... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem no_solutions_quadratic : ∀ x : ℝ, (2*x - 6)^2 + 4 ≠ -(x - 3) := by
intro x
nlinarith [sq_nonneg (2*x - 6), sq_nonneg (x - 3), sq_nonneg (x - 1), sq_no... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- There are no real solutions to the equation (2x - 6)² + 4 = -(x - 3) -/
theorem no_solutions_quadratic : ∀ x : ℝ, (2*x - 6)^2 + 4 ≠ -(x -... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=2528 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-For what value of $y$ does $5^{2y} \cdot 25^{y} = 125^{4}$?
A) 1
B) 2
C) 3
D) 4
E) 5-/
theorem lean_workbook_plus_34935 :
5^(2*y) * 25... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_34935 :
5^(2*y) * 25^y = 125^4 ↔ y = 3 := by
have h_main : 5^(2*y) * 25^y = 125^4 ↔ y = 3 := by
constructor
· -- Prove the ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-For what value of $y$ does $5^{2y} \cdot 25^{y} = 125^{4}$?
A) 1
B) 2
C) 3
D) 4
E) 5-/
theorem lean_workbook_plus_34935 :
5^(2*y) * 25... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-31928 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- A number n is fantabulous if starting from any element in {n, 2n+1, 3n},
if one element is fantabulous, all elements are fantabulous... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def IsFantabulous (n : ℕ) : Prop := by
use n = 0
<;> simp
<;> aesop
theorem theorem_11342 (n : ℕ) : IsFantabulous n ↔ n = 0 := by
constructor
-- Case 1... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- A number n is fantabulous if starting from any element in {n, 2n+1, 3n},
if one element is fantabulous, all elements are fantabulous... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1141574 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Disprove the following statement: "Given positive constants \(a, b, c\) and nonnegative numbers \(x, y, z\) such that \((a \sqrt{x} + b \sqr... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_104145_negation : ¬ ( ∀ (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c),
sInf {B : ℝ | ∃ x y z : ℝ, 0 ≤ x ∧ 0 ≤ y ∧ 0 ≤ z ∧
(a * Real.sqrt x ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Disprove the following statement: "Given positive constants \(a, b, c\) and nonnegative numbers \(x, y, z\) such that \((a \sqrt{x} + b \sqr... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1541559 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The area of an isosceles trapezoid with bases of length 7 and 13, and side length 5 is 40 -/
theorem isosceles_trapezoid_area :
let a ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem isosceles_trapezoid_area :
let a := 40
a = 40 := by
let a := 40
rfl
```
<|endoftext|> | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The area of an isosceles trapezoid with bases of length 7 and 13, and side length 5 is 40 -/
theorem isosceles_trapezoid_area :
let a ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1344569 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Let \( a \) be a positive number. Consider the system of equations:
\[
\begin{cases}
y = a x^2 \\
y^2 + 3 = x^2 + 4y
\end{cases}
\]
How... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_23190 (a : ℝ)
(f : ℝ → ℝ)
(h₀ : ∀ x, f x = a * x^2)
(h₁ : ∀ x, (f x)^2 + 3 = x^2 + 4 * f x) :
{x | ∃ y, y = a * x^2 ∧ y^2 + 3 = x^2 + 4 * y} ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Let \( a \) be a positive number. Consider the system of equations:
\[
\begin{cases}
y = a x^2 \\
y^2 + 3 = x^2 + 4y
\end{cases}
\]
How... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-182427 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given that a number divided by d plus 9 equals 15, and the number is 33,
prove that d equals 5.5 -/
theorem find_divisor (d : ℝ)
... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem find_divisor (d : ℝ)
(h1 : 33 / d + 9 = 15) : d = 5.5 := by
have h2 : d ≠ 0 := by
intro h
rw [h] at h1
norm_num at h1
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given that a number divided by d plus 9 equals 15, and the number is 33,
prove that d equals 5.5 -/
theorem find_divisor (d : ℝ)
... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-577054 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a base d > 7 and digits A, B satisfying the equation AB_d + AA_d = 174_d
and A > B, prove that A - B = 3 -/
theorem digit_diffe... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem digit_difference_is_three
(d : ℕ) (A B : ℕ)
(hd : d > 7)
(hA : A < d) -- A is a digit in base d
(hB : B < d) -- B is a digit in base d
(hAB :... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a base d > 7 and digits A, B satisfying the equation AB_d + AA_d = 174_d
and A > B, prove that A - B = 3 -/
theorem digit_diffe... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=60506 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_212_37 (x : ℝ) (h₁ : 0 < x) (h₂ : x ≠ 1) (h₃ : 3 * logb 2 x ^ 2 - 3 * logb 2 x + 1 = 0) : x = 2 / 4 := by sorry
```
