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/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | rw [sum_range_succ, ih] | case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ ∑ i in range (n + 1 + 1), ↑(i ^ 3) = (↑(n + 1) * (↑(n + 1) + 1) / 2) ^ 2 | case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + ↑((n + 1) ^ 3) = (↑(n + 1) * (↑(n + 1) + 1) / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ ∑ i in range (n + 1 + 1), ↑(i ^ 3) = (↑(n + 1) * (↑(n + 1) + 1) / 2) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | push_cast | case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + ↑((n + 1) ^ 3) = (↑(n + 1) * (↑(n + 1) + 1) / 2) ^ 2 | case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + (↑n + 1) ^ 3 = ((↑n + 1) * (↑n + 1 + 1) / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + ↑((n + 1) ^ 3) = (↑(n + 1) * (↑(n + 1) + 1) / 2) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | ring | case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + (↑n + 1) ^ 3 = ((↑n + 1) * (↑n + 1 + 1) / 2) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
ih : ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
⊢ (↑n * (↑n + 1) / 2) ^ 2 + (↑n + 1) ^ 3 = ((↑n + 1) * (↑n + 1 + 1) / 2) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | have h := wf.wf.has_min | ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i | ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | have lem1 : ∀ i j, i < j → Disjoint (C i) (C j) | ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ ∀ (i j : ι), i < j → Disjoint (C i) (C j)
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | constructor | ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i | case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C)
case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i = ⋃ i, A i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | intro i j hij | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ ∀ (i j : ι), i < j → Disjoint (C i) (C j) | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ Disjoint (C i) (C j) | Please generate a tactic in lean4 to solve the state.
STATE:
case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
⊢ ∀ (i j : ι), i < j → Disjoint (C i) (C j)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | simp [disjoint_left, hC] | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ Disjoint (C i) (C j) | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ ∀ ⦃a : α⦄, a ∈ A i → (∀ x < i, a ∉ A x) → a ∈ A j → ∃ i < j, a ∈ A i | Please generate a tactic in lean4 to solve the state.
STATE:
case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ Disjoint (C i) (C j)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | tauto | case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ ∀ ⦃a : α⦄, a ∈ A i → (∀ x < i, a ∉ A x) → a ∈ A j → ∃ i < j, a ∈ A i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case lem1
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
i j : ι
hij : i < j
⊢ ∀ ⦃a : α⦄, a ∈ A i → (∀ x < i, a ∉ A x) → a ∈ A j → ∃ i < j, a ∈ A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | intro i j hij | case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C) | case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
⊢ (Disjoint on C) i j | Please generate a tactic in lean4 to solve the state.
STATE:
case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ Pairwise (Disjoint on C)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | obtain h|h := hij.lt_or_lt | case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
⊢ (Disjoint on C) i j | case left.inl
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : i < j
⊢ (Disjoint on C) i j
case left.inr
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : j < i
⊢ (Disjoint on C) i j | Please generate a tactic in lean4 to solve the state.
STATE:
case left
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
⊢ (Disjoint on C) i j
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | exact lem1 i j h | case left.inl
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : i < j
⊢ (Disjoint on C) i j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.inl
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : i < j
⊢ (Disjoint on C) i j
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | exact (lem1 j i h).symm | case left.inr
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : j < i
⊢ (Disjoint on C) i j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.inr
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h✝ : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i j : ι
hij : i ≠ j
h : j < i
⊢ (Disjoint on C) i j
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | apply subset_antisymm | case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i = ⋃ i, A i | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i ⊆ ⋃ i, A i
case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, A i ⊆ ⋃ i, C i | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i = ⋃ i, A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | gcongr with i | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i ⊆ ⋃ i, A i | case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ C i ⊆ A i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, C i ⊆ ⋃ i, A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | rw [hC] | case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ C i ⊆ A i | case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ A i \ ⋃ j, ⋃ (_ : j < i), A j ⊆ A i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ C i ⊆ A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | exact? | case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ A i \ ⋃ j, ⋃ (_ : j < i), A j ⊆ A i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a.h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
i : ι
⊢ A i \ ⋃ j, ⋃ (_ : j < i), A j ⊆ A i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | simp [subset_def] | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, A i ⊆ ⋃ i, C i | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ∀ (x : α) (x_1 : ι), x ∈ A x_1 → ∃ i, x ∈ C i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ⋃ i, A i ⊆ ⋃ i, C i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | intros x i hx | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ∀ (x : α) (x_1 : ι), x ∈ A x_1 → ∃ i, x ∈ C i | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
⊢ ∃ i, x ∈ C i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
⊢ ∀ (x : α) (x_1 : ι), x ∈ A x_1 → ∃ i, x ∈ C i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | obtain ⟨i₀, h1i₀, h2i₀⟩ := h { i : ι | x ∈ A i } ⟨i, hx⟩ | case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
⊢ ∃ i, x ∈ C i | case right.a.intro.intro
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∃ i, x ∈ C i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
⊢ ∃ i, x ∈ C i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | use i₀ | case right.a.intro.intro
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∃ i, x ∈ C i | case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ C i₀ | Please generate a tactic in lean4 to solve the state.
