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/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | eq_bot_iff_card | [75, 1] | [84, 41] | use 1, H.one_mem | 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 | no goals | 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)
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] | rintro ⟨y, -, hy'⟩ x hx | 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 | case mpr.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
y : G
hy' : ∀ a ∈ H, a = y
x : G
hx : x ∈ H
⊢ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
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
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] | calc x = y := hy' x hx
_ = 1 := (hy' 1 H.one_mem).symm | case mpr.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
y : G
hy' : ∀ a ∈ H, a = y
x : G
hx : x ∈ H
⊢ x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
y : G
hy' : ∀ a ∈ H, a = y
x : G
hx : x ∈ H
⊢ x = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | inf_bot_of_coprime | [86, 1] | [90, 63] | have D₁ : card (H ⊓ K : Subgroup G) ∣ card H := card_dvd_of_le inf_le_left | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
⊢ H ⊓ K = ⊥ | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
⊢ H ⊓ K = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
⊢ H ⊓ K = ⊥
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | inf_bot_of_coprime | [86, 1] | [90, 63] | have D₂ : card (H ⊓ K : Subgroup G) ∣ card K := card_dvd_of_le inf_le_right | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
⊢ H ⊓ K = ⊥ | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
D₂ : card (↥(H ⊓ K)) ∣ card (↥K)
⊢ H ⊓ K = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
⊢ H ⊓ K = ⊥
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | inf_bot_of_coprime | [86, 1] | [90, 63] | exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂) | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
D₂ : card (↥(H ⊓ K)) ∣ card (↥K)
⊢ H ⊓ K = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
D₁ : card (↥(H ⊓ K)) ∣ card (↥H)
D₂ : card (↥(H ⊓ K)) ∣ card (↥K)
⊢ H ⊓ K = ⊥
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | conjugate_one | [96, 1] | [98, 19] | ext x | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ conjugate 1 H = H | case h
G : Type u_1
inst✝ : Group G
H : Subgroup G
x : G
⊢ x ∈ conjugate 1 H ↔ x ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ conjugate 1 H = H
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | conjugate_one | [96, 1] | [98, 19] | simp [conjugate] | case h
G : Type u_1
inst✝ : Group G
H : Subgroup G
x : G
⊢ x ∈ conjugate 1 H ↔ x ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝ : Group G
H : Subgroup G
x : G
⊢ x ∈ conjugate 1 H ↔ x ∈ H
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | aux_card_eq | [131, 1] | [136, 49] | have := calc
card (G ⧸ H) * card H = card G := by rw [← H.index_eq_card, H.index_mul_card]
_ = card K * card H := by rw [h', mul_comm] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) = card (↥K) | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
this : card (G ⧸ H) * card (↥H) = card (↥K) * card (↥H)
⊢ card (G ⧸ H) = card (↥K) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) = card (↥K)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | aux_card_eq | [131, 1] | [136, 49] | exact Nat.eq_of_mul_eq_mul_right card_pos this | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
this : card (G ⧸ H) * card (↥H) = card (↥K) * card (↥H)
⊢ card (G ⧸ H) = card (↥K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
this : card (G ⧸ H) * card (↥H) = card (↥K) * card (↥H)
⊢ card (G ⧸ H) = card (↥K)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | aux_card_eq | [131, 1] | [136, 49] | rw [← H.index_eq_card, H.index_mul_card] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) * card (↥H) = card G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) * card (↥H) = card G
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/solutions/Solutions_S01_Groups.lean | aux_card_eq | [131, 1] | [136, 49] | rw [h', mul_comm] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card G = card (↥K) * card (↥H) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card G = card (↥K) * card (↥H)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/S03_Topological_Spaces.lean | aux | [104, 1] | [108, 8] | sorry | X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/solutions/Solutions_S01_Structures.lean | C06S01.Point.add_assoc | [19, 11] | [20, 24] | simp [add, add_assoc] | a b c : Point
