url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment4.lean | exercise4_5 | [66, 1] | [89, 8] | sorry | a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 2 ^ 0 ≤ 2 ^ (a * b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
⊢ 2 ^ 0 ≤ 2 ^ (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment4.lean | exercise4_5 | [66, 1] | [89, 8] | norm_cast at h | case h'
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h'
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
h2 : 2 ^ 2 ≤ 2 ^ a
h3 : 1 ≤ 2 ^ a
h4 : 2 ^ a - 1 ≠ 1
h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1
h6' : 2 ^ 0 ≤ 2 ^ (a * b)
h6 : 1 ≤ 2 ^ (a * b)
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment4.lean | exercise4_6 | [94, 1] | [95, 64] | sorry | a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | even_of_even_sqr | [42, 1] | [44, 25] | rw [pow_two, Nat.prime_two.dvd_mul] at h | m : ℕ
h : 2 ∣ m ^ 2
⊢ 2 ∣ m | m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
h : 2 ∣ m ^ 2
⊢ 2 ∣ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | even_of_even_sqr | [42, 1] | [44, 25] | cases h <;> assumption | m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
h : 2 ∣ m ∨ 2 ∣ m
⊢ 2 ∣ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | factorization_mul' | [78, 1] | [81, 6] | rw [Nat.factorization_mul mnez nnez] | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization (m * n)) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization (m * n)) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | factorization_mul' | [78, 1] | [81, 6] | rfl | m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
mnez : m ≠ 0
nnez : n ≠ 0
p : ℕ
⊢ ↑(Nat.factorization m + Nat.factorization n) p = ↑(Nat.factorization m) p + ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | factorization_pow' | [83, 1] | [86, 6] | rw [Nat.factorization_pow] | n k p : ℕ
⊢ ↑(Nat.factorization (n ^ k)) p = k * ↑(Nat.factorization n) p | n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p | Please generate a tactic in lean4 to solve the state.
STATE:
n k p : ℕ
⊢ ↑(Nat.factorization (n ^ k)) p = k * ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | factorization_pow' | [83, 1] | [86, 6] | rfl | n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k p : ℕ
⊢ ↑(k • Nat.factorization n) p = k * ↑(Nat.factorization n) p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | Nat.Prime.factorization' | [88, 1] | [91, 7] | rw [prime_p.factorization] | p : ℕ
prime_p : Prime p
⊢ ↑(Nat.factorization p) p = 1 | p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
prime_p : Prime p
⊢ ↑(Nat.factorization p) p = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean | Nat.Prime.factorization' | [88, 1] | [91, 7] | simp | p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment1.lean | exercise1_1 | [22, 1] | [22, 78] | sorry | a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment1.lean | exercise1_2 | [27, 1] | [27, 78] | sorry | a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment1.lean | exercise1_3 | [48, 1] | [48, 72] | sorry | a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment1.lean | exercise1_4 | [52, 1] | [52, 72] | sorry | a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | rw [Metric.cauchySeq_iff'] | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ CauchySeq u | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ CauchySeq u
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | intro ε ε_pos | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | use N | case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | intro n hn | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε | case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | calc
dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry
_ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry
_ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry
_ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := sorry
_ ≤ 1 / 2 ^ N * 2 := sorry
_ < ε := sorry | case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | sorry | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, 1 / 2 ^ N * 2 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, 1 / 2 ^ N * 2 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Extra.lean | odd_succ_even | [12, 1] | [14, 11] | refine Even.add_odd h ?_ | n : ℕ
h : EvenNat n
⊢ OddNat (n + 1) | n : ℕ
h : EvenNat n
⊢ Odd 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : EvenNat n
⊢ OddNat (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Extra.lean | odd_succ_even | [12, 1] | [14, 11] | norm_num | n : ℕ
h : EvenNat n
⊢ Odd 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : EvenNat n
⊢ Odd 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Extra.lean | even_succ_odd | [16, 1] | [18, 11] | refine Odd.add_odd h ?_ | n : ℕ
h : OddNat n
⊢ EvenNat (n + 1) | n : ℕ
h : OddNat n
⊢ Odd 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : OddNat n
⊢ EvenNat (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Extra.lean | even_succ_odd | [16, 1] | [18, 11] | norm_num | n : ℕ
h : OddNat n
⊢ Odd 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : OddNat n
⊢ Odd 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_zero | [39, 1] | [39, 67] | rw [add_comm, zero_add] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + 0 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a + 0 = a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_neg | [41, 1] | [41, 77] | rw [add_comm, add_left_neg] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + -a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a + -a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_add_cancel_left | [51, 1] | [52, 43] | rw [← add_assoc, add_left_neg, zero_add] | R : Type u_1
inst✝ : Ring R
a b : R
⊢ -a + (a + b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
⊢ -a + (a + b) = b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_neg_cancel_right | [55, 1] | [56, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_left_cancel | [58, 1] | [59, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_cancel | [61, 1] | [62, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | have h : a * 0 + a * 0 = a * 0 + 0 := by
rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 = 0 | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | rw [add_left_cancel h] | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 + a * 0 = a * 0 + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 + a * 0 = a * 0 + 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [69, 1] | [70, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_eq_of_add_eq_zero | [72, 1] | [73, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [75, 1] | [76, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [78, 1] | [80, 16] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
⊢ -0 = 0 | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ -0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [78, 1] | [80, 16] | rw [add_zero] | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_neg | [82, 1] | [83, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.self_sub | [105, 1] | [106, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.one_add_one_eq_two | [108, 1] | [109, 11] | norm_num | R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.two_mul | [111, 1] | [112, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [134, 1] | [135, 8] | sorry | G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_one | [137, 1] | [138, 8] | sorry | G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_inv_rev | [140, 1] | [141, 8] | sorry | G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [33, 1] | [38, 30] | induction' n with n ih | n : ℕ
⊢ 0 < fac n | case zero
⊢ 0 < fac 0
case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < fac (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 0 < fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [33, 1] | [38, 30] | rw [fac] | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < fac (n + 1) | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < (n + 1) * fac n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < fac (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [33, 1] | [38, 30] | exact mul_pos n.succ_pos ih | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < (n + 1) * fac n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [33, 1] | [38, 30] | rw [fac] | case zero
⊢ 0 < fac 0 | case zero
⊢ 0 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 0 < fac 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [33, 1] | [38, 30] | exact zero_lt_one | case zero
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 0 < 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | induction' n with n ih | i n : ℕ
ipos : 0 < i
ile : i ≤ n
⊢ i ∣ fac n | case zero
i : ℕ
ipos : 0 < i
ile : i ≤ 0
⊢ i ∣ fac 0
case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ fac (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
i n : ℕ
ipos : 0 < i
ile : i ≤ n
⊢ i ∣ fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | rw [fac] | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ fac (n + 1) | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ (n + 1) * fac n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ fac (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | rcases Nat.of_le_succ ile with h | h | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ (n + 1) * fac n | case succ.inl
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i ≤ n
⊢ i ∣ (n + 1) * fac n
case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ i ∣ (n + 1) * fac n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | rw [h] | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ i ∣ (n + 1) * fac n | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ Nat.succ n ∣ (n + 1) * fac n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ i ∣ (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | apply dvd_mul_right | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ Nat.succ n ∣ (n + 1) * fac n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = Nat.succ n
⊢ Nat.succ n ∣ (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | exact absurd ipos (not_lt_of_ge ile) | case zero
i : ℕ
ipos : 0 < i
ile : i ≤ 0
⊢ i ∣ fac 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
ipos : 0 < i
ile : i ≤ 0
⊢ i ∣ fac 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [40, 1] | [47, 22] | apply dvd_mul_of_dvd_right (ih h) | case succ.inl
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i ≤ n
⊢ i ∣ (n + 1) * fac n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.inl
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i ≤ n
⊢ i ∣ (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [49, 1] | [52, 8] | rcases n with _ | n | n : ℕ
⊢ 2 ^ (n - 1) ≤ fac n | case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero
case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 2 ^ (n - 1) ≤ fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [49, 1] | [52, 8] | sorry | case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [49, 1] | [52, 8] | simp [fac] | case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | symm | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i = n * (n + 1) / 2 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in range (n + 1), i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i = n * (n + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in range (n + 1), i | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) = 2 * ∑ i in range (n + 1), i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in range (n + 1), i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | induction' n with n ih | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) = 2 * ∑ i in range (n + 1), i | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) = 2 * ∑ i in range (0 + 1), i
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = 2 * ∑ i in range (n + 1 + 1), i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) = 2 * ∑ i in range (n + 1), i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | rw [Finset.sum_range_succ, mul_add 2, ← ih] | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = 2 * ∑ i in range (n + 1 + 1), i | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = n * (n + 1) + 2 * (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = 2 * ∑ i in range (n + 1 + 1), i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | ring | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = n * (n + 1) + 2 * (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i in range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = n * (n + 1) + 2 * (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | norm_num | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [95, 1] | [100, 7] | simp | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) = 2 * ∑ i in range (0 + 1), i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) = 2 * ∑ i in range (0 + 1), i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [102, 1] | [103, 8] | sorry | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [120, 1] | [123, 15] | induction' n with n ih | n : MyNat