STATE:
case right.a.intro.intro
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∃ i, x ∈ C i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | simp [hC] | case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ C i₀ | case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ A i₀ ∧ ∀ x_1 < i₀, x ∉ A x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ C i₀
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | use h1i₀ | case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ A i₀ ∧ ∀ x_1 < i₀, x ∉ A x_1 | case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∀ x_1 < i₀, x ∉ A x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ x ∈ A i₀ ∧ ∀ x_1 < i₀, x ∉ A x_1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | intro j hj h2j | case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∀ x_1 < i₀, x ∉ A x_1 | case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
j : ι
hj : j < i₀
h2j : x ∈ A j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
⊢ ∀ x_1 < i₀, x ∉ A x_1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_2 | [49, 1] | [72, 26] | exact h2i₀ j h2j hj | case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
j : ι
hj : j < i₀
h2j : x ∈ A j
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ι : Type u_1
α : Type u_2
inst✝ : LinearOrder ι
wf : WellFoundedLT ι
A C : ι → Set α
hC : ∀ (i : ι), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j
h : ∀ (s : Set ι), Set.Nonempty s → ∃ a ∈ s, ∀ x ∈ s, ¬x < a
lem1 : ∀ (i j : ι), i < j → Disjoint (C i) (C j)
x : α
i : ι
hx : x ∈ A i
i₀ : ι
h1i₀ : i₀ ∈ {i | x ∈ A i}
h2i₀ : ∀ x_1 ∈ {i | x ∈ A i}, ¬x_1 < i₀
j : ι
hj : j < i₀
h2j : x ∈ A j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | ext | x y z : PosReal
⊢ x * y * z = x * (y * z) | case h
x y z : PosReal
⊢ ↑(x * y * z) = ↑(x * (y * z)) | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : PosReal
⊢ x * y * z = x * (y * z)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | apply mul_assoc | case h
x y z : PosReal
⊢ ↑(x * y * z) = ↑(x * (y * z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y z : PosReal
⊢ ↑(x * y * z) = ↑(x * (y * z))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | norm_num | ⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | ext | x : PosReal
⊢ 1 * x = x | case h
x : PosReal
⊢ ↑(1 * x) = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
x : PosReal
⊢ 1 * x = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | apply one_mul | case h
x : PosReal
⊢ ↑(1 * x) = ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : PosReal
⊢ ↑(1 * x) = ↑x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | ext | x : PosReal
⊢ x * 1 = x | case h
x : PosReal
⊢ ↑(x * 1) = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
x : PosReal
⊢ x * 1 = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | apply mul_one | case h
x : PosReal
⊢ ↑(x * 1) = ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : PosReal
⊢ ↑(x * 1) = ↑x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | ext | x : PosReal
⊢ x⁻¹ * x = 1 | case h
x : PosReal
⊢ ↑(x⁻¹ * x) = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : PosReal
⊢ x⁻¹ * x = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | apply inv_mul_cancel | case h
x : PosReal
⊢ ↑(x⁻¹ * x) = ↑1 | case h.h
x : PosReal
⊢ ↑x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : PosReal
⊢ ↑(x⁻¹ * x) = ↑1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_3 | [85, 1] | [92, 64] | exact x.2.ne' | case h.h
x : PosReal
⊢ ↑x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
x : PosReal
⊢ ↑x ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | rw [← or_assoc, ← Nat.le_one_iff_eq_zero_or_eq_one, ← not_lt, ← imp_iff_not_or, Nat.prime_def_lt'] | n : ℕ
⊢ ¬Nat.Prime n ↔ n = 0 ∨ n = 1 ∨ ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b | n : ℕ
⊢ ¬(2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m < n → ¬m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ¬Nat.Prime n ↔ n = 0 ∨ n = 1 ∨ ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | push_neg | n : ℕ
⊢ ¬(2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m < n → ¬m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b | n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ¬(2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m < n → ¬m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | simp | n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b | n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply imp_congr | n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | case h₁
