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... |
End of preview. Expand in Data Studio
APE training data
This dataset is derived from the train split of
formalmathatepfl/sft_classic
at revision 21027890d0208fdc3881bb79b969e7bdf01dabbf.
All rows with proof_repair == true were removed. The source contained
2,678,749 train rows; 879,381 repair rows were excluded, leaving 1,799,368
examples. The source test and sft_replay_classic splits are not included.
Each question requests an informal solution using [Problem Restatement],
[Useful Observations], [Reasoning Path], [Key Derivation], [Proof Type],
[Formal Anchors], and [Final Solution], matching the prompt contract used by
formalmathatepfl/reasoning_data:APE_warm_start_data.
The uuid, data_source, and answer values are copied unchanged from the
source. prompt_planner contains the modified question followed by the exact
continuation whitespace. prompt_abstractor contains that question and the
verified Lean 4 proof extracted from answer.
- Downloads last month
- 28