⊢ add (add a b) c = add a (add b c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Point
⊢ add (add a b) c = add a (add b c)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/solutions/Solutions_S01_Structures.lean | C06S01.Point.smul_distrib | [25, 1] | [27, 28] | simp [add, smul, mul_add] | r : ℝ
a b : Point
⊢ add (smul r a) (smul r b) = smul r (add a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r : ℝ
a b : Point
⊢ add (smul r a) (smul r b) = smul r (add a b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rcases sosx with ⟨a, b, xeq⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rcases sosy with ⟨c, d, yeq⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | rw [xeq, yeq] | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [105, 1] | [111, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | rcases sosx with ⟨a, b, rfl⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | rcases sosy with ⟨c, d, rfl⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [113, 1] | [118, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [152, 1] | [156, 8] | 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/S01_Groups.lean | eq_bot_iff_card | [152, 1] | [156, 8] | sorry | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype (↥H)
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x | 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)
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [152, 1] | [156, 8] | 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/S01_Groups.lean | inf_bot_of_coprime | [160, 1] | [162, 10] | sorry | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
⊢ H ⊓ K = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype (↥H)
inst✝ : Fintype (↥K)
h : Nat.Coprime (card (↥H)) (card (↥K))
⊢ H ⊓ K = ⊥
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | compat_myMap | [189, 1] | [192, 7] | rintro _ rfl | ⊢ ∀ r ∈ {FreeGroup.of () ^ 3}, ↑(↑FreeGroup.lift myMap) r = 1 | ⊢ ↑(↑FreeGroup.lift myMap) (FreeGroup.of () ^ 3) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ r ∈ {FreeGroup.of () ^ 3}, ↑(↑FreeGroup.lift myMap) r = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | compat_myMap | [189, 1] | [192, 7] | simp | ⊢ ↑(↑FreeGroup.lift myMap) (FreeGroup.of () ^ 3) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ↑(↑FreeGroup.lift myMap) (FreeGroup.of () ^ 3) = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | conjugate_one | [229, 1] | [230, 10] | sorry | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ conjugate 1 H = H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ conjugate 1 H = H
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C08_Groups_and_Rings/S01_Groups.lean | aux_card_eq | [274, 1] | [275, 8] | sorry | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) = card (↥K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card (↥H) * card (↥K)
⊢ card (G ⧸ H) = card (↥K)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [48, 1] | [49, 8] | sorry | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [51, 1] | [52, 8] | sorry | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_1 | [35, 1] | [43, 8] | have hab : a ≠ b | ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔ Tendsto (fun i => if p i then a else b) L (if q then F else G) | case hab
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
⊢ a ≠ b
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔ Tendsto (fun i => if p i then a else b) L (if q then F else G) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔ Tendsto (fun i => if p i then a else b) L (if q then F else G)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_1 | [35, 1] | [43, 8] | rw [tendsto_iff_eventually] | ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔ Tendsto (fun i => if p i then a else b) L (if q then F else G) | ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔
∀ ⦃p_1 : α → Prop⦄, (∀ᶠ (y : α) in if q then F else G, p_1 y) → ∀ᶠ (x : ι) in L, p_1 (if p x then a else b) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔ Tendsto (fun i => if p i then a else b) L (if q then F else G)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_1 | [35, 1] | [43, 8] | sorry | ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔
∀ ⦃p_1 : α → Prop⦄, (∀ᶠ (y : α) in if q then F else G, p_1 y) → ∀ᶠ (x : ι) in L, p_1 (if p x then a else b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
hab : a ≠ b
⊢ (∀ᶠ (i : ι) in L, p i ↔ q) ↔