n : ℕ
⊢ 2 ≤ n ↔ 1 < n
case h₂
n : ℕ
⊢ (∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (2 ≤ n → ∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ 1 < n → ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply exists_congr | case h₂
n : ℕ
⊢ (∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | case h₂.h
n : ℕ
⊢ ∀ (a : ℕ), 2 ≤ a ∧ a < n ∧ a ∣ n ↔ 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
n : ℕ
⊢ (∃ m, 2 ≤ m ∧ m < n ∧ m ∣ n) ↔ ∃ a, 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | intro k | case h₂.h
n : ℕ
⊢ ∀ (a : ℕ), 2 ≤ a ∧ a < n ∧ a ∣ n ↔ 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x | case h₂.h
n k : ℕ
⊢ 2 ≤ k ∧ k < n ∧ k ∣ n ↔ 2 ≤ k ∧ ∃ x, 2 ≤ x ∧ n = k * x | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h
n : ℕ
⊢ ∀ (a : ℕ), 2 ≤ a ∧ a < n ∧ a ∣ n ↔ 2 ≤ a ∧ ∃ x, 2 ≤ x ∧ n = a * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply and_congr_right | case h₂.h
n k : ℕ
⊢ 2 ≤ k ∧ k < n ∧ k ∣ n ↔ 2 ≤ k ∧ ∃ x, 2 ≤ x ∧ n = k * x | case h₂.h.h
n k : ℕ
⊢ 2 ≤ k → (k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h
n k : ℕ
⊢ 2 ≤ k ∧ k < n ∧ k ∣ n ↔ 2 ≤ k ∧ ∃ x, 2 ≤ x ∧ n = k * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | intro hk | case h₂.h.h
n k : ℕ
⊢ 2 ≤ k → (k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x) | case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h
n k : ℕ
⊢ 2 ≤ k → (k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | rw [dvd_def, ← exists_and_left] | case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x | case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ (∃ x, k < n ∧ n = k * x) ↔ ∃ x, 2 ≤ x ∧ n = k * x | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ k < n ∧ k ∣ n ↔ ∃ x, 2 ≤ x ∧ n = k * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply exists_congr | case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ (∃ x, k < n ∧ n = k * x) ↔ ∃ x, 2 ≤ x ∧ n = k * x | case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
⊢ ∀ (a : ℕ), k < n ∧ n = k * a ↔ 2 ≤ a ∧ n = k * a | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h
n k : ℕ
hk : 2 ≤ k
⊢ (∃ x, k < n ∧ n = k * x) ↔ ∃ x, 2 ≤ x ∧ n = k * x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | intro l | case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
⊢ ∀ (a : ℕ), k < n ∧ n = k * a ↔ 2 ≤ a ∧ n = k * a | case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < n ∧ n = k * l ↔ 2 ≤ l ∧ n = k * l | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
⊢ ∀ (a : ℕ), k < n ∧ n = k * a ↔ 2 ≤ a ∧ n = k * a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply and_congr_left | case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < n ∧ n = k * l ↔ 2 ≤ l ∧ n = k * l | case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ n = k * l → (k < n ↔ 2 ≤ l) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < n ∧ n = k * l ↔ 2 ≤ l ∧ n = k * l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | intro h | case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ n = k * l → (k < n ↔ 2 ≤ l) | case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
h : n = k * l
⊢ k < n ↔ 2 ≤ l | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ n = k * l → (k < n ↔ 2 ≤ l)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | subst h | case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
h : n = k * l
⊢ k < n ↔ 2 ≤ l | case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 2 ≤ l | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h.h
n k : ℕ
hk : 2 ≤ k
l : ℕ
h : n = k * l
⊢ k < n ↔ 2 ≤ l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | rw [succ_le] | case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 2 ≤ l | case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 1 < l | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 2 ≤ l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | apply lt_mul_iff_one_lt_right | case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 1 < l | case h₂.h.h.h.h.a0
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ 0 < k | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h.h
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ k < k * l ↔ 1 < l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | linarith | case h₂.h.h.h.h.a0
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ 0 < k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.h.h.h.a0
k : ℕ
hk : 2 ≤ k
l : ℕ
⊢ 0 < k
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_4 | [102, 1] | [123, 11] | rw? | case h₁
n : ℕ
⊢ 2 ≤ n ↔ 1 < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