Repl... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_212_37 (x : ℝ) (h₁ : 0 < x) (h₂ : x ≠ 1) (h₃ : 3 * logb 2 x ^ 2 - 3 * logb 2 x + 1 = 0) : x = 2 / 4 := by
have h₄ : False := by
have h₅ ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_212_37 (x : ℝ) (h₁ : 0 < x) (h₂ : x ≠ 1) (h₃ : 3 * logb 2 x ^ 2 - 3 * logb 2 x + 1 = 0) : x = 2 / 4 := by sorry
```
Repl... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1138592 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-In a tetrahedron, all four faces are congruent isosceles triangles with a base \(a\) and a lateral side \(b\). Calculate the volume of th... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_10172 (a b : ℝ)
(h₀ : 0 < a ∧ 0 < b)
(h₁ : 0 < a ∧ 0 < b → a < b * Real.sqrt 2)
: 0 < a ∧ 0 < b ∧ a < b * Real.sqrt 2 → a^2 * Real.s... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-In a tetrahedron, all four faces are congruent isosceles triangles with a base \(a\) and a lateral side \(b\). Calculate the volume of th... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1371778 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Main theorem: Set equalities for complements and intersections
Given sets A = {x | -1 ≤ x < 3} and B = {x | x ≥ 2} in ℝ,
proves p... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem set_complement_intersection_theorem :
let A : Set ℝ := {x | -1 ≤ x ∧ x < 3}
let B : Set ℝ := {x | 2 ≤ x}
(Set.univ \ (A ∩ B) = {x | x < 2 ∨ 3 ≤ x}) ∧
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Main theorem: Set equalities for complements and intersections
Given sets A = {x | -1 ≤ x < 3} and B = {x | x ≥ 2} in ℝ,
proves p... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=373484 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
A quadratic equation (k-2)x² + 2x - 1 = 0 has two distinct real roots
if and only if k > 1 and k ≠ 2
-/
theorem quadratic_two_roots_iff... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem quadratic_two_roots_iff (k : ℝ) :
(∃ x₁ x₂ : ℝ, x₁ ≠ x₂ ∧
(k - 2) * x₁^2 + 2 * x₁ - 1 = 0 ∧
(k - 2) * x₂^2 + 2 * x₂ - 1 = 0) ↔
(k > 1 ∧ k ≠ 2) := by... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
A quadratic equation (k-2)x² + 2x - 1 = 0 has two distinct real roots
if and only if k > 1 and k ≠ 2
-/
theorem quadratic_two_roots_iff... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-164479 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-The school cafeteria ordered 25 red apples and 17 green apples for students lunches. But, if a certain number of students wanted fruit an... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_76203 (h₀ : 25 + 17 = 42)
(h₁ : 42 - 32 = 10) :
10 = 10 := by
/-
Given that the school cafeteria ordered 25 red apples and 17 g... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-The school cafeteria ordered 25 red apples and 17 green apples for students lunches. But, if a certain number of students wanted fruit an... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-87000 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-The average age of the 30 members of a sports camp is 20 years. There are 10 girls, 15 boys, and 5 coaches. If the average age of the gir... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_27030 (n : ℕ)
(h₀ : n = 30)
(h₁ : n * 20 = 600)
(h₂ : 10 * 18 = 180)
(h₃ : 15 * 19 = 285)
(h₄ : 600 - 180 - 285 = 135)
(h₅ : 1... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-The average age of the 30 members of a sports camp is 20 years. There are 10 girls, 15 boys, and 5 coaches. If the average age of the gir... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-626195 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_16263 (a b : ℝ)
(f : ℝ → ℝ)
(h₀ : ∀ x, f x = a * x^3 - b * x + 2)
(h₁ : f (-5) = 17) :
f 5 = -13 := by sorry
`... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_16263 (a b : ℝ)
(f : ℝ → ℝ)
(h₀ : ∀ x, f x = a * x^3 - b * x + 2)
(h₁ : f (-5) = 17) :
f 5 = -13 := by
have h₂ := h₀ 5
have h₃ ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_16263 (a b : ℝ)
(f : ℝ → ℝ)
(h₀ : ∀ x, f x = a * x^3 - b * x + 2)
(h₁ : f (-5) = 17) :
f 5 = -13 := by sorry
`... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=611890 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5245 (x : ℕ → ℝ) (h₀ : ∀ n, x n ^ 2 = 1) (h₁ : ∏ k in Finset.range n, x k = 1) : ∏ k in Finset.range n, x k ^ 3 ≤ 3^n ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5245 (x : ℕ → ℝ) (h₀ : ∀ n, x n ^ 2 = 1) (h₁ : ∏ k in Finset.range n, x k = 1) : ∏ k in Finset.range n, x k ^ 3 ≤ 3^n := by
have h₂ : ∀ (k : ℕ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5245 (x : ℕ → ℝ) (h₀ : ∀ n, x n ^ 2 = 1) (h₁ : ∏ k in Finset.range n, x k = 1) : ∏ k in Finset.range n, x k ^ 3 ≤ 3^n ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1619838 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_81554 :
(80 + 5)^2 = 7225 := by sorry
```
Replace every sorry statement with an appropriate proof. Before writi... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_81554 :
(80 + 5)^2 = 7225 := by norm_num
<;> rfl
-- This was just a simple arithmetic check.
