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_S02_Rings.lean | chineseMap_surj | [53, 1] | [75, 30] | simp [(he j).2 i hj.symm] | case h.h
ι : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Fintype ι
I : ι → Ideal R
hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)
g : Π (i : ι), R ⧸ I i
f : ι → R
hf : ∀ (i : ι), ↑(mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), ↑(mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → ↑(mk (I j)) (e i) = 0
i j : ι
hj : j ≠ i
⊢ ↑(mk (I i)) (f j * e j) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
ι : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Fintype ι
I : ι → Ideal R
hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)
g : Π (i : ι), R ⧸ I i
f : ι → R
hf : ∀ (i : ι), ↑(mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), ↑(mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → ↑(mk (I j)) (e i) = 0
i j : ι
hj : j ≠ i
⊢ ↑(mk (I i)) (f j * e j) = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C09_Topology/solutions/Solutions_S03_Topological_Spaces.lean | aux | [112, 1] | [116, 89] | simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in | 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/C04_Sets_and_Functions/S02_Functions.lean | inverse_spec | [178, 1] | [180, 32] | rw [inverse, dif_pos h] | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (inverse f y) = y | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (inverse f y) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | inverse_spec | [178, 1] | [180, 32] | exact Classical.choose_spec h | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | intro f surjf | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | let S := { i | i ∉ f i } | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | rcases surjf S with ⟨j, h⟩ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | have h₁ : j ∉ f j := by
intro h'
have : j ∉ f j := by rwa [h] at h'
contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | have h₂ : j ∈ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | sorry | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | have h₃ : j ∉ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | sorry | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | intro h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | have : j ∉ f j := by rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | contradiction | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [198, 1] | [210, 16] | rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_1 | [22, 1] | [23, 45] | rw [add_comm a b, add_assoc d, add_comm d] | a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_2 | [29, 1] | [33, 47] | calc a + b + c + d
= b + a + c + d := by rw [add_comm a b]
_ = d + (b + a + c) := by rw [add_comm d]
_ = d + (b + a) + c := by rw [add_assoc d] | a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ a + b + c + d = d + (b + a) + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_2 | [29, 1] | [33, 47] | rw [add_comm a b] | a b c d : ℝ
⊢ a + b + c + d = b + a + c + d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ a + b + c + d = b + a + c + d
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_2 | [29, 1] | [33, 47] | rw [add_comm d] | a b c d : ℝ
⊢ b + a + c + d = d + (b + a + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ b + a + c + d = d + (b + a + c)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_2 | [29, 1] | [33, 47] | rw [add_assoc d] | a b c d : ℝ
⊢ d + (b + a + c) = d + (b + a) + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ d + (b + a + c) = d + (b + a) + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [pow_two a] | a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2 | a b : ℝ
⊢ (a + b) * (a - b) = a * a - b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [pow_two b] | a b : ℝ
⊢ (a + b) * (a - b) = a * a - b ^ 2 | a b : ℝ
⊢ (a + b) * (a - b) = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a * a - b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [mul_sub (a+b) a b] | a b : ℝ
⊢ (a + b) * (a - b) = a * a - b * b | a b : ℝ
⊢ (a + b) * a - (a + b) * b = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [add_mul a b a] | a b : ℝ
⊢ (a + b) * a - (a + b) * b = a * a - b * b | a b : ℝ
⊢ a * a + b * a - (a + b) * b = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * a - (a + b) * b = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [add_mul a b b] | a b : ℝ
⊢ a * a + b * a - (a + b) * b = a * a - b * b | a b : ℝ
⊢ a * a + b * a - (a * b + b * b) = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + b * a - (a + b) * b = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [mul_comm b a] | a b : ℝ
⊢ a * a + b * a - (a * b + b * b) = a * a - b * b | a b : ℝ
⊢ a * a + a * b - (a * b + b * b) = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + b * a - (a * b + b * b) = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [← sub_sub] | a b : ℝ
⊢ a * a + a * b - (a * b + b * b) = a * a - b * b | a b : ℝ
⊢ a * a + a * b - a * b - b * b = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + a * b - (a * b + b * b) = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [← add_sub] | a b : ℝ
⊢ a * a + a * b - a * b - b * b = a * a - b * b | a b : ℝ
⊢ a * a + (a * b - a * b) - b * b = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + a * b - a * b - b * b = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [sub_self] | a b : ℝ
⊢ a * a + (a * b - a * b) - b * b = a * a - b * b | a b : ℝ
⊢ a * a + 0 - b * b = a * a - b * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + (a * b - a * b) - b * b = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_3 | [54, 1] | [64, 16] | rw [add_zero] | a b : ℝ