∀ ⦃p_1 : α → Prop⦄, (∀ᶠ (y : α) in if q then F else G, p_1 y) → ∀ᶠ (x : ι) in L, p_1 (if p x then a else b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_1 | [35, 1] | [43, 8] | sorry | case hab
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
⊢ a ≠ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
ι : Type u_1
α : Type u_2
p : ι → Prop
q : Prop
a b : α
L : Filter ι
F G : Filter α
hbF : ∀ᶠ (x : α) in F, x ≠ b
haG : ∀ᶠ (x : α) in G, x ≠ a
haF : pure a ≤ F
hbG : pure b ≤ G
⊢ a ≠ b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_2 | [49, 1] | [54, 8] | rw [tendsto_pi_nhds] | ι : Type u_1
L : Filter ι
s : ι → Set ℝ
t : Set ℝ
f : ℝ → ℝ
ha : ∀ (x : ℝ), f x ≠ 0
⊢ (∀ (x : ℝ), ∀ᶠ (i : ι) in L, x ∈ s i ↔ x ∈ t) ↔ Tendsto (fun i => indicator (s i) f) L (𝓝 (indicator t f)) | ι : Type u_1
L : Filter ι
s : ι → Set ℝ
t : Set ℝ
f : ℝ → ℝ
ha : ∀ (x : ℝ), f x ≠ 0
⊢ (∀ (x : ℝ), ∀ᶠ (i : ι) in L, x ∈ s i ↔ x ∈ t) ↔
∀ (x : ℝ), Tendsto (fun i => indicator (s i) f x) L (𝓝 (indicator t f x)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
L : Filter ι
s : ι → Set ℝ
t : Set ℝ
f : ℝ → ℝ
ha : ∀ (x : ℝ), f x ≠ 0
⊢ (∀ (x : ℝ), ∀ᶠ (i : ι) in L, x ∈ s i ↔ x ∈ t) ↔ Tendsto (fun i => indicator (s i) f) L (𝓝 (indicator t f))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_2 | [49, 1] | [54, 8] | sorry | ι : Type u_1
L : Filter ι
s : ι → Set ℝ
t : Set ℝ
f : ℝ → ℝ
ha : ∀ (x : ℝ), f x ≠ 0
⊢ (∀ (x : ℝ), ∀ᶠ (i : ι) in L, x ∈ s i ↔ x ∈ t) ↔
∀ (x : ℝ), Tendsto (fun i => indicator (s i) f x) L (𝓝 (indicator t f x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
L : Filter ι
s : ι → Set ℝ
t : Set ℝ
f : ℝ → ℝ
ha : ∀ (x : ℝ), f x ≠ 0
⊢ (∀ (x : ℝ), ∀ᶠ (i : ι) in L, x ∈ s i ↔ x ∈ t) ↔
∀ (x : ℝ), Tendsto (fun i => indicator (s i) f x) L (𝓝 (indicator t f x))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_3 | [72, 1] | [74, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → α
hf : Continuous f
h2f : Injective f
a b x : α
hab : a ≤ b
h2ab : f a < f b
hx : a ≤ x
⊢ f a ≤ f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → α
hf : Continuous f
h2f : Injective f
a b x : α
hab : a ≤ b
h2ab : f a < f b
hx : a ≤ x
⊢ f a ≤ f x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_4 | [78, 1] | [80, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → α
hf : Continuous f
h2f : Injective f
a b : α
hab : a ≤ b
h2ab : f a < f b
⊢ StrictMonoOn f (Ici a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → α
hf : Continuous f
h2f : Injective f
a b : α
hab : a ≤ b
h2ab : f a < f b
⊢ StrictMonoOn f (Ici a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | have h3f : ∀ {a b : ℝ} (hab : a ≤ b) (h2ab : f a < f b), StrictMonoOn f (Iic b) | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
⊢ StrictMono f ∨ StrictAnti f | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
⊢ ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b)
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
h3f : ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b)
⊢ StrictMono f ∨ StrictAnti f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
⊢ StrictMono f ∨ StrictAnti f
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
h3f : ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b)
⊢ StrictMono f ∨ StrictAnti f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
h3f : ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b)
⊢ StrictMono f ∨ StrictAnti f
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | intro a b hab h2ab | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
⊢ ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b) | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
⊢ StrictMonoOn f (Iic b) | Please generate a tactic in lean4 to solve the state.
STATE:
case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
⊢ ∀ {a b : ℝ}, a ≤ b → f a < f b → StrictMonoOn f (Iic b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | have := exercise6_4 (OrderDual ℝ) hf h2f hab h2ab | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
⊢ StrictMonoOn f (Iic b) | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b) | Please generate a tactic in lean4 to solve the state.
STATE:
case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
⊢ StrictMonoOn f (Iic b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | convert this using 0 | case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b) | case a
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b) ↔ StrictMonoOn f (Ici b) | Please generate a tactic in lean4 to solve the state.