n : ℕ
⊢ 2 ≤ n ↔ 1 < n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | by_contra h2n | n : ℕ
hn : Nat.Prime (2 ^ n - 1)
⊢ Nat.Prime n | n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : ¬Nat.Prime n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : Nat.Prime (2 ^ n - 1)
⊢ Nat.Prime n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | rw [exercise4_4] at h2n | n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : ¬Nat.Prime n
⊢ False | n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : n = 0 ∨ n = 1 ∨ ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : ¬Nat.Prime n
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | obtain rfl|rfl|⟨a, b, ha, hb, rfl⟩ := h2n | n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : n = 0 ∨ n = 1 ∨ ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
⊢ False | case inl
hn : Nat.Prime (2 ^ 0 - 1)
⊢ False
case inr.inl
hn : Nat.Prime (2 ^ 1 - 1)
⊢ False
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : Nat.Prime (2 ^ n - 1)
h2n : n = 0 ∨ n = 1 ∨ ∃ a b, 2 ≤ a ∧ 2 ≤ b ∧ n = a * b
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h : (2 : ℤ) ^ a - 1 ∣ (2 : ℤ) ^ (a * b) - 1 | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ False | case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h2 : 2 ^ 2 ≤ 2 ^ a := by gcongr; norm_num | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | case inr.inr.intro.intro.intro.intro
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
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h3 : 1 ≤ 2 ^ a := by linarith | case inr.inr.intro.intro.intro.intro
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
⊢ False | case inr.inr.intro.intro.intro.intro
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
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h4 : 2 ^ a - 1 ≠ 1 := by zify; simp [h3]; linarith | case inr.inr.intro.intro.intro.intro
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
⊢ False | case inr.inr.intro.intro.intro.intro
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
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h6' : 2 ^ 0 ≤ 2 ^ (a * b) := by gcongr; norm_num; exact? | case inr.inr.intro.intro.intro.intro
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
⊢ False | case inr.inr.intro.intro.intro.intro
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)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h6 : 1 ≤ 2 ^ (a * b) := h6' | case inr.inr.intro.intro.intro.intro
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)
⊢ False | case inr.inr.intro.intro.intro.intro
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)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | have h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 | case inr.inr.intro.intro.intro.intro
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)
⊢ False | 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
case inr.inr.intro.intro.intro.intro
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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | rw [Nat.prime_def_lt] at hn | case inr.inr.intro.intro.intro.intro
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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | apply h4 | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | apply hn.2 _ h5 | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 = 1 | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | norm_cast at h | case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : 2 ≤ 2 ^ (a * b) - 1 ∧ ∀ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 → m = 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)
h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | simp at hn | case inl
hn : Nat.Prime (2 ^ 0 - 1)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
hn : Nat.Prime (2 ^ 0 - 1)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | norm_num at hn | case inr.inl
hn : Nat.Prime (2 ^ 1 - 1)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
hn : Nat.Prime (2 ^ 1 - 1)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | rw [← Int.modEq_zero_iff_dvd] | case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 | case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ (a * b) - 1 ≡ 0 [ZMOD 2 ^ a - 1] | 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)
⊢ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | calc (2 : ℤ) ^ (a * b) - 1
≡ ((2 : ℤ) ^ a) ^ b - 1 [ZMOD (2 : ℤ) ^ a - 1] := by rw [pow_mul]
_ ≡ (1 : ℤ) ^ b - 1 [ZMOD (2 : ℤ) ^ a - 1] := by gcongr; exact?