⊢ a * a + 0 - b * b = a * a - b * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ a * a + 0 - b * b = a * a - b * b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions1.lean | exercise1_4 | [68, 1] | [68, 71] | ring | a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ (a + b) * (a - b) = a ^ 2 - b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [61, 1] | [62, 8] | sorry | x✝ y x : ℝ
⊢ x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [64, 1] | [65, 8] | sorry | x✝ y x : ℝ
⊢ -x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_add | [67, 1] | [68, 8] | sorry | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [70, 1] | [71, 8] | sorry | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [73, 1] | [74, 8] | sorry | x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [36, 1] | [39, 8] | intro x y ε epos ele1 xlt ylt | ⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [36, 1] | [39, 8] | sorry | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [41, 1] | [48, 19] | intro x y ε epos ele1 xlt ylt | ⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [41, 1] | [48, 19] | calc
|x * y| = |x| * |y| := sorry
_ ≤ |x| * ε := sorry
_ < 1 * ε := sorry
_ = ε := sorry | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.Subset.trans | [146, 1] | [147, 8] | sorry | α : Type u_1
r s t : Set α
⊢ r ⊆ s → s ⊆ t → r ⊆ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
r s t : Set α
⊢ r ⊆ s → s ⊆ t → r ⊆ t
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [39, 1] | [44, 11] | have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [39, 1] | [44, 11] | calc
a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [39, 1] | [44, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [39, 1] | [44, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact1 | [39, 1] | [44, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a - b) ^ 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ (a - b) ^ 2 ≥ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [46, 1] | [51, 11] | have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [46, 1] | [51, 11] | calc
a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [46, 1] | [51, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [46, 1] | [51, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean | fact2 | [46, 1] | [51, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a + b) ^ 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d e : ℝ
⊢ (a + b) ^ 2 ≥ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.add_comm | [58, 11] | [61, 25] | rw [add, add] | a b : Point
⊢ add a b = add b a | a b : Point
⊢ { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } = { x := b.x + a.x, y := b.y + a.y, z := b.z + a.z } | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Point
⊢ add a b = add b a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.add_comm | [58, 11] | [61, 25] | ext <;> dsimp | a b : Point
⊢ { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } = { x := b.x + a.x, y := b.y + a.y, z := b.z + a.z } | case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Point
⊢ { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } = { x := b.x + a.x, y := b.y + a.y, z := b.z + a.z }
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.add_comm | [58, 11] | [61, 25] | repeat' apply add_comm | case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.add_comm | [58, 11] | [61, 25] | apply add_comm | case z
a b : Point
⊢ a.z + b.z = b.z + a.z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case z
a b : Point
⊢ a.z + b.z = b.z + a.z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.addAlt_x | [74, 1] | [75, 6] | rfl | a b : Point
⊢ (addAlt a b).x = a.x + b.x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Point
⊢ (addAlt a b).x = a.x + b.x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.addAlt_comm | [77, 1] | [81, 25] | rw [addAlt, addAlt] | a b : Point
⊢ addAlt a b = addAlt b a | a b : Point
⊢ (match a, b with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }) =
match b, a with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ } | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Point
⊢ addAlt a b = addAlt b a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.addAlt_comm | [77, 1] | [81, 25] | ext <;> dsimp | a b : Point
⊢ (match a, b with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }) =
match b, a with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ } | case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Point
⊢ (match a, b with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }) =
match b, a with
| { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.addAlt_comm | [77, 1] | [81, 25] | repeat' apply add_comm | case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case x
a b : Point
⊢ a.x + b.x = b.x + a.x
case y
a b : Point
⊢ a.y + b.y = b.y + a.y
case z
a b : Point
⊢ a.z + b.z = b.z + a.z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.addAlt_comm | [77, 1] | [81, 25] | apply add_comm | case z
a b : Point
⊢ a.z + b.z = b.z + a.z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case z
a b : Point
⊢ a.z + b.z = b.z + a.z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C06_Structures/S01_Structures.lean | C06S01.Point.add_assoc | [83, 11] | [84, 8] | sorry | 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/S01_Structures.lean | C06S01.Point.smul_distrib | [89, 1] | [91, 8] | sorry | 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/Assignments/Assignment3.lean | exercise3_1 | [25, 1] | [25, 51] | sorry | p : Prop
⊢ p ∨ ¬p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : Prop
⊢ p ∨ ¬p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_2 | [43, 1] | [45, 42] | sorry | s : ℕ → ℝ
r : ℕ → ℕ
a : ℝ
hs : SequentialLimit s a
hr : ∀ (m : ℕ), ∃ N, ∀ n ≥ N, r n ≥ m
⊢ SequentialLimit (s ∘ r) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
r : ℕ → ℕ
a : ℝ
hs : SequentialLimit s a