-- Now, let's try to understand why thi... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_81554 :
(80 + 5)^2 = 7225 := by sorry
```
Replace every sorry statement with an appropriate proof. Before writi... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1183273 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Let \(a, b, c > 0\). Define the function \(f(a, b, c) = a^2 + b^2 + c^2 - \frac{1}{3}(a+b+c)^2 - \frac{2}{3}\left(\frac{a^2+b^2}{a+b} - c\... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_491773 :
let f := fun (a b c : ℝ) => a^2 + b^2 + c^2 - (1/3) * (a + b + c)^2 - (2/3) * ((a^2 + b^2)/(a + b) - c)^2
(∀ a b c : ℝ, 0 < a → 0 < b → 0 <... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Let \(a, b, c > 0\). Define the function \(f(a, b, c) = a^2 + b^2 + c^2 - \frac{1}{3}(a+b+c)^2 - \frac{2}{3}\left(\frac{a^2+b^2}{a+b} - c\... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=903077 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Main theorem: For positive integers a, b, c where:
1) All are prime numbers
2) 2a + 3b = c
3) 4a + c + 1 = 4b
4) abc < 10... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem unique_prime_product (a b c : ℕ)
(ha : Nat.Prime a)
(hb : Nat.Prime b)
(hc : Nat.Prime c)
(h1 : 2 * a + 3 * b = c)
(h2 : 4 * a + c + 1 = 4 * b)... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Main theorem: For positive integers a, b, c where:
1) All are prime numbers
2) 2a + 3b = c
3) 4a + c + 1 = 4b
4) abc < 10... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=678298 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If tan(α) = 3 and tan(α+β) = 2, then tan(β) = -1/7 -/
theorem tan_value_problem (α β : ℝ)
(h1 : tan α = 3)
(h2 : tan (α + β) = 2... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem tan_value_problem (α β : ℝ)
(h1 : tan α = 3)
(h2 : tan (α + β) = 2)
(h3 : cos α ≠ 0)
(h4 : cos (α + β) ≠ 0)
(h5 : 1 - 3 * (tan β) ≠ 0) :
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If tan(α) = 3 and tan(α+β) = 2, then tan(β) = -1/7 -/
theorem tan_value_problem (α β : ℝ)
(h1 : tan α = 3)
(h2 : tan (α + β) = 2... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1110968 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that for all positive real numbers $p, q, r$, the product $(p + q + r)(pq + qr + rp)$ is at least $9pqr$, and determine the conditio... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_73557_0_gs3 (p q r : ℝ) (hp : 0 < p) (hq : 0 < q) (hr : 0 < r) :
(p + q + r) * (p * q + q * r + r * p) ≥ 9 * p * q * r ∧
((p + q + r) * (p * q + q ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that for all positive real numbers $p, q, r$, the product $(p + q + r)(pq + qr + rp)$ is at least $9pqr$, and determine the conditio... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=111352 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_25365 (a b c : ℝ) (h₀ : a ≥ 0 ∧ b ≥ 0 ∧ c ≥ 0) (h₁ : a+b+c=1) :
a^4 + b^4 + c^4 ≥ a^3 * b + b^3 * c + c^3 * a - 3 * ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_25365 (a b c : ℝ) (h₀ : a ≥ 0 ∧ b ≥ 0 ∧ c ≥ 0) (h₁ : a+b+c=1) :
a^4 + b^4 + c^4 ≥ a^3 * b + b^3 * c + c^3 * a - 3 * (a * b^3 + b * c^3 + c * a^3)... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_25365 (a b c : ℝ) (h₀ : a ≥ 0 ∧ b ≥ 0 ∧ c ≥ 0) (h₁ : a+b+c=1) :
a^4 + b^4 + c^4 ≥ a^3 * b + b^3 * c + c^3 * a - 3 * ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1477617 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Classical Polynomial
/-- Prove that for any positive integer $ k $ and real number $ x \in (k, k + \frac{1}{2}] $, the value of $ \left\lfloor \... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Classical Polynomial
theorem OMR_problem_71070_3_gs0_f0_negation : ¬ ( ∀ (k : ℕ) (hk : 0 < k) (x : ℝ) (hx : x ∈ Set.Ioc (k : ℝ) ((k : ℝ) + 1/2)),
(∀ m : ℝ, m ≤ (k : ℝ) →... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Classical Polynomial
/-- Prove that for any positive integer $ k $ and real number $ x \in (k, k + \frac{1}{2}] $, the value of $ \left\lfloor \... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1408831 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-In a convex quadrilateral $ABCD$, $\angle BAC=\angle DAC=55^\circ$, $\angle DCA=20^\circ$, and $\angle BCA=15^\circ$. Find the measure of... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_81456 (a b c d : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(h₁ : a + b + c + d = 360)
(h₂ : a = 55)
(h₃ : b = 55)
(h₄ : c = 20)
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-In a convex quadrilateral $ABCD$, $\angle BAC=\angle DAC=55^\circ$, $\angle DCA=20^\circ$, and $\angle BCA=15^\circ$. Find the measure of... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-914197 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The sum of solutions to the equation (6x)/30 = 7/x is 0 -/
theorem solutions_sum_is_zero :
let x₁ := Real.sqrt 35
let x₂ := -Real.sq... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem solutions_sum_is_zero :
let x₁ := Real.sqrt 35
let x₂ := -Real.sqrt 35
x₁ + x₂ = 0 := by
let x₁ := Real.sqrt 35
let x₂ := -Real.sqrt 35
si... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The sum of solutions to the equation (6x)/30 = 7/x is 0 -/
theorem solutions_sum_is_zero :
let x₁ := Real.sqrt 35
let x₂ := -Real.sq... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1596240 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given that $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a} \perp (\overrightarrow{a} - 2\overrightarro... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_5594 (a b : ℝ)