STATE:
case h3f
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_5 | [91, 1] | [98, 8] | exact strict_mono_on_dual_iff.symm | case a
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b) ↔ StrictMonoOn f (Ici b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : ℝ → ℝ
hf : Continuous f
h2f : Injective f
a b : ℝ
hab : a ≤ b
h2ab : f a < f b
this : StrictMonoOn f (Ici b)
⊢ StrictMonoOn f (Iic b) ↔ StrictMonoOn f (Ici b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | intro h | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
⊢ ¬DifferentiableAt ℝ (fun x => |x|) 0 | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
⊢ ¬DifferentiableAt ℝ (fun x => |x|) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | have h1 : HasDerivWithinAt (fun x : ℝ ↦ |x|) 1 (Ici 0) 0 | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ False | case h1
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | have h2 : HasDerivWithinAt (fun x : ℝ ↦ |x|) (-1) (Iic 0) 0 | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ False | case h2
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | have h3 : UniqueDiffWithinAt ℝ (Ici (0 : ℝ)) 0 := by
sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
⊢ False | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | have h4 : UniqueDiffWithinAt ℝ (Iic (0 : ℝ)) 0 := by
sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
⊢ False | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
h4 : UniqueDiffWithinAt ℝ (Iic 0) 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
h4 : UniqueDiffWithinAt ℝ (Iic 0) 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
h4 : UniqueDiffWithinAt ℝ (Iic 0) 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | sorry | case h1
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
⊢ HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | sorry | case h2
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
⊢ HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
⊢ UniqueDiffWithinAt ℝ (Ici 0) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
⊢ UniqueDiffWithinAt ℝ (Ici 0) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment6.lean | exercise6_6 | [118, 1] | [128, 8] | sorry | α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
⊢ UniqueDiffWithinAt ℝ (Iic 0) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
h : DifferentiableAt ℝ (fun x => |x|) 0
h1 : HasDerivWithinAt (fun x => |x|) 1 (Ici 0) 0
h2 : HasDerivWithinAt (fun x => |x|) (-1) (Iic 0) 0
h3 : UniqueDiffWithinAt ℝ (Ici 0) 0
⊢ UniqueDiffWithinAt ℝ (Iic 0) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | induction n | n : ℕ
⊢ 0 < fac n | case zero
⊢ 0 < fac 0
case succ
n✝ : ℕ
a✝ : 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/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | case zero =>
rw [fac]
norm_num | ⊢ 0 < fac 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < fac 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | case succ k ih =>
rw [fac]
positivity | k : ℕ
ih : 0 < fac k
⊢ 0 < fac (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 0 < fac k
⊢ 0 < fac (k + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | rw [fac] | ⊢ 0 < fac 0 | ⊢ 0 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < fac 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | 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/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | rw [fac] | k : ℕ
ih : 0 < fac k
⊢ 0 < fac (k + 1) | k : ℕ
ih : 0 < fac k
⊢ 0 < (k + 1) * fac k | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 0 < fac k
⊢ 0 < fac (k + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_pos | [45, 1] | [62, 2] | positivity | k : ℕ
ih : 0 < fac k
⊢ 0 < (k + 1) * fac k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 0 < fac k
⊢ 0 < (k + 1) * fac k
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | induction n | n : ℕ
⊢ 2 ^ n ≤ fac (n + 1) | case zero
⊢ 2 ^ 0 ≤ fac (0 + 1)
case succ
n✝ : ℕ
a✝ : 2 ^ n✝ ≤ fac (n✝ + 1)
⊢ 2 ^ (n✝ + 1) ≤ fac (n✝ + 1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 2 ^ n ≤ fac (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | case zero =>
norm_num | ⊢ 2 ^ 0 ≤ fac (0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 2 ^ 0 ≤ fac (0 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | case succ k ih =>
rw [fac, pow_add, mul_comm]
gcongr
