url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | intro x 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
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x | 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
x : β
y : α
⊢ g x ≠ f y | 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
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | symm | 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
x : β
y : α
⊢ g x ≠ f 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
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ f y ≠ g x | 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
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ g x ≠ f y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment3.lean | exercise3_9 | [129, 1] | [146, 8] | apply h1' | 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
x : β
y : α
⊢ f y ≠ g x | no goals | 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
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ f y ≠ g x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean | inverse_spec | [201, 1] | [203, 32] | rw [inverse, dif_pos h] | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
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
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
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/solutions/Solutions_S02_Functions.lean | inverse_spec | [201, 1] | [203, 32] | exact Classical.choose_spec h | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
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
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | intro f surjf | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | let S := { i | i ∉ f i } | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | rcases surjf S with ⟨j, h⟩ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 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✝
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✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | have h₂ : j ∈ S := h₁ | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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 intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | have h₃ : j ∉ S := by rwa [h] at h₁ | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | intro h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | have : j ∉ f j := by rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | contradiction | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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✝
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/MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean | Cantor | [239, 1] | [249, 16] | rwa [h] at h₁ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | have : x ∈ g '' univ := by
contrapose! hx
rw [sbSet, mem_iUnion]
use 0
rw [sbAux, mem_diff]
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | have : ∃ y, g y = x := by
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | contrapose! hx | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | rw [sbSet, mem_iUnion] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | use 0 | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | rw [sbAux, mem_diff] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | set A := sbSet f g with A_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | set h := sbFun f g with h_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro x₁ x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro (hxeq : h x₁ = h x₂) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | simp only [h_def, sbFun, ← A_def] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | by_cases xA : x₁ ∈ A ∨ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : ¬(x₁ ∈ A ∨ x₂ ∈ A)
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | push_neg at xA | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : ¬(x₁ ∈ A ∨ x₂ ∈ A)
⊢ x₁ = x₂ | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : ¬(x₁ ∈ A ∨ x₂ ∈ A)
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂ | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₁ = x₂
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | have x₂A : x₂ ∈ A := by
apply not_imp_self.mp
intro (x₂nA : x₂ ∉ A)
rw [if_pos x₁A, if_neg x₂nA] at hxeq
rw [A_def, sbSet, mem_iUnion] at x₁A
have x₂eq : x₂ = g (f x₁) := by
sorry
rcases x₁A with ⟨n, hn⟩
rw [A_def, sbSet, mem_iUnion]
use n + 1
simp [sbAux]
exact ⟨x₁, hn, x₂eq.symm⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | symm | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₁ = x₂ | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₂ = x₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | apply this hxeq.symm xA.symm (xA.resolve_left x₁A) | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₂ = x₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ : α⦄,
((if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂) → x₁ ∈ A ∨ x₂ ∈ A → x₁ ∈ A → x₁ = x₂
x₁A : x₁ ∉ A
⊢ x₂ = x₁
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | apply not_imp_self.mp | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro (x₂nA : x₂ ∉ A) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [if_pos x₁A, if_neg x₂nA] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [A_def, sbSet, mem_iUnion] at x₁A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | have x₂eq : x₂ = g (f x₁) := by
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rcases x₁A with ⟨n, hn⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [A_def, sbSet, mem_iUnion] | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | use n + 1 | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux f g i | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ sbAux f g (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux f g i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | simp [sbAux] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ sbAux f g (n + 1) | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ sbAux f g (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | exact ⟨x₁, hn, x₂eq.symm⟩ | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ a ∈ sbAux f g n, g (f a) = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ = g (f x₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ = g (f x₁)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | set A := sbSet f g with A_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Surjective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Surjective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | set h := sbFun f g with h_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | intro y | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | by_cases gyA : g y ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∉ A
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | sorry | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∉ A
⊢ ∃ a, h a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∉ A
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rw [A_def, sbSet, mem_iUnion] at gyA | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases gyA with ⟨n, hn⟩ | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y | case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases n with _ | n | case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y | case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g (Nat.succ n)
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp [sbAux] at hn | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g (Nat.succ n)
⊢ ∃ a, h a = y | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g (Nat.succ n)
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases hn with ⟨x, xmem, hx⟩ | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
⊢ ∃ a, h a = y | case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | use x | case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ a, h a = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | have : x ∈ A := by
rw [A_def, sbSet, mem_iUnion]
exact ⟨n, xmem⟩ | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h x = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ h x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp only [h_def, sbFun, if_pos this] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ h x = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ f x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ h x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | exact hg hx | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ f x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ f x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp [sbAux] at hn | case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rw [A_def, sbSet, mem_iUnion] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ x ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ x ∈ A
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | exact ⟨n, xmem⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈ sbAux f g i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈ sbAux f g i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5Before.lean | sequentialLimit_unique | [79, 1] | [80, 68] | sorry | u : ℕ → ℝ
l l' : ℝ
⊢ SequentialLimit u l → SequentialLimit u l' → l = l' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l l' : ℝ
⊢ SequentialLimit u l → SequentialLimit u l' → l = l'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5Before.lean | convergesTo_const | [90, 1] | [90, 80] | sorry | a : ℝ
⊢ SequentialLimit (fun n => a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
⊢ SequentialLimit (fun n => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5Before.lean | SequentialLimit.add | [93, 1] | [95, 60] | sorry | s t : ℕ → ℝ
a b : ℝ
hs : SequentialLimit s a
ht : SequentialLimit t b
⊢ SequentialLimit (fun n => s n + t n) (a + b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
hs : SequentialLimit s a
ht : SequentialLimit t b
⊢ SequentialLimit (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | intro ε εpos | a : ℝ
⊢ ConvergesTo (fun x => a) a | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
⊢ ConvergesTo (fun x => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | use 0 | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | intro n nge | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | rw [sub_self, abs_zero] | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | apply εpos | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | intro ε εpos | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | dsimp | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ε2pos : 0 < ε / 2 := by linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | use max Ns Nt | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | intro n hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ngeNs : n ≥ Ns := le_of_max_le_left hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ngeNt : n ≥ Nt := le_of_max_le_right hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | calc
|s n + t n - (a + b)| = |s n - a + (t n - b)| := by
congr
ring
_ ≤ |s n - a| + |t n - b| := (abs_add _ _)
_ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))
_ = ε := by norm_num | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | congr | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| = |s n - a + (t n - b)| | case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a + b) = s n - a + (t n - b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| = |s n - a + (t n - b)|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | ring | case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a + b) = s n - a + (t n - b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a + b) = s n - a + (t n - b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | norm_num | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ ε / 2 + ε / 2 = ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ ε / 2 + ε / 2 = ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | by_cases h : c = 0 | s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a) | case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | have acpos : 0 < |c| := abs_pos.mpr h | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | intro ε εpos | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : 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:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | dsimp | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩ | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | use Ns | case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | intro n ngt | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
n : ℕ
ngt : n ≥ Ns
⊢ |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
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | calc
|c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub]
_ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)
_ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
n : ℕ
ngt : n ≥ Ns
⊢ |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
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
n : ℕ
ngt : n ≥ Ns
⊢ |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | convert convergesTo_const 0 | case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ c * s x✝ = 0
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rw [h] | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0 | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0
TACTIC:
|
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