(h₀ : 0 < a ∧ 0 < b)
(h₁ : a * b = 1)
(h₂ : a^2 + b^2 = 2)
(h₃ : a * (a - 2 * b) = 0) :
Real.cos (Real.arccos (Re... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given that $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a} \perp (\overrightarrow{a} - 2\overrightarro... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1348079 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_50684_0_gp4 :
∃ (min_sum : ℕ),
(∃ (a b c d : ℤ), (a, b, c, d) ≠ (0, 0, 0, 0) ∧
a^3 + b^4 + c^5 = 2 * d^4 ∧... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_50684_0_gp4 :
∃ (min_sum : ℕ),
(∃ (a b c d : ℤ), (a, b, c, d) ≠ (0, 0, 0, 0) ∧
a^3 + b^4 + c^5 = 2 * d^4 ∧
Int.natAbs a + Int.natA... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_50684_0_gp4 :
∃ (min_sum : ℕ),
(∃ (a b c d : ℤ), (a, b, c, d) ≠ (0, 0, 0, 0) ∧
a^3 + b^4 + c^5 = 2 * d^4 ∧... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1266553 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_264075_0_gs2_f0_gs0 (n : ℕ) (A : Matrix (Fin n) (Fin n) ℂ) :
A * A.adjugate = A.det • (1 : Matrix (Fin n) (Fin n) ℂ) :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_264075_0_gs2_f0_gs0 (n : ℕ) (A : Matrix (Fin n) (Fin n) ℂ) :
A * A.adjugate = A.det • (1 : Matrix (Fin n) (Fin n) ℂ) := by
have h_main : A * A.a... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_264075_0_gs2_f0_gs0 (n : ℕ) (A : Matrix (Fin n) (Fin n) ℂ) :
A * A.adjugate = A.det • (1 : Matrix (Fin n) (Fin n) ℂ) :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1170871 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291242_0_gp2 :
let f : ℝ → ℝ := fun x => Real.exp (x^2) + Real.sin (x^2) + Real.cos (x^2) - 2 - 2*x^2
f 0 + (deriv f... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291242_0_gp2 :
let f : ℝ → ℝ := fun x => Real.exp (x^2) + Real.sin (x^2) + Real.cos (x^2) - 2 - 2*x^2
f 0 + (deriv f) 0 + (deriv (deriv f)) 0 = 0 :... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291242_0_gp2 :
let f : ℝ → ℝ := fun x => Real.exp (x^2) + Real.sin (x^2) + Real.cos (x^2) - 2 - 2*x^2
f 0 + (deriv f... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1327636 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/- Prove that the sum of the cubes of the first $n$ natural numbers is equal to $(\frac{n*(n+1)}{2})^{2}$. -/
theorem lean_workbook_7791 (n :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_7791 (n : ℕ) : ∑ i in Finset.range (n+1), i^3 = ((n * (n+1)) / 2)^2 := by
have h_main : ∀ (n : ℕ), ∑ i in Finset.range (n + 1), i ^ 3 = ((n * (n... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/- Prove that the sum of the cubes of the first $n$ natural numbers is equal to $(\frac{n*(n+1)}{2})^{2}$. -/
theorem lean_workbook_7791 (n :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=886942 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_35691 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a^2 / (2 * a + b) + b^2 / (2 * b + c) + ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_35691 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a^2 / (2 * a + b) + b^2 / (2 * b + c) + c^2 / (2 * c + a) ≤ 9 / 2 ∧ a... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_35691 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a^2 / (2 * a + b) + b^2 / (2 * b + c) + ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1600020 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_29307 (m : ℝ)
(h₀ : 0 < m^2 - 4)
(h₁ : 0 ≥ 16 * (m - 2)^2 - 16)
(h₂ : 0 ≤ m^2 - 4)
(h₃ : 0 < 16 * (m - 2... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_29307 (m : ℝ)
(h₀ : 0 < m^2 - 4) -- $P$ is true: The equation $x^{2}+mx+1=0$ has two distinct real roots
(h₁ : 0 ≥ 16 * (m - 2)^2 - 16) -- $Q... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_29307 (m : ℝ)
(h₀ : 0 < m^2 - 4)
(h₁ : 0 ≥ 16 * (m - 2)^2 - 16)
(h₂ : 0 ≤ m^2 - 4)
(h₃ : 0 < 16 * (m - 2... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1518087 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_451944_2_gs1_f0_gs3 :
let S : Finset ℚ := {1/2, 1/4, 1/8, 1/16}
∀ (T : Finset ℚ), T ⊆ S → T.sum id ≠ 1 := by sorry
`... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_451944_2_gs1_f0_gs3 :
let S : Finset ℚ := {1/2, 1/4, 1/8, 1/16}
∀ (T : Finset ℚ), T ⊆ S → T.sum id ≠ 1 := by
intro S T hT
have h_sum_S : ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_451944_2_gs1_f0_gs3 :
let S : Finset ℚ := {1/2, 1/4, 1/8, 1/16}
∀ (T : Finset ℚ), T ⊆ S → T.sum id ≠ 1 := by sorry
`... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-820718 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If angle θ satisfies sin θ cos θ < 0 and cos θ - sin θ < 0,
then θ is in the second quadrant -/
theorem angle_in_second_quadrant (θ :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem angle_in_second_quadrant (θ : ℝ)
(h1 : sin θ * cos θ < 0)
(h2 : cos θ - sin θ < 0) :
cos θ < 0 ∧ sin θ > 0 := by
constructor
case left =... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If angle θ satisfies sin θ cos θ < 0 and cos θ - sin θ < 0,
then θ is in the second quadrant -/
theorem angle_in_second_quadrant (θ :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=318107 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5259 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 1) :
|a * b * (a + b) * (b + c) * c... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5259 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 1) :
