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/C05_Elementary_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean | Nat.Prime.factorization' | [71, 1] | [74, 7] | simp | p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
prime_p : Prime p
⊢ (↑fun₀ | p => 1) p = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | Point.add_comm | [154, 1] | [155, 47] | simp [add, add_comm] | n : ℕ
a b : Point
⊢ add a b = add b a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
a b : Point
⊢ add a b = add b a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | Point.add_x | [164, 1] | [164, 68] | rfl | n : ℕ
a b : Point
⊢ (a + b).x = a.x + b.x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
a b : Point
⊢ (a + b).x = a.x + b.x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | Point.add_y | [165, 1] | [165, 68] | rfl | n : ℕ
a b : Point
⊢ (a + b).y = a.y + b.y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
a b : Point
⊢ (a + b).y = a.y + b.y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | Point.add_z | [166, 1] | [166, 68] | rfl | n : ℕ
a b : Point
⊢ (a + b).z = a.z + b.z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
a b : Point
⊢ (a + b).z = a.z + b.z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | AbelianGroup.zero_add | [330, 1] | [332, 26] | rw [g.comm, g.add_zero] | n : ℕ
g : AbelianGroup
x : g.G
⊢ add g g.zero x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
g : AbelianGroup
x : g.G
⊢ add g g.zero x = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture8.lean | PointedFunction.comp | [516, 1] | [518, 25] | sorry | n : ℕ
X : PointedType
Y : PointedType
Z : PointedType
g : Y →. Z
f : X →. Y
⊢ (↑g ∘ ↑f) X.pt = Z.pt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
X : PointedType
Y : PointedType
Z : PointedType
g : Y →. Z
f : X →. Y
⊢ (↑g ∘ ↑f) X.pt = Z.pt
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | fac_zero | [31, 1] | [31, 39] | sorry | n : ℕ
⊢ fac 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ fac 0 = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | fac_succ | [33, 1] | [33, 67] | sorry | n✝ n : ℕ
⊢ fac (n + 1) = (n + 1) * fac n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ fac (n + 1) = (n + 1) * fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | fac_pos | [37, 1] | [37, 48] | sorry | n✝ n : ℕ
⊢ 0 < fac n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ 0 < fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | pow_two_le_fac | [45, 1] | [45, 65] | sorry | n✝ n : ℕ
⊢ 2 ^ n ≤ fac (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ 2 ^ n ≤ fac (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | sum_id | [65, 1] | [65, 81] | sorry | n✝ n : ℕ
⊢ ∑ i in Finset.range (n + 1), i = n * (n + 1) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ ∑ i in Finset.range (n + 1), i = n * (n + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | coe_fib_eq | [125, 1] | [125, 79] | sorry | n✝ n : ℕ
⊢ ↑(fib n) = (ϕ ^ n - ψ ^ n) / (ϕ - ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ ↑(fib n) = (ϕ ^ n - ψ ^ n) / (ϕ - ψ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | fac_dvd_fac | [158, 1] | [158, 70] | sorry | n✝ n m : ℕ
h : n ≤ m
⊢ fac n ∣ fac m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n m : ℕ
h : n ≤ m
⊢ fac n ∣ fac m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7Before.lean | le_generateFrom_iff_subset_isOpen | [169, 1] | [171, 71] | sorry | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
⊢ {s | IsOpen s} ⊆ {s | IsOpen s} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
⊢ {s | IsOpen s} ⊆ {s | IsOpen s}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [41, 1] | [46, 21] | apply le_min | a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c) | case h₁
a b c d : ℝ
⊢ min a b + c ≤ a + c
case h₂
a b c d : ℝ
⊢ min a b + c ≤ b + c | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [41, 1] | [46, 21] | apply add_le_add_right | case h₂
a b c d : ℝ
⊢ min a b + c ≤ b + c | case h₂.bc
a b c d : ℝ
⊢ min a b ≤ b | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
a b c d : ℝ
⊢ min a b + c ≤ b + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [41, 1] | [46, 21] | apply min_le_right | case h₂.bc
a b c d : ℝ
⊢ min a b ≤ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.bc
a b c d : ℝ
⊢ min a b ≤ b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [41, 1] | [46, 21] | apply add_le_add_right | case h₁
a b c d : ℝ
⊢ min a b + c ≤ a + c | case h₁.bc
a b c d : ℝ
⊢ min a b ≤ a | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
a b c d : ℝ
⊢ min a b + c ≤ a + c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [41, 1] | [46, 21] | apply min_le_left | case h₁.bc
a b c d : ℝ
⊢ min a b ≤ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.bc
a b c d : ℝ
⊢ min a b ≤ a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.le_def | [367, 1] | [367, 85] | simp | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ U ≤ V ↔ ↑U ⊆ ↑V | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ U ≤ V ↔ ↑U ⊆ ↑V
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.cl_le_iff | [402, 1] | [403, 37] | sorry | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V U : RegularOpens X
C : Closeds X
⊢ cl U ≤ C ↔ U ≤ Closeds.int C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V U : RegularOpens X
C : Closeds X
⊢ cl U ≤ C ↔ U ≤ Closeds.int C
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.cl_int | [405, 1] | [405, 48] | sorry | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ Closeds.int (cl U) = U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ Closeds.int (cl U) = U