linarith | k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ (k + 1) ≤ fac (k + 1 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ (k + 1) ≤ fac (k + 1 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | norm_num | ⊢ 2 ^ 0 ≤ fac (0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 2 ^ 0 ≤ fac (0 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | rw [fac, pow_add, mul_comm] | k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ (k + 1) ≤ fac (k + 1 + 1) | k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 * 2 ^ k ≤ (k + 1 + 1) * fac (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ (k + 1) ≤ fac (k + 1 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | gcongr | k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 * 2 ^ k ≤ (k + 1 + 1) * fac (k + 1) | case h₁
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 ≤ k + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 * 2 ^ k ≤ (k + 1 + 1) * fac (k + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | pow_two_le_fac | [71, 1] | [78, 13] | linarith | case h₁
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 ≤ k + 1 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
k : ℕ
ih : 2 ^ k ≤ fac (k + 1)
⊢ 2 ^ 1 ≤ k + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | symm | n : ℕ
⊢ ∑ i in Finset.range (n + 1), i = n * (n + 1) / 2 | n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in Finset.range (n + 1), i | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∑ i in Finset.range (n + 1), i = n * (n + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | rw [Nat.div_eq_of_eq_mul_left] | n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in Finset.range (n + 1), i | case H1
n : ℕ
⊢ 0 < 2
case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ n * (n + 1) / 2 = ∑ i in Finset.range (n + 1), i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | norm_num | case H1
n : ℕ
⊢ 0 < 2
case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2 | case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
n : ℕ
⊢ 0 < 2
case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | symm | case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2 | case H2
n : ℕ
⊢ (∑ i in Finset.range (n + 1), i) * 2 = n * (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
n : ℕ
⊢ n * (n + 1) = (∑ i in Finset.range (n + 1), i) * 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | induction n | case H2
n : ℕ
⊢ (∑ i in Finset.range (n + 1), i) * 2 = n * (n + 1) | case H2.zero
⊢ (∑ i in Finset.range (0 + 1), i) * 2 = 0 * (0 + 1)
case H2.succ
n✝ : ℕ
a✝ : (∑ i in Finset.range (n✝ + 1), i) * 2 = n✝ * (n✝ + 1)
⊢ (∑ i in Finset.range (n✝ + 1 + 1), i) * 2 = (n✝ + 1) * (n✝ + 1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
n : ℕ
⊢ (∑ i in Finset.range (n + 1), i) * 2 = n * (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | case zero => simp | ⊢ (∑ i in Finset.range (0 + 1), i) * 2 = 0 * (0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (∑ i in Finset.range (0 + 1), i) * 2 = 0 * (0 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | case succ k ih =>
rw [sum_range_succ, add_mul, ih]
ring | k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ (∑ i in Finset.range (k + 1 + 1), i) * 2 = (k + 1) * (k + 1 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ (∑ i in Finset.range (k + 1 + 1), i) * 2 = (k + 1) * (k + 1 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | simp | ⊢ (∑ i in Finset.range (0 + 1), i) * 2 = 0 * (0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (∑ i in Finset.range (0 + 1), i) * 2 = 0 * (0 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | rw [sum_range_succ, add_mul, ih] | k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ (∑ i in Finset.range (k + 1 + 1), i) * 2 = (k + 1) * (k + 1 + 1) | k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ k * (k + 1) + (k + 1) * 2 = (k + 1) * (k + 1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ (∑ i in Finset.range (k + 1 + 1), i) * 2 = (k + 1) * (k + 1 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | sum_id | [102, 1] | [114, 2] | ring | k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ k * (k + 1) + (k + 1) * 2 = (k + 1) * (k + 1 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
ih : (∑ i in Finset.range (k + 1), i) * 2 = k * (k + 1)
⊢ k * (k + 1) + (k + 1) * 2 = (k + 1) * (k + 1 + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sub_ψ_ne_zero | [185, 1] | [188, 11] | simp [ϕ, ψ, sub_eq_zero] | n : ℕ
⊢ ϕ - ψ ≠ 0 | n : ℕ
⊢ ¬1 + sqrt 5 = 1 - sqrt 5 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ϕ - ψ ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sub_ψ_ne_zero | [185, 1] | [188, 11] | simp [sub_eq_add_neg] | n : ℕ
⊢ ¬1 + sqrt 5 = 1 - sqrt 5 | n : ℕ
⊢ 0 < 5 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ¬1 + sqrt 5 = 1 - sqrt 5
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sub_ψ_ne_zero | [185, 1] | [188, 11] | norm_num | n : ℕ