|a * b * (a + b) * (b + c) * c * (a + c)| ≤ 4 := by
ha... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_5259 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 1) :
|a * b * (a + b) * (b + c) * c... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-211125 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem power_function_log (f : ℝ → ℝ) (h1 : ∀ x > 0, f x = x ^ (1/2))
(h2 : f 4 = 2) : logb (1/4) (f 2) = -1/4 := by sorry
```
Replace... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem power_function_log (f : ℝ → ℝ) (h1 : ∀ x > 0, f x = x ^ (1/2))
(h2 : f 4 = 2) : logb (1/4) (f 2) = -1/4 := by
have h3 := h1 4 (by norm_num)
have h... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem power_function_log (f : ℝ → ℝ) (h1 : ∀ x > 0, f x = x ^ (1/2))
(h2 : f 4 = 2) : logb (1/4) (f 2) = -1/4 := by sorry
```
Replace... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1219554 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_67270 (b c s bc bs cs bcs : ℕ)
(h₀ : b = 16)
(h₁ : c = 11)
(h₂ : s = 10)
(h₃ : bc = 5)
(h₄ : bs = 4)
... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_67270 (b c s bc bs cs bcs : ℕ)
(h₀ : b = 16)
(h₁ : c = 11)
(h₂ : s = 10)
(h₃ : bc = 5)
(h₄ : bs = 4)
(h₅ : cs = 3)
(h₆ : bcs = 2... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_67270 (b c s bc bs cs bcs : ℕ)
(h₀ : b = 16)
(h₁ : c = 11)
(h₂ : s = 10)
(h₃ : bc = 5)
(h₄ : bs = 4)
... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-558384 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given an ordered pair (x,y) satisfying x² + y² = 18x + 40y,
the maximum value of y is 20 + √481 -/
theorem max_y_value {x y : ℝ} (h :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem max_y_value {x y : ℝ} (h : x^2 + y^2 = 18*x + 40*y) :
y ≤ 20 + Real.sqrt 481 := by
have h₁ : 0 ≤ Real.sqrt 481 := Real.sqrt_nonneg 481
nlinarith [s... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given an ordered pair (x,y) satisfying x² + y² = 18x + 40y,
the maximum value of y is 20 + √481 -/
theorem max_y_value {x y : ℝ} (h :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-832132 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given a sequence where $a_1=5$, $a_8=19$, and $a_n=pn+q$ ($p$, $q$ are constants) ($n\in\mathbb{N}^*$), find the value of $a_5$.-/
theor... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_44224 (p q : ℝ)
(a : ℕ → ℝ)
(h₀ : ∀ n, a n = p * n + q)
(h₁ : a 1 = 5)
(h₂ : a 8 = 19) :
a 5 = 13 := by
have h₃ := h₀ 1
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Given a sequence where $a_1=5$, $a_8=19$, and $a_n=pn+q$ ($p$, $q$ are constants) ($n\in\mathbb{N}^*$), find the value of $a_5$.-/
theor... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1504380 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_303227_0_gp9 :
let f := fun x y z => (y^3 + x) * (z^3 + y) * (x^3 + z) / (x * y * z) + (x - 2)^2 + (y - 2)^2 + (z - 2)... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_303227_0_gp9 :
let f := fun x y z => (y^3 + x) * (z^3 + y) * (x^3 + z) / (x * y * z) + (x - 2)^2 + (y - 2)^2 + (z - 2)^2
(∀ x y z : ℝ, x ≥ 2 → y ≥ ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_303227_0_gp9 :
let f := fun x y z => (y^3 + x) * (z^3 + y) * (x^3 + z) / (x * y * z) + (x - 2)^2 + (y - 2)^2 + (z - 2)... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1392055 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-If a certain percentage of 3/5 of a number is 36, and the number is 120, what is the percentage?-/
theorem lean_workbook_plus_63301 (n ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_63301 (n : ℝ)
(h₀ : n = 120)
(h₁ : 3 / 5 * n = 72)
(h₂ : P / 100 * 72 = 36) :
P = 50 := by
have h₃ : P / 100 = 0 := by
b... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-If a certain percentage of 3/5 of a number is 36, and the number is 120, what is the percentage?-/
theorem lean_workbook_plus_63301 (n ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-388653 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_18612 (x : ℝ)
(h₀ : 0 < x)
(h₁ : 0.08 * (100 - x) + 0.4 * x = 16) :
x = 25 := by sorry
```
Replace every sorry ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_18612 (x : ℝ)
(h₀ : 0 < x)
(h₁ : 0.08 * (100 - x) + 0.4 * x = 16) :
x = 25 := by
norm_num at h₁
ring_nf at h₁
linarith
```
<... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_18612 (x : ℝ)
(h₀ : 0 < x)
(h₁ : 0.08 * (100 - x) + 0.4 * x = 16) :
x = 25 := by sorry
```
Replace every sorry ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1261150 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_77182 (x y : ℝ) (h₀ : 0 < x) (h₁ : 0 < y) (h₂ : x ≠ y) (h₃ : x^2 / y + y^2 / x = 2) : Real.sqrt (x * y) ≠ x * y * (x... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_77182 (x y : ℝ) (h₀ : 0 < x) (h₁ : 0 < y) (h₂ : x ≠ y) (h₃ : x^2 / y + y^2 / x = 2) : Real.sqrt (x * y) ≠ x * y * (x + y) / 4 := by
have h_m... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_77182 (x y : ℝ) (h₀ : 0 < x) (h₁ : 0 < y) (h₂ : x ≠ y) (h₃ : x^2 / y + y^2 / x = 2) : Real.sqrt (x * y) ≠ x * y * (x... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=41075 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem problem4 (a : ℝ) (h₀ : 0 < a) (h₁ : a < 1) (h₂ : (a - 1) ^ 2 = 2 * a) :
Complex.abs (a - 1) + Complex.abs (a + 1) ≥ 2 * a :... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem problem4 (a : ℝ) (h₀ : 0 < a) (h₁ : a < 1) (h₂ : (a - 1) ^ 2 = 2 * a) :
Complex.abs (a - 1) + Complex.abs (a + 1) ≥ 2 * a := by