⊢ add zero n = n | case zero
⊢ add zero zero = zero
case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : MyNat
⊢ add zero n = n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [120, 1] | [123, 15] | rw [add, ih] | case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [120, 1] | [123, 15] | rfl | case zero
⊢ add zero zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ add zero zero = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [125, 1] | [129, 6] | induction' n with n ih | m n : MyNat
⊢ add (succ m) n = succ (add m n) | case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero)
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ add (succ m) n = succ (add m n)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [125, 1] | [129, 6] | rw [add, ih] | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n)) | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [125, 1] | [129, 6] | rfl | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [125, 1] | [129, 6] | rfl | case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [131, 1] | [135, 25] | induction' n with n ih | m n : MyNat
⊢ add m n = add n m | case zero
m : MyNat
⊢ add m zero = add zero m
case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ add m n = add n m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [131, 1] | [135, 25] | rw [add, succ_add, ih] | case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [131, 1] | [135, 25] | rw [zero_add] | case zero
m : MyNat
⊢ add m zero = add zero m | case zero
m : MyNat
⊢ add m zero = m | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add m zero = add zero m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [131, 1] | [135, 25] | rfl | case zero
m : MyNat
⊢ add m zero = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add m zero = m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_assoc | [137, 1] | [138, 8] | sorry | m n k : MyNat
⊢ add (add m n) k = add m (add n k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : MyNat
⊢ add (add m n) k = add m (add n k)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_add | [139, 1] | [140, 8] | sorry | m n k : MyNat
⊢ mul m (add n k) = add (mul m n) (mul m k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : MyNat
⊢ mul m (add n k) = add (mul m n) (mul m k)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_mul | [141, 1] | [142, 8] | sorry | n : MyNat
⊢ mul zero n = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : MyNat
⊢ mul zero n = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_mul | [143, 1] | [144, 8] | sorry | m n : MyNat
⊢ mul (succ m) n = add (mul m n) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ mul (succ m) n = add (mul m n) n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_comm | [145, 1] | [146, 8] | sorry | m n : MyNat
⊢ mul m n = mul n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ mul m n = mul n m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | rw [Metric.cauchySeq_iff'] | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ CauchySeq u | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ CauchySeq u
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | intro ε ε_pos | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | use N | case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | intro n hn | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
⊢ ∀ n ≥ N, dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn | case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε | case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
n : ℕ
hn : n ≥ N
⊢ dist (u n) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | calc
dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry
_ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry
_ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry
_ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := sorry
_ ≤ 1 / 2 ^ N * 2 := sorry
_ < ε := sorry | case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
N : ℕ
hN : 1 / 2 ^ N * 2 < ε
k : ℕ
hn : N + k ≥ N
⊢ dist (u (N + k)) (u N) < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [185, 1] | [199, 19] | sorry | X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, 1 / 2 ^ N * 2 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : MetricSpace X
a b c : X
r : ℝ
u : ℕ → X
hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n
ε : ℝ
ε_pos : ε > 0
⊢ ∃ N, 1 / 2 ^ N * 2 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [45, 1] | [46, 8] | sorry | a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | eq_bot_iff_card | [75, 1] | [84, 41] | suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by
simpa [eq_bot_iff_forall, card_eq_one_iff] | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ H = ⊥ ↔ card (↥H) = 1 | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ H = ⊥ ↔ card (↥H) = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | eq_bot_iff_card | [75, 1] | [84, 41] | constructor | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) → ∃ x ∈ H, ∀ a ∈ H, a = x
case mpr
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∃ x ∈ H, ∀ a ∈ H, a = x) → ∀ x ∈ H, x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | eq_bot_iff_card | [75, 1] | [84, 41] | simpa [eq_bot_iff_forall, card_eq_one_iff] | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
this : (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x
⊢ H = ⊥ ↔ card (↥H) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
this : (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x
⊢ H = ⊥ ↔ card (↥H) = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | eq_bot_iff_card | [75, 1] | [84, 41] | intro h | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) → ∃ x ∈ H, ∀ a ∈ H, a = x | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
h : ∀ x ∈ H, x = 1
⊢ ∃ x ∈ H, ∀ a ∈ H, a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) → ∃ x ∈ H, ∀ a ∈ H, a = x
TACTIC:
|
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