_ ≡ 0 [ZMOD (2 : ℤ) ^ a - 1] := by simp | case h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ (a * b) - 1 ≡ 0 [ZMOD 2 ^ a - 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)
⊢ 2 ^ (a * b) - 1 ≡ 0 [ZMOD 2 ^ a - 1]
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | rw [pow_mul] | a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ (a * b) - 1 ≡ (2 ^ a) ^ b - 1 [ZMOD 2 ^ a - 1] | 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)
⊢ 2 ^ (a * b) - 1 ≡ (2 ^ a) ^ b - 1 [ZMOD 2 ^ a - 1]
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | gcongr | a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ (2 ^ a) ^ b - 1 ≡ 1 ^ b - 1 [ZMOD 2 ^ a - 1] | case h.h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a ≡ 1 [ZMOD 2 ^ a - 1] | 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)
⊢ (2 ^ a) ^ b - 1 ≡ 1 ^ b - 1 [ZMOD 2 ^ a - 1]
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | exact? | case h.h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a ≡ 1 [ZMOD 2 ^ a - 1] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 2 ^ a ≡ 1 [ZMOD 2 ^ a - 1]
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | simp | a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
⊢ 1 ^ b - 1 ≡ 0 [ZMOD 2 ^ a - 1] | 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)
⊢ 1 ^ b - 1 ≡ 0 [ZMOD 2 ^ a - 1]
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | gcongr | a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 2 ^ 2 ≤ 2 ^ a | case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 1 ≤ 2 | 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
⊢ 2 ^ 2 ≤ 2 ^ a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | norm_num | case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 1 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
a b : ℕ
ha : 2 ≤ a
hb : 2 ≤ b
hn : Nat.Prime (2 ^ (a * b) - 1)
h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
⊢ 1 ≤ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | linarith | 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
⊢ 1 ≤ 2 ^ a | 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
⊢ 1 ≤ 2 ^ a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | zify | 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
⊢ 2 ^ a - 1 ≠ 1 | 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
⊢ ↑(2 ^ a - 1) ≠ 1 | 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
⊢ 2 ^ a - 1 ≠ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | simp [h3] | 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
⊢ ↑(2 ^ a - 1) ≠ 1 | 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
⊢ ¬2 ^ a - 1 = 1 | 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
⊢ ↑(2 ^ a - 1) ≠ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | linarith | 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
⊢ ¬2 ^ a - 1 = 1 | 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
⊢ ¬2 ^ a - 1 = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | apply tsub_lt_tsub_right_of_le h3 | 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
⊢ 2 ^ a - 1 < 2 ^ (a * b) - 1 | 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
⊢ 2 ^ a < 2 ^ (a * b) | 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
⊢ 2 ^ a - 1 < 2 ^ (a * b) - 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | gcongr | 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
⊢ 2 ^ a < 2 ^ (a * b) | 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
⊢ 1 < 2
case h2
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
⊢ a < a * b | 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
⊢ 2 ^ a < 2 ^ (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | refine lt_mul_right ?h2.hn hb | case h2
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
⊢ a < a * b | case h2.hn
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
⊢ 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
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
⊢ a < a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | linarith | case h2.hn
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
⊢ 0 < a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2.hn
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
⊢ 0 < a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | norm_num | 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
⊢ 1 < 2 | 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
⊢ 1 < 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | gcongr | 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) | case ha
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
⊢ 1 ≤ 2
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
⊢ 0 ≤ a * b | 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/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | norm_num | case ha
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
⊢ 1 ≤ 2
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
⊢ 0 ≤ a * b | 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
⊢ 0 ≤ a * b | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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
⊢ 1 ≤ 2
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
⊢ 0 ≤ a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | exact? | 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
⊢ 0 ≤ a * b | 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
⊢ 0 ≤ a * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_5 | [125, 1] | [153, 17] | 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/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | have h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬ IsSquare (b ^ 2 + a) | a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a) | case h1
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a) | 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/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | obtain h|h := le_total a b | a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a) | case inl
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : a ≤ b
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
case inr
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
h : b ≤ a
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
ha : 0 < a
hb : 0 < b
h1 : ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
⊢ ¬IsSquare (a ^ 2 + b) ∨ ¬IsSquare (b ^ 2 + a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | intro a b ha hab ⟨c, hc⟩ | case h1
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a) | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b ^ 2 + a = c * c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a b : ℕ
ha : 0 < a
hb : 0 < b
⊢ ∀ {a b : ℕ}, 0 < a → a ≤ b → ¬IsSquare (b ^ 2 + a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | rw [pow_two] at hc | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b ^ 2 + a = c * c
⊢ False | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b ^ 2 + a = c * c
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | have : b * b < c * c := by linarith | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ False | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | have : b < c := by exact? | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ False | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this : b * b < c * c
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | have : c * c < (b + 1) * (b + 1) := by linarith | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ False | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝ : b * b < c * c
this : b < c
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | have : c < b + 1 := by exact? | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ False | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝² : b * b < c * c
this✝¹ : b < c
this✝ : c * c < (b + 1) * (b + 1)
this : c < b + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝¹ : b * b < c * c
this✝ : b < c
this : c * c < (b + 1) * (b + 1)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | exact? | case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝² : b * b < c * c
this✝¹ : b < c
this✝ : c * c < (b + 1) * (b + 1)
this : c < b + 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
this✝² : b * b < c * c
this✝¹ : b < c
this✝ : c * c < (b + 1) * (b + 1)
this : c < b + 1
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | linarith | a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ b * b < c * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ b✝ : ℕ
ha✝ : 0 < a✝
hb : 0 < b✝
a b : ℕ
ha : 0 < a
hab : a ≤ b
c : ℕ
hc : b * b + a = c * c
⊢ b * b < c * c
TACTIC:
|
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