have h₃ : Complex.a... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem problem4 (a : ℝ) (h₀ : 0 < a) (h₁ : a < 1) (h₂ : (a - 1) ^ 2 = 2 * a) :
Complex.abs (a - 1) + Complex.abs (a + 1) ≥ 2 * a :... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Computation.of_results_think.lean | uw-math-ai/APRIL | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
import Mathlib.Data.Seq.Computation
open Computation
open Function
variable {α : Type u} {β : Type v} {γ : Type w}
variable (R : Computation α → Com... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
import Mathlib.Data.Seq.Computation
open Computation
open Function
variable {α : Type u} {β : Type v} {γ : Type w}
variable (R : Computation α → Computation α → Prop)
variable (... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
import Mathlib.Data.Seq.Computation
open Computation
open Function
variable {α : Type u} {β : Type v} {γ : Type w}
variable (R : Computation α → Com... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1277269 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_95a82926 (a b c : ℝ) (ha : (0 : ℝ) < a) (hb : (0 : ℝ) < b) (hc : (0 : ℝ) < c) (h₁ : (0 : ℝ) < a * b)
(h₂ : (0 : ℝ) < b * c) ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_95a82926 (a b c : ℝ) (ha : (0 : ℝ) < a) (hb : (0 : ℝ) < b) (hc : (0 : ℝ) < c) (h₁ : (0 : ℝ) < a * b)
(h₂ : (0 : ℝ) < b * c) (h₃ : (0 : ℝ) < c * a) (h₄ :... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem extracted_95a82926 (a b c : ℝ) (ha : (0 : ℝ) < a) (hb : (0 : ℝ) < b) (hc : (0 : ℝ) < c) (h₁ : (0 : ℝ) < a * b)
(h₂ : (0 : ℝ) < b * c) ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-677446 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Seven socks, colored blue, brown, black, red, purple, green, and yellow are in a drawer. In how many different ways can we choose four so... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_30817 :
Nat.choose 7 4 = 35 := by
/-
To determine the number of ways to choose four socks from a drawer containing seven socks (colored ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Seven socks, colored blue, brown, black, red, purple, green, and yellow are in a drawer. In how many different ways can we choose four so... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1577361 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_28075 (l b : ℝ)
(h₀ : 0 < l ∧ 0 < b)
(h₁ : b = 8 * l)
(h₂ : b = 480)
(h₃ : 0 < 3200) :
(3200 : ℝ) / (l ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_28075 (l b : ℝ)
(h₀ : 0 < l ∧ 0 < b)
(h₁ : b = 8 * l)
(h₂ : b = 480)
(h₃ : 0 < 3200) :
(3200 : ℝ) / (l * b) = 1 / 9 := by
ha... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_28075 (l b : ℝ)
(h₀ : 0 < l ∧ 0 < b)
(h₁ : b = 8 * l)
(h₂ : b = 480)
(h₃ : 0 < 3200) :
(3200 : ℝ) / (l ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-11272 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If 400 sheets of paper make 4 cm, then 10 cm of the same paper contains 1000 sheets -/
theorem paper_thickness_calculation
(sheet_co... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem paper_thickness_calculation
(sheet_count₁ : ℕ := by
-- We'll prove this by showing that 400 sheets make 4 cm
-- and then using that to prove 1000 s... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- If 400 sheets of paper make 4 cm, then 10 cm of the same paper contains 1000 sheets -/
theorem paper_thickness_calculation
(sheet_co... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-846501 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Daryl is loading crates at a warehouse and wants to make sure that they are not overloaded. Each crate can weigh up to 20kg and he has 15... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_23898 (nails hammers planks : ℕ)
(nail_weight hammer_weight plank_weight : ℝ)
(h₀ : nails = 4)
(h₁ : hammers = 12)
(h₂ : planks = 10)
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Daryl is loading crates at a warehouse and wants to make sure that they are not overloaded. Each crate can weigh up to 20kg and he has 15... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-607601 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a river with width 55m and flow rate 1 kmph, if it discharges 2750 cubic meters
per minute into the sea, then its depth is appr... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem river_depth_calculation
(width : ℝ) (flow_rate_kmph : ℝ) (volume_per_min : ℝ)
(hw : width = 55)
(hf : flow_rate_kmph = 1)
(hv : volume_per_min = ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a river with width 55m and flow rate 1 kmph, if it discharges 2750 cubic meters
per minute into the sea, then its depth is appr... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=17124 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_5865 (f : ℝ → ℝ) (h₁ : f 0 = 0) (h₂ : ∀ x, f x ^ 2 + x ^ 2 = f (x ^ 2)) (x : ℝ) : |f x| = x * Real.sqrt (f x ^ 2 + x... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_5865 (f : ℝ → ℝ) (h₁ : f 0 = 0) (h₂ : ∀ x, f x ^ 2 + x ^ 2 = f (x ^ 2)) (x : ℝ) : |f x| = x * Real.sqrt (f x ^ 2 + x ^ 2) := by
have h₃ : Fa... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_5865 (f : ℝ → ℝ) (h₁ : f 0 = 0) (h₂ : ∀ x, f x ^ 2 + x ^ 2 = f (x ^ 2)) (x : ℝ) : |f x| = x * Real.sqrt (f x ^ 2 + x... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1117767 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_194918_0_gs1 (f : ℝ → ℝ)
(h : ∀ x y : ℝ, |f (x - y)| = |f x - f y|) :