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_inf | [421, 1] | [423, 14] | have : U ⊓ V = (U.cl ⊓ V.cl).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ ↑(U ⊓ V) = ↑U ∩ ↑V | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊓ V = Closeds.int (cl U ⊓ cl V)
⊢ ↑(U ⊓ V) = ↑U ∩ ↑V | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ ↑(U ⊓ V) = ↑U ∩ ↑V
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_inf | [421, 1] | [423, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊓ V = Closeds.int (cl U ⊓ cl V)
⊢ ↑(U ⊓ V) = ↑U ∩ ↑V | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊓ V = Closeds.int (cl U ⊓ cl V)
⊢ ↑(U ⊓ V) = ↑U ∩ ↑V
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sup | [425, 1] | [427, 14] | have : U ⊔ V = (U.cl ⊔ V.cl).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ ↑(U ⊔ V) = interior (closure (↑U ∪ ↑V)) | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊔ V = Closeds.int (cl U ⊔ cl V)
⊢ ↑(U ⊔ V) = interior (closure (↑U ∪ ↑V)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
⊢ ↑(U ⊔ V) = interior (closure (↑U ∪ ↑V))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sup | [425, 1] | [427, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊔ V = Closeds.int (cl U ⊔ cl V)
⊢ ↑(U ⊔ V) = interior (closure (↑U ∪ ↑V)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V✝ U V : RegularOpens X
this : U ⊔ V = Closeds.int (cl U ⊔ cl V)
⊢ ↑(U ⊔ V) = interior (closure (↑U ∪ ↑V))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_top | [429, 1] | [431, 14] | have : (⊤ : RegularOpens X) = (⊤ : Closeds X).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ ↑⊤ = univ | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊤ = Closeds.int ⊤
⊢ ↑⊤ = univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ ↑⊤ = univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_top | [429, 1] | [431, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊤ = Closeds.int ⊤
⊢ ↑⊤ = univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊤ = Closeds.int ⊤
⊢ ↑⊤ = univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_bot | [433, 1] | [435, 14] | have : (⊥ : RegularOpens X) = (⊥ : Closeds X).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ ↑⊥ = ∅ | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊥ = Closeds.int ⊥
⊢ ↑⊥ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
⊢ ↑⊥ = ∅
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_bot | [433, 1] | [435, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊥ = Closeds.int ⊥
⊢ ↑⊥ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
this : ⊥ = Closeds.int ⊥
⊢ ↑⊥ = ∅
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sInf | [437, 1] | [441, 14] | have : sInf U = (sInf (cl '' U)).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
⊢ ↑(sInf U) = interior (⋂₀ ((fun u => closure ↑u) '' U)) | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
⊢ ↑(sInf U) = interior (⋂₀ ((fun u => closure ↑u) '' U)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
⊢ ↑(sInf U) = interior (⋂₀ ((fun u => closure ↑u) '' U))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sInf | [437, 1] | [441, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
⊢ ↑(sInf U) = interior (⋂₀ ((fun u => closure ↑u) '' U)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
⊢ ↑(sInf U) = interior (⋂₀ ((fun u => closure ↑u) '' U))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.Closeds.coe_sSup | [443, 1] | [446, 19] | have : sSup C = Closeds.closure (sSup ((↑) '' C)) := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
⊢ ↑(sSup C) = closure (⋃₀ (SetLike.coe '' C)) | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(sSup C) = closure (⋃₀ (SetLike.coe '' C)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
⊢ ↑(sSup C) = closure (⋃₀ (SetLike.coe '' C))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.Closeds.coe_sSup | [443, 1] | [446, 19] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(sSup C) = closure (⋃₀ (SetLike.coe '' C)) | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(Closeds.closure (⋃ x ∈ C, ↑x)) = closure (⋃ x ∈ C, ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(sSup C) = closure (⋃₀ (SetLike.coe '' C))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.Closeds.coe_sSup | [443, 1] | [446, 19] | rfl | X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(Closeds.closure (⋃ x ∈ C, ↑x)) = closure (⋃ x ∈ C, ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
⊢ ↑(Closeds.closure (⋃ x ∈ C, ↑x)) = closure (⋃ x ∈ C, ↑x)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sSup | [448, 1] | [452, 14] | have : sSup U = (sSup (cl '' U)).int := rfl | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
⊢ ↑(sSup U) = interior (closure (⋃₀ ((fun u => closure ↑u) '' U))) | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
⊢ ↑(sSup U) = interior (closure (⋃₀ ((fun u => closure ↑u) '' U))) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
⊢ ↑(sSup U) = interior (closure (⋃₀ ((fun u => closure ↑u) '' U)))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_sSup | [448, 1] | [452, 14] | simp [this] | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
⊢ ↑(sSup U) = interior (closure (⋃₀ ((fun u => closure ↑u) '' U))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
⊢ ↑(sSup U) = interior (closure (⋃₀ ((fun u => closure ↑u) '' U)))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11Before.lean | RegularOpens.coe_compl | [466, 1] | [467, 79] | sorry | X : Type u_1
inst✝ : TopologicalSpace X
U✝ V U : RegularOpens X
⊢ ↑Uᶜ = interior (↑U)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
U✝ V U : RegularOpens X
