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
|
|---|---|---|---|---|---|---|---|---|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
intro ε ε0
|
ι : Type
U : ι → Set ℝ
⊢ IsCauSeq abs (nestedIntervalSeq U)
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
have ⟨i, hi⟩ : ∃ i : ℕ, ε⁻¹ < 2 ^ i := pow_unbounded_of_one_lt ε⁻¹ (by linarith)
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ε⁻¹ < 2 ^ i
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
have hi : (2 ^ i : ℝ)⁻¹ < ε := inv_lt_of_inv_lt ε0 hi
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ε⁻¹ < 2 ^ i
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
exists i
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
intro j hj
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
j : ℕ
hj : j ≥ i
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
calc |nestedIntervalSeq U j - nestedIntervalSeq U i|
_ ≤ (2 ^ i : ℝ)⁻¹ := nestedIntervalSeq_isCauSeq_aux' U hj
_ < ε := hi
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
j : ℕ
hj : j ≥ i
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[214, 1]
|
[222, 28]
|
linarith
|
ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ 1 < 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_tendsto
|
[232, 1]
|
[234, 47]
|
apply (nestedIntervalCauSeq U).tendsto_limit
|
ι : Type
U : ι → Set ℝ
⊢ Tendsto (nestedIntervalSeq U) atTop (𝓝 (CauSeq.lim (nestedIntervalCauSeq U)))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
by_contra H
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
⊢ HasFinSubCover U (Icc 0 1)
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
⊢ False
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
set c := (nestedIntervalCauSeq U).lim
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
⊢ False
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
⊢ False
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
rcases cover (nestedIntervalLim_mem U 0) with ⟨_, ⟨i, rfl⟩, hU' : c ∈ U i⟩
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
⊢ False
|
case intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
⊢ False
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
rcases Metric.isOpen_iff.mp (hU i) c hU' with ⟨ε, ε0, hε⟩
|
case intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
⊢ False
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ False
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
have ⟨n, hn⟩ : ∃ n : ℕ, (ε / 2)⁻¹ < 2 ^ n := by
apply pow_unbounded_of_one_lt (ε / 2)⁻¹ (by linarith)
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ False
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ False
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
suffices HasFinSubCover U I(n) by
apply nestedInterval_not_HasFinSubCover H n this
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ False
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
suffices I(n) ⊆ U i by
exists {i}
simpa using this
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2)
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
suffices ∀ x, x ∈ I(n) → |x - c| < ε by
intro x hx
apply hε (this x hx)
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
intro x hx
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
have hba : β n - α n = (2 ^ n : ℝ)⁻¹ := nestedInterval_len U n
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ |x - c| < ε
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
have hn := inv_lt_of_inv_lt (by linarith) hn
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ |x - c| < ε
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
⊢ |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
have hc : α n ≤ c ∧ c ≤ β n := nestedIntervalLim_mem U n
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
⊢ |x - c| < ε
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
⊢ |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
have hx : α n ≤ x ∧ x ≤ β n := hx
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
⊢ |x - c| < ε
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
calc |x - c|
_ = |(x - α n) - (c - α n)| := by simp
_ ≤ |x - α n| + |c - α n| := by apply abs_sub
_ = (x - α n) + (c - α n) := by
apply congrArg₂ <;> rw [abs_eq_self] <;> linarith
_ < ε / 2 + ε / 2 := by linarith
_ = ε := by ring
|
case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| < ε
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
apply pow_unbounded_of_one_lt (ε / 2)⁻¹ (by linarith)
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ ∃ n, (ε / 2)⁻¹ < 2 ^ n
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
linarith
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ 1 < 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
apply nestedInterval_not_HasFinSubCover H n this
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2)
⊢ False
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
exists {i}
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2)
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ ⋃ i_1 ∈ {i}, U i_1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
simpa using this
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ ⋃ i_1 ∈ {i}, U i_1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
intro x hx
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ x ∈ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
apply hε (this x hx)
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
linarith
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ 0 < ε / 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
simp
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| = |x - (nestedInterval U n).1 - (c - (nestedInterval U n).1)|
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
apply abs_sub
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - (nestedInterval U n).1 - (c - (nestedInterval U n).1)| ≤
|x - (nestedInterval U n).1| + |c - (nestedInterval U n).1|
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
apply congrArg₂ <;> rw [abs_eq_self] <;> linarith
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - (nestedInterval U n).1| + |c - (nestedInterval U n).1| =
x - (nestedInterval U n).1 + (c - (nestedInterval U n).1)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
linarith
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ x - (nestedInterval U n).1 + (c - (nestedInterval U n).1) < ε / 2 + ε / 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[254, 1]
|
[291, 21]
|
ring
|
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ ε / 2 + ε / 2 = ε
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Category/Lecture2.lean
|
Tutorial.Category.Initial.uniq'
|
[28, 1]
|
[30, 25]
|
sorry
|
C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ f = h.fromInitial b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Category/Lecture2.lean
|
Tutorial.Category.Initial.uniq'
|
[28, 1]
|
[30, 25]
|
sorry
|
C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ h.fromInitial b = g
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Category/Lecture2.lean
|
Tutorial.Coequalizer.hom_id
|
[309, 1]
|
[309, 71]
|
cases i <;> rfl
|
J : Type u₁
inst✝¹ : Category J
C : Type u₂
inst✝ : Category C
F : Functor J C
i : Shape
⊢ ShapeHom.id i = 𝟙 i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.comp
|
[38, 1]
|
[48, 11]
|
rw [Injective]
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ Injective (g ∘ f)
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.comp
|
[38, 1]
|
[48, 11]
|
intro x₁ x₂ hgf
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.comp
|
[38, 1]
|
[48, 11]
|
have hf := hginj hgf
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.comp
|
[38, 1]
|
[48, 11]
|
apply hfinj
|
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ x₁ = x₂
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ f x₁ = f x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.comp
|
[38, 1]
|
[48, 11]
|
apply hf
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ f x₁ = f x₂
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
rw [Injective]
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ Injective f
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
intro x₁ x₂ hf
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
have h : g (f x₁) = g (f x₂) := by
rw [hf]
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ x₁ = x₂
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ x₁ = x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
