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/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
intro x
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' a : ℝ
n : ℕ
ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a
⊢ ∀ (x : ℝ),
(fun x => x ^ (n + 1) * x) x - (fun x => x ^ (n + 1) * x) a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) =
(fun x => x ^ (n + 1 + 1)) x - (fun x => x ^ (n + 1 + 1)) a - (x - a) * ((↑(n + 1) + 1) * a ^ (n + 1))
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' a : ℝ
n : ℕ
ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a
x : ℝ
⊢ (fun x => x ^ (n + 1) * x) x - (fun x => x ^ (n + 1) * x) a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) =
(fun x => x ^ (n + 1 + 1)) x - (fun x => x ^ (n + 1 + 1)) a - (x - a) * ((↑(n + 1) + 1) * a ^ (n + 1))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
simp
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' a : ℝ
n : ℕ
ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a
x : ℝ
⊢ (fun x => x ^ (n + 1) * x) x - (fun x => x ^ (n + 1) * x) a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) =
(fun x => x ^ (n + 1 + 1)) x - (fun x => x ^ (n + 1 + 1)) a - (x - a) * ((↑(n + 1) + 1) * a ^ (n + 1))
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' a : ℝ
n : ℕ
ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a
x : ℝ
⊢ x ^ (n + 1) * x - a ^ (n + 1) * a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1)) =
x ^ (n + 1 + 1) - a ^ (n + 1 + 1) - (x - a) * ((↑n + 1 + 1) * a ^ (n + 1))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
ring
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' a : ℝ
n : ℕ
ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a
x : ℝ
⊢ x ^ (n + 1) * x - a ^ (n + 1) * a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1)) =
x ^ (n + 1 + 1) - a ^ (n + 1 + 1) - (x - a) * ((↑n + 1 + 1) * a ^ (n + 1))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.abs_of_ten_inv
|
[18, 1]
|
[19, 55]
|
linarith
|
i : ℕ
⊢ 0 < 10
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
calc _ = Real.ofCauchy (Quotient.mk CauSeq.equiv (CauSeq.const abs 1)) := ?_
_ = (1 : ℝ) := Real.ofCauchy_one
|
⊢ { cauchy := ⟦«0.9999999»⟧ } = 1
|
⊢ { cauchy := ⟦«0.9999999»⟧ } = { cauchy := ⟦CauSeq.const abs 1⟧ }
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
rw [«0.9999999»]
|
⊢ { cauchy := ⟦«0.9999999»⟧ } = { cauchy := ⟦CauSeq.const abs 1⟧ }
|
⊢ { cauchy := ⟦{ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 }⟧ } =
{ cauchy := ⟦CauSeq.const abs 1⟧ }
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
congr 1
|
⊢ { cauchy := ⟦{ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 }⟧ } =
{ cauchy := ⟦CauSeq.const abs 1⟧ }
|
case e_cauchy
⊢ ⟦{ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 }⟧ = ⟦CauSeq.const abs 1⟧
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
apply Quotient.sound
|
case e_cauchy
⊢ ⟦{ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 }⟧ = ⟦CauSeq.const abs 1⟧
|
case e_cauchy.a
⊢ { val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 } ≈ CauSeq.const abs 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
intro ε ε0
|
case e_cauchy.a
⊢ { val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 } ≈ CauSeq.const abs 1
|
case e_cauchy.a
ε : ℚ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |↑({ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 } - CauSeq.const abs 1) j| < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
suffices ∃ i, ∀ (j : ℕ), j ≥ i → (10 ^ j : ℚ)⁻¹ < ε by simpa [abs_of_ten_inv]
|
case e_cauchy.a
ε : ℚ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |↑({ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 } - CauSeq.const abs 1) j| < ε
|
case e_cauchy.a
ε : ℚ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
sorry
|
case e_cauchy.a
ε : ℚ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.«0.9999999 = 1»
|
[39, 1]
|
[48, 8]
|
simpa [abs_of_ten_inv]
|
ε : ℚ
ε0 : ε > 0
this : ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε
⊢ ∃ i, ∀ j ≥ i, |↑({ val := fun n => 1 - (10 ^ n)⁻¹, property := «0.9999999».proof_1 } - CauSeq.const abs 1) j| < ε
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
rcases hac with ⟨ι_ac, cover_ac⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
hac : HasFinSubCover U (Icc a c)
hcb : HasFinSubCover U (Icc c b)
⊢ HasFinSubCover U (Icc a b)
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
hcb : HasFinSubCover U (Icc c b)
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