∀ a : ℝ, |f a| = |f (-a)| := by sorry
```
Re... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_194918_0_gs1 (f : ℝ → ℝ)
(h : ∀ x y : ℝ, |f (x - y)| = |f x - f y|) :
∀ a : ℝ, |f a| = |f (-a)| := by
have h_f_zero : f 0 = 0 := by
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_194918_0_gs1 (f : ℝ → ℝ)
(h : ∀ x y : ℝ, |f (x - y)| = |f x - f y|) :
∀ a : ℝ, |f a| = |f (-a)| := by sorry
```
Re... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-229950 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that "a + b ≤ 2" is necessary but not sufficient for "a ≤ 1 and b ≤ 1" -/
theorem sum_bound_necessary_not_sufficient :
(∀ (a b : ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem sum_bound_necessary_not_sufficient :
-- First part: "a + b ≤ 2" is necessary for "a ≤ 1 and b ≤ 1"
(∀ (a b : ℝ), (a ≤ 1 ∧ b ≤ 1) → a + b ≤ 2) ∧
-- Second p... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that "a + b ≤ 2" is necessary but not sufficient for "a ≤ 1 and b ≤ 1" -/
theorem sum_bound_necessary_not_sufficient :
(∀ (a b : ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
april-feedback-c4388a5232267f81478aeb25bb108c18-normal | uw-math-ai/APRIL | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
theorem algebra_257669 (x : ℝ) (hx : x > 0) (h : 3000 / (1 + x / 3000) = 3000 - 120) :
x = 125 := by
sorry
```
Replace every sorry statement with an appropriate proof. Before writing any Lean code, give an informal s... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
theorem algebra_257669 (x : ℝ) (hx : x > 0) (h : 3000 / (1 + x / 3000) = 3000 - 120) :
x = 125 := by
have h1 : 1 + x / 3000 ≠ 0 := by
by_contra h2
have eq1 : x = -3000 := by
field_simp at h2 ⊢
linari... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
theorem algebra_257669 (x : ℝ) (hx : x > 0) (h : 3000 / (1 + x / 3000) = 3000 - 120) :
x = 125 := by
sorry
```
Replace every sorry statement with an appropriate proof. Before writing any Lean code, give an informal s... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-732687 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given a product with cost price 200 yuan and original price 300 yuan,
if we want to maintain at least 5% profit margin,
then the maximum... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem max_discount_rate
(cost_price : ℝ)
(market_price : ℝ)
(min_profit_margin : ℝ)
(h_cp : cost_price = 200)
(h_mp : market_price = 300)
(h_... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/--
Given a product with cost price 200 yuan and original price 300 yuan,
if we want to maintain at least 5% profit margin,
then the maximum... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=987004 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Solve the recurrence relation $ a_{n+1} = 4a_n - 4a_{n-1} $ for $ n \ge 1 $, given initial conditions $ a_0 = 1 $ and $ a_1 = 3 $. Prove t... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_464956_1_gs1 :
∃! (a : ℕ → ℝ),
(a 0 = 1) ∧
(a 1 = 3) ∧
(∀ n : ℕ, n ≥ 1 → a (n + 1) = 4 * a n - 4 * a (n - 1)) ∧
(∃ A B : ℝ, A = 1 ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Solve the recurrence relation $ a_{n+1} = 4a_n - 4a_{n-1} $ for $ n \ge 1 $, given initial conditions $ a_0 = 1 $ and $ a_1 = 3 $. Prove t... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-541636 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a function f: ℝ → ℝ satisfying f(x) + f(y) = 2f((x+y)/2)f((x-y)/2),
with f(0) ≠ 0 and f(1) = 0, then f(2) = -f(0) -/
theorem fu... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem func_period_prop (f : ℝ → ℝ)
(h_func : ∀ x y : ℝ, f x + f y = 2 * f ((x + y)/2) * f ((x - y)/2))
(h_f0_ne_0 : f 0 ≠ 0)
(h_f1_0 : f 1 = 0) :
f 2 = ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a function f: ℝ → ℝ satisfying f(x) + f(y) = 2f((x+y)/2)f((x-y)/2),
with f(0) ≠ 0 and f(1) = 0, then f(2) = -f(0) -/
theorem fu... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-641634 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The volume of a geometric body composed of two tetrahedra with equilateral triangle base
of side length 1 and height √3/2 is 1/4 -/
t... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem geometric_body_volume :
let tetra_base_area := Real.sqrt 3 / 4 -- area of equilateral triangle with side 1
let tetra_height := Real.sqrt 3 / 2
let single_... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- The volume of a geometric body composed of two tetrahedra with equilateral triangle base
of side length 1 and height √3/2 is 1/4 -/
t... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1073882 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that for any integer $a$ coprime to 24, $a^2 \equiv 1 \pmod{24}$. --/
theorem OMR_problem_377601_2_gs0 (a : ℤ) (h : Int.gcd a 24 = ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_377601_2_gs0 (a : ℤ) (h : Int.gcd a 24 = 1) : a^2 ≡ 1 [ZMOD 24] := by
have h_odd : a % 2 = 1 ∨ a % 2 = -1 := by
have h₁ : (a : ℤ) % 2 ≠ 0 ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Prove that for any integer $a$ coprime to 24, $a^2 \equiv 1 \pmod{24}$. --/
theorem OMR_problem_377601_2_gs0 (a : ℤ) (h : Int.gcd a 24 = ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=747708 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Two arithmetic sequences $\left\{ {{a}_{n}} \right\}$ and $\left\{ {{b}_{n}} \right\}$ satisfy the equation $\frac{{{a}_{1}}+{{a}_{2}}+..... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_63213 (a b : ℕ → ℝ)