apply hgfinj
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ x₁ = x₂
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ (g ∘ f) x₁ = (g ∘ f) x₂
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
simp
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ (g ∘ f) x₁ = (g ∘ f) x₂
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ g (f x₁) = g (f x₂)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
apply h
|
case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ g (f x₁) = g (f x₂)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Injective.of_comp
|
[78, 1]
|
[89, 10]
|
rw [hf]
|
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ g (f x₁) = g (f x₂)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
rw [Surjective]
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ Surjective (g ∘ f)
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
intro z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
⊢ ∃ x, (g ∘ f) x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
have ⟨y, hy⟩ := hgsurj z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
⊢ ∃ x, (g ∘ f) x = z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
have ⟨x, hx⟩ := hfsurj y
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
exists x
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ (g ∘ f) x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
simp
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ (g ∘ f) x = z
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ g (f x) = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.comp
|
[137, 1]
|
[148, 14]
|
rw [hx, hy]
|
X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ g (f x) = z
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.of_comp
|
[160, 1]
|
[164, 13]
|
intro z
|
X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
⊢ Surjective g
|
X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
⊢ ∃ x, g x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.of_comp
|
[160, 1]
|
[164, 13]
|
have ⟨x, hx⟩ := h z
|
X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
⊢ ∃ x, g x = z
|
X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
x : X
hx : (g ∘ f) x = z
⊢ ∃ x, g x = z
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Basic/Lecture4.lean
|
Tutorial.Surjective.of_comp
|
[160, 1]
|
[164, 13]
|
exists f x
|
X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
x : X
hx : (g ∘ f) x = z
⊢ ∃ x, g x = z
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[51, 28]
|
have h : f 1 * f 1 = f 1 * 1 := by
rw [← map_mul, mul_one, mul_one]
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 = 1
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[51, 28]
|
exact mul_left_cancel _ h
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[51, 28]
|
rw [← map_mul, mul_one, mul_one]
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 * f 1 = f 1 * 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_inv
|
[57, 1]
|
[60, 40]
|
apply eq_inv_of_mul_eq_one_left
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ = (f a)⁻¹
|
case a
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ * f a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_inv
|
[57, 1]
|
[60, 40]
|
rw [← map_mul, inv_mul_self, map_one]
|
case a
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ * f a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f ↔ ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
intro hf a ha
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
apply hf
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ a = 1
|
case mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ f a = f 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
rw [ha, map_one]
|
case mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ f a = f 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
intro h a b hab
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a = f b
⊢ a = b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
rw [← mul_inv_eq_one] at *
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a = f b
⊢ a = b
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ a * b⁻¹ = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
apply h
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ a * b⁻¹ = 1
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[219, 1]
|
[230, 31]
|
rw [map_mul, map_inv, hab]
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
rw [injective_iff_map_eq_one]
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ Function.Injective ⇑f
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
intro h a (hf : a ∈ f.ker)
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ker f = ⊥
a : G₁
hf : a ∈ ker f
⊢ a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
simpa [h] using hf
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ker f = ⊥
a : G₁
hf : a ∈ ker f
⊢ a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
intro h
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
⊢ ker f = ⊥
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
ext a
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
⊢ ker f = ⊥
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a ∈ ker f ↔ a ∈ ⊥
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
simp only [mem_ker, mem_bot]
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a ∈ ker f ↔ a ∈ ⊥
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 ↔ a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
constructor
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 ↔ a = 1
|
case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 → a = 1
case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a = 1 → f a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
intro ha
|
case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 → a = 1
|
case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
apply h
|
case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ a = 1
|
case mpr.a.mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ f a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
apply ha
|
case mpr.a.mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ f a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
intro ha
|
case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a = 1 → f a = 1
|
case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : a = 1
⊢ f a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[236, 1]
|
[253, 23]
|
rw [ha, map_one]
|
case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : a = 1
⊢ f a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ ↔ Function.Surjective ⇑f
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
intro hrange y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
have hy : y ∈ (⊤ : Subgroup G₂) := by
simp
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
rw [← hrange] at hy
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
have ⟨x, hx⟩ := hy
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
⊢ ∃ a, f a = y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
x : G₁
hx : f x = y
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
exists x
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
x : G₁
hx : f x = y
⊢ ∃ a, f a = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
simp
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ y ∈ ⊤
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
intro hsurj
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
ext y
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ y ∈ range f ↔ y ∈ ⊤
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
simp only [mem_range, mem_top, iff_true]
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ y ∈ range f ↔ y ∈ ⊤
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[257, 1]
|
[273, 18]
|
apply hsurj y
|
case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ ∃ a, f a = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[346, 1]
|
[360, 15]
|
rw [injective_iff_map_eq_one]
|
G : Type
inst✝ : Group G
⊢ Function.Injective ⇑(homToPerm G)
|
G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[346, 1]
|
[360, 15]
|
intro a h
|
G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[346, 1]
|
[360, 15]
|
calc
a = a * 1 := by simp
_ = (homToPerm G a) 1 := by
rfl
_ = 1 := by
simp [h]
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1
|
no goals
|
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