hr : ∀ (m : ℕ), ∃ N, ∀ n ≥ N, r n ≥ m
⊢ SequentialLimit (s ∘ r) a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_3 | [51, 1] | [54, 37] | sorry | s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
⊢ SequentialLimit s₂ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
⊢ SequentialLimit s₂ a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_4 | [67, 1] | [67, 70] | sorry | ⊢ SequentialLimit (fun n => 1 / (↑n + 1)) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => 1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | intro ε hε | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
⊢ SequentialLimit (fun n => c * s n) (c * a) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
⊢ SequentialLimit (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | obtain ⟨N, hN⟩ := hs (ε / max |c| 1) (by positivity) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | use N | case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | intro n hn | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | specialize hN n hn | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | simp | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | calc |c * s n - c * a|
= |c * (s n - a)| := by ring
_ = |c| * |s n - a| := by exact abs_mul c (s n - a)
_ ≤ max |c| 1 * |s n - a| := by gcongr; exact le_max_left |c| 1
_ < max |c| 1 * (ε / max |c| 1) := by gcongr
_ = ε := by refine mul_div_cancel' ε ?hb; positivity | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | positivity | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ε / max |c| 1 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ε / max |c| 1 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | ring | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| = |c * (s n - a)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| = |c * (s n - a)|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | exact abs_mul c (s n - a) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * (s n - a)| = |c| * |s n - a| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * (s n - a)| = |c| * |s n - a|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | gcongr | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| * |s n - a| ≤ max |c| 1 * |s n - a| | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| * |s n - a| ≤ max |c| 1 * |s n - a|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | exact le_max_left |c| 1 | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | gcongr | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * |s n - a| < max |c| 1 * (ε / max |c| 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * |s n - a| < max |c| 1 * (ε / max |c| 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | refine mul_div_cancel' ε ?hb | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * (ε / max |c| 1) = ε | case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * (ε / max |c| 1) = ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | convergesTo_mul_const | [73, 1] | [86, 57] | positivity | case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | use_me | [88, 1] | [92, 23] | have : SequentialLimit (fun n ↦ (-1) * (1 / (n+1))) (-1 * 0) :=
convergesTo_mul_const (-1) exercise3_4 | ⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | use_me | [88, 1] | [92, 23] | simp at this | this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | use_me | [88, 1] | [92, 23] | simp [neg_div, this] | this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_5 | [94, 1] | [94, 74] | sorry | ⊢ SequentialLimit (fun n => sin ↑n / (↑n + 1)) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => sin ↑n / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_6 | [101, 1] | [102, 29] | sorry | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
⊢ ∀ (n : ℕ), u n ≤ l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
⊢ ∀ (n : ℕ), u n ≤ l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_7 | [109, 1] | [110, 48] | sorry | α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
⊢ f '' s ∩ t = f '' (s ∩ f ⁻¹' t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
⊢ f '' s ∩ t = f '' (s ∩ f ⁻¹' t)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_8 | [112, 1] | [113, 93] | sorry | ⊢ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} = {x | x ≤ -2} ∪ {x | x ≥ 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} = {x | x ≤ -2} ∪ {x | x ≥ 2}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | have h1' : ∀ x y, f x ≠ g y | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | have h1'' : ∀ y x, g y ≠ f x | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | have h2' : ∀ x, x ∈ range f ∪ range g := eq_univ_iff_forall.1 h2 | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | have hf' : ∀ x x', f x = f x' ↔ x = x' := fun x x' ↦ hf.eq_iff | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | let L : Set α × Set β → Set γ :=
fun (s, t) ↦ sorry | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | let R : Set γ → Set α × Set β :=
fun s ↦ sorry | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
R : Set γ → Set α × Set β := fun s => sorryAx (Set α × Set β)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | sorry | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
R : Set γ → Set α × Set β := fun s => sorryAx (Set α × Set β)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => sorryAx (Set γ)
R : Set γ → Set α × Set β := fun s => sorryAx (Set α × Set β)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | intro x y h | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | apply h1.subset | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False | case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | exact ⟨⟨x, h⟩, ⟨y, rfl⟩⟩ | case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ
TACTIC:
|
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