⊢ ↑Uᶜ = interior (↑U)ᶜ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture9.lean | PointedFunction.comp | [87, 1] | [89, 24] | simp | X : PointedType
Y : PointedType
Z : PointedType
g : Y →. Z
f : X →. Y
⊢ (↑g ∘ ↑f) X.pt = Z.pt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : PointedType
Y : PointedType
Z : PointedType
g : Y →. Z
f : X →. Y
⊢ (↑g ∘ ↑f) X.pt = Z.pt
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture9.lean | conjugate_one | [370, 1] | [370, 69] | sorry | G : Type u_1
H✝ : Type u_2
K : Type u_3
inst✝² : Group G
inst✝¹ : Group H✝
inst✝ : Group K
H : Subgroup G
⊢ conjugate 1 H = H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
H✝ : Type u_2
K : Type u_3
inst✝² : Group G
inst✝¹ : Group H✝
inst✝ : Group K
H : Subgroup G
⊢ conjugate 1 H = H
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture9.lean | conjugate_mul | [372, 1] | [373, 66] | sorry | G : Type u_1
H✝ : Type u_2
K : Type u_3
inst✝² : Group G
inst✝¹ : Group H✝
inst✝ : Group K
x y : G
H : Subgroup G
⊢ conjugate (x * y) H = conjugate x (conjugate y H) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
H✝ : Type u_2
K : Type u_3
inst✝² : Group G
inst✝¹ : Group H✝
inst✝ : Group K
x y : G
H : Subgroup G
⊢ conjugate (x * y) H = conjugate x (conjugate y H)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | have : x ∈ g '' univ := by
contrapose! hx
rw [sbSet, mem_iUnion]
use 0
rw [sbAux, mem_diff]
exact ⟨mem_univ _, 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
⊢ 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | have : ∃ y, g y = x := by
simp at this
assumption | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | exact invFun_eq this | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | exact ⟨mem_univ _, hx⟩ | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | simp at this | 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 | 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 : ∃ y, g y = x
⊢ ∃ y, g y = 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
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [35, 23] | assumption | 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 : ∃ y, g y = x
⊢ ∃ 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 : ∃ y, g y = x
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [if_neg xA.1, if_neg xA.2] at hxeq | 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 : invFun g x₁ = 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] | 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 : invFun g x₁ = 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 : invFun g x₁ = invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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
rw [hxeq, sb_right_inv f g x₂nA]
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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [if_pos x₁A, if_pos x₂A] 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
x₂A : x₂ ∈ 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 : f x₁ = f 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
x₂A : x₂ ∈ A
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | exact hf 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 : f x₁ = f 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 : f x₁ = f 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | have x₂eq : x₂ = g (f x₁) := by
rw [hxeq, sb_right_inv f g x₂nA] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [37, 1] | [64, 60] | rw [hxeq, sb_right_inv f g x₂nA] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | use g 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 | 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
gyA : g y ∉ A
⊢ h (g y) = y | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | simp only [h_def, sbFun, if_neg gyA] | 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
gyA : g y ∉ A
⊢ h (g y) = 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
gyA : g y ∉ A
⊢ invFun g (g y) = 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
gyA : g y ∉ A
⊢ h (g y) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | apply leftInverse_invFun hg | 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
gyA : g y ∉ A
⊢ invFun g (g y) = 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
gyA : g y ∉ A
⊢ invFun g (g y) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C04_Sets_and_Functions/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/solutions/Solutions_S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [66, 1] | [85, 30] | 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/Lecture10Before.lean | units_ne_neg_self | [305, 1] | [305, 78] | sorry | R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CharZero R
u : Rˣ
⊢ u ≠ -u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CharZero R
u : Rˣ
⊢ u ≠ -u
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture10Before.lean | iterate_frobeniusMorphism | [341, 1] | [341, 96] | sorry | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
n : ℕ
⊢ (↑(frobeniusMorphism p K))^[n] x = x ^ p ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
n : ℕ
⊢ (↑(frobeniusMorphism p K))^[n] x = x ^ p ^ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture10Before.lean | frobeniusMorphism_injective | [343, 1] | [345, 8] | have : ∀ x : K, x ^ p = 0 → x = 0 := by exact? | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
⊢ Injective ↑(frobeniusMorphism p K) | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
this : ∀ (x : K), x ^ p = 0 → x = 0
⊢ Injective ↑(frobeniusMorphism p K) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
⊢ Injective ↑(frobeniusMorphism p K)
TACTIC:
|
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