⊢ 0 < 5 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 0 < 5
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sq | [190, 1] | [193, 7] | simp [ϕ, add_sq] | n : ℕ
⊢ ϕ ^ 2 = ϕ + 1 | n : ℕ
⊢ (1 + 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 + sqrt 5) / 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ϕ ^ 2 = ϕ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sq | [190, 1] | [193, 7] | field_simp | n : ℕ
⊢ (1 + 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 + sqrt 5) / 2 + 1 | n : ℕ
⊢ (1 + 2 * sqrt 5 + 5) * 2 = (1 + sqrt 5 + 2) * 2 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (1 + 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 + sqrt 5) / 2 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ϕ_sq | [190, 1] | [193, 7] | ring | n : ℕ
⊢ (1 + 2 * sqrt 5 + 5) * 2 = (1 + sqrt 5 + 2) * 2 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (1 + 2 * sqrt 5 + 5) * 2 = (1 + sqrt 5 + 2) * 2 ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ψ_sq | [195, 1] | [198, 7] | simp [ψ, sub_sq] | n : ℕ
⊢ ψ ^ 2 = ψ + 1 | n : ℕ
⊢ (1 - 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 - sqrt 5) / 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ψ ^ 2 = ψ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ψ_sq | [195, 1] | [198, 7] | field_simp | n : ℕ
⊢ (1 - 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 - sqrt 5) / 2 + 1 | n : ℕ
⊢ (1 - 2 * sqrt 5 + 5) * 2 = (1 - sqrt 5 + 2) * 2 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (1 - 2 * sqrt 5 + sqrt 5 ^ 2) / 2 ^ 2 = (1 - sqrt 5) / 2 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | ψ_sq | [195, 1] | [198, 7] | ring | n : ℕ
⊢ (1 - 2 * sqrt 5 + 5) * 2 = (1 - sqrt 5 + 2) * 2 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (1 - 2 * sqrt 5 + 5) * 2 = (1 - sqrt 5 + 2) * 2 ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | induction n using Nat.two_step_induction | n✝ n : ℕ
⊢ ↑(fib n) = (ϕ ^ n - ψ ^ n) / (ϕ - ψ) | case zero
n : ℕ
⊢ ↑(fib 0) = (ϕ ^ 0 - ψ ^ 0) / (ϕ - ψ)
case one
n : ℕ
⊢ ↑(fib 1) = (ϕ ^ 1 - ψ ^ 1) / (ϕ - ψ)
case step
n k✝ : ℕ
IH0✝ : ↑(fib k✝) = (ϕ ^ k✝ - ψ ^ k✝) / (ϕ - ψ)
IH1✝ : ↑(fib (k✝ + 1)) = (ϕ ^ (k✝ + 1) - ψ ^ (k✝ + 1)) / (ϕ - ψ)
⊢ ↑(fib (k✝ + 2)) = (ϕ ^ (k✝ + 2) - ψ ^ (k✝ + 2)) / (ϕ - ψ) | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ ↑(fib n) = (ϕ ^ n - ψ ^ n) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | case zero => simp | n : ℕ
⊢ ↑(fib 0) = (ϕ ^ 0 - ψ ^ 0) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ↑(fib 0) = (ϕ ^ 0 - ψ ^ 0) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | case one => simp | n : ℕ
⊢ ↑(fib 1) = (ϕ ^ 1 - ψ ^ 1) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ↑(fib 1) = (ϕ ^ 1 - ψ ^ 1) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | case step k ih1 ih2 =>
simp [fib, ih1, ih2]
field_simp
simp [pow_add]
ring | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ↑(fib (k + 2)) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ↑(fib (k + 2)) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | simp | n : ℕ
⊢ ↑(fib 0) = (ϕ ^ 0 - ψ ^ 0) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ↑(fib 0) = (ϕ ^ 0 - ψ ^ 0) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | simp | n : ℕ
⊢ ↑(fib 1) = (ϕ ^ 1 - ψ ^ 1) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ↑(fib 1) = (ϕ ^ 1 - ψ ^ 1) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | simp [fib, ih1, ih2] | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ↑(fib (k + 2)) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ) | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ) + (ϕ ^ k - ψ ^ k) / (ϕ - ψ) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ) | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ↑(fib (k + 2)) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | field_simp | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ) + (ϕ ^ k - ψ ^ k) / (ϕ - ψ) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ) | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ (k + 1) - ψ ^ (k + 1) + (ϕ ^ k - ψ ^ k) = ϕ ^ (k + 2) - ψ ^ (k + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ) + (ϕ ^ k - ψ ^ k) / (ϕ - ψ) = (ϕ ^ (k + 2) - ψ ^ (k + 2)) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | simp [pow_add] | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ (k + 1) - ψ ^ (k + 1) + (ϕ ^ k - ψ ^ k) = ϕ ^ (k + 2) - ψ ^ (k + 2) | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ k * ϕ - ψ ^ k * ψ + (ϕ ^ k - ψ ^ k) = ϕ ^ k * (ϕ + 1) - ψ ^ k * (ψ + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ (k + 1) - ψ ^ (k + 1) + (ϕ ^ k - ψ ^ k) = ϕ ^ (k + 2) - ψ ^ (k + 2)
TACTIC:
|
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