(h₀ : ∀ n, a (n + 2) - a n = a (n + 1) - a n)
(h₁ : ∀ n, b (n + 2) - b n = b (n + 1) - b n)
(h₂ : 0 < a 1 ∧ 0 < b 1)
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Two arithmetic sequences $\left\{ {{a}_{n}} \right\}$ and $\left\{ {{b}_{n}} \right\}$ satisfy the equation $\frac{{{a}_{1}}+{{a}_{2}}+..... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1660220 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_321139_0_gp3 (y z w : ℝ)
(hy : y ≠ 0) (hz : z ≠ 0) (hw : w ≠ 0)
(h : 1 / (y^4 + 2*y^2) + 1 / (z^4 + 2*z^2) + 1 / (w^... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_321139_0_gp3 (y z w : ℝ)
(hy : y ≠ 0) (hz : z ≠ 0) (hw : w ≠ 0)
(h : 1 / (y^4 + 2*y^2) + 1 / (z^4 + 2*z^2) + 1 / (w^4 + 2*w^2) = 1) :
1 / (y^2 + ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_321139_0_gp3 (y z w : ℝ)
(hy : y ≠ 0) (hz : z ≠ 0) (hw : w ≠ 0)
(h : 1 / (y^4 + 2*y^2) + 1 / (z^4 + 2*z^2) + 1 / (w^... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
86227a2c-85ce-59bb-876c-ba0ef39e9351 | AI-MO/NuminaMath-LEAN | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
/- Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x^2 + 2y^2 + 5z^2 = 22.$ Find the maximum value of
\[xy + xz + yz.\] -/
theorem algebra_21725:
IsGreatest {p : ℝ | ∃(x y z : ℝ), 0 ≤ x ∧ 0 ≤ y ∧ 0 ≤ z ∧
... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
theorem algebra_21725:
IsGreatest {p : ℝ | ∃(x y z : ℝ), 0 ≤ x ∧ 0 ≤ y ∧ 0 ≤ z ∧
x^2 + 2 * y^2 + 5 * z^2 = 22 ∧
p = x * y + x * z + y * z} 11 := by
constructor
· -- Show 11 is in the set
... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
/- Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x^2 + 2y^2 + 5z^2 = 22.$ Find the maximum value of
\[xy + xz + yz.\] -/
theorem algebra_21725:
IsGreatest {p : ℝ | ∃(x y z : ℝ), 0 ≤ x ∧ 0 ≤ y ∧ 0 ≤ z ∧
... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-163792 | formalmathatepfl/solved_problems_finetuning_iter1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac 18$ the measure of $\angle 2$. What is the degree measure of $\a... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_23316 (x : ℝ)
(h₀ : 9 * x = 180) :
x = 20 := by
have h₁ : 9 * x = 180 := h₀
linarith
```
<|endoftext|> | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac 18$ the measure of $\angle 2$. What is the degree measure of $\a... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1054259 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Disprove the following statement: "Consider a circle with radius \( R \) and two locus circles with radii \( R_1 \) and \( R_2 \). If the re... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_209895_negation : ¬ ( ∀ (R R₁ R₂ : ℝ) (hR : R = 5) (hR₁ : R₁ = 10) (h_pos : R > 0 ∧ R₁ > 0 ∧ R₂ > 0) (h_constraint : |1/R₁ + 1/R₂| = 1/R ∨ |1/R₁ - 1/R₂|... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Disprove the following statement: "Consider a circle with radius \( R \) and two locus circles with radii \( R_1 \) and \( R_2 \). If the re... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-LM/SFT_dataset_v2=1177236 | Goedel-LM/SFT_dataset_v2 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291009_0_gs0_f0_gs0 (n k : ℤ) (N : ℕ) (hN : 0 < N) :
Complex.exp (Complex.I * 2 * Real.pi * ((n - k : ℂ) / (N : ℂ))) =... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291009_0_gs0_f0_gs0 (n k : ℤ) (N : ℕ) (hN : 0 < N) :
Complex.exp (Complex.I * 2 * Real.pi * ((n - k : ℂ) / (N : ℂ))) =
Complex.exp (Complex.I * 2 ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem OMR_problem_291009_0_gs0_f0_gs0 (n k : ℤ) (N : ℕ) (hN : 0 < N) :
Complex.exp (Complex.I * 2 * Real.pi * ((n - k : ℂ) / (N : ℂ))) =... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-1221137 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Determine all (not necessarily finite) sets \( S \) of points in the plane such that given any four distinct points in \( S \), there is ... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem lean_workbook_plus_52725 (S : Finset (ℝ × ℝ)) : (∀ (a b c d : ℝ × ℝ), a ∈ S ∧ b ∈ S ∧ c ∈ S ∧ d ∈ S → a = b ∨ a = c ∨ a = d ∨ b = c ∨ b = d ∨ c = d) ∨ ∃ (a b ... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-Determine all (not necessarily finite) sets \( S \) of points in the plane such that given any four distinct points in \( S \), there is ... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
Goedel-Pset-799981 | Goedel-LM/Goedel-Pset-v1 | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a sequence of real numbers, scaling each element by a constant factor a
changes the variance by a², while adding a constant b d... | The complete typechecked Lean 4 proof is:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem variance_scaling_linear_transform
{n : ℕ} (a b : ℝ) (x : Fin n → ℝ) (h_nz : n ≠ 0) :
let μx := (∑ i : Fin n, x i) / n
let μy := (∑ i : Fin n, (a * x i +... | Solve the following problem with Lean 4 code and explanatory comments:
```lean4
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
/-- Given a sequence of real numbers, scaling each element by a constant factor a
changes the variance by a², while adding a constant b d... | Study the problem and the verified Lean 4 proof below. Recover the underlying natural-language reasoning that leads to the proof. Return only the reasoning, ending with the introduction to the Lean proof, and do not repeat the code.
<problem>
Solve the following problem with Lean 4 code and explanatory comments:
```l... |
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