⊢ HasFinSubCover U (Icc a b)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
rcases hcb with ⟨ι_cb, cover_cb⟩
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
hcb : HasFinSubCover U (Icc c b)
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
⊢ HasFinSubCover U (Icc a b)
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
⊢ HasFinSubCover U (Icc a b)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
exists ι_ac ∪ ι_cb
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
⊢ HasFinSubCover U (Icc a b)
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
⊢ Icc a b ⊆ ⋃ i ∈ ι_ac ∪ ι_cb, U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
intro x hx
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
⊢ Icc a b ⊆ ⋃ i ∈ ι_ac ∪ ι_cb, U i
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
⊢ x ∈ ⋃ i ∈ ι_ac ∪ ι_cb, U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
suffices ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i by
simpa using this
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
⊢ x ∈ ⋃ i ∈ ι_ac ∪ ι_cb, U i
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
cases le_total x c
|
case intro.intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
case intro.intro.inl
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
h✝ : x ≤ c
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
case intro.intro.inr
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
h✝ : c ≤ x
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
case inl hxc =>
obtain ⟨i, hi⟩ : ∃ i, i ∈ ι_ac ∧ x ∈ U i := by simpa using cover_ac ⟨hx.left, hxc⟩
exact ⟨i, Or.inl hi.1, hi.2⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : x ≤ c
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
case inr hxc =>
obtain ⟨i, hi⟩ : ∃ i, i ∈ ι_cb ∧ x ∈ U i := by simpa using cover_cb ⟨hxc, hx.right⟩
exact ⟨i, Or.inr hi.1, hi.2⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : c ≤ x
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
simpa using this
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
this : ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
⊢ x ∈ ⋃ i ∈ ι_ac ∪ ι_cb, U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
obtain ⟨i, hi⟩ : ∃ i, i ∈ ι_ac ∧ x ∈ U i := by simpa using cover_ac ⟨hx.left, hxc⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : x ≤ c
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : x ≤ c
i : ι
hi : i ∈ ι_ac ∧ x ∈ U i
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
exact ⟨i, Or.inl hi.1, hi.2⟩
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : x ≤ c
i : ι
hi : i ∈ ι_ac ∧ x ∈ U i
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
simpa using cover_ac ⟨hx.left, hxc⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : x ≤ c
⊢ ∃ i ∈ ι_ac, x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
obtain ⟨i, hi⟩ : ∃ i, i ∈ ι_cb ∧ x ∈ U i := by simpa using cover_cb ⟨hxc, hx.right⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : c ≤ x
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : c ≤ x
i : ι
hi : i ∈ ι_cb ∧ x ∈ U i
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
exact ⟨i, Or.inr hi.1, hi.2⟩
|
case intro
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : c ≤ x
i : ι
hi : i ∈ ι_cb ∧ x ∈ U i
⊢ ∃ i, (i ∈ ι_ac ∨ i ∈ ι_cb) ∧ x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.hasFinSubCover_concat
|
[102, 1]
|
[116, 33]
|
simpa using cover_cb ⟨hxc, hx.right⟩
|
ι : Type
U : ι → Set ℝ
a c b : ℝ
ι_ac : Finset ι
cover_ac : Icc a c ⊆ ⋃ i ∈ ι_ac, U i
ι_cb : Finset ι
cover_cb : Icc c b ⊆ ⋃ i ∈ ι_cb, U i
x : ℝ
hx : x ∈ Icc a b
hxc : c ≤ x
⊢ ∃ i ∈ ι_cb, x ∈ U i
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.not_HasFinSubCover_concat
|
[118, 1]
|
[121, 48]
|
contrapose!
|
ι : Type
U : ι → Set ℝ
a b c : ℝ
⊢ ¬HasFinSubCover U (Icc a b) → HasFinSubCover U (Icc a c) → ¬HasFinSubCover U (Icc c b)
|
ι : Type
U : ι → Set ℝ
a b c : ℝ
⊢ HasFinSubCover U (Icc a c) ∧ HasFinSubCover U (Icc c b) → HasFinSubCover U (Icc a b)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.not_HasFinSubCover_concat
|
[118, 1]
|
[121, 48]
|
apply (fun H ↦ hasFinSubCover_concat H.1 H.2)
|
ι : Type
U : ι → Set ℝ
a b c : ℝ
⊢ HasFinSubCover U (Icc a c) ∧ HasFinSubCover U (Icc c b) → HasFinSubCover U (Icc a b)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSucc_eq_or_eq
|
[133, 1]
|
[136, 21]
|
apply ite_eq_or_eq
|
ι : Type
U : ι → Set ℝ
a b : ℝ
⊢ nestedIntervalSucc U a b = (a, (a + b) / 2) ∨ nestedIntervalSucc U a b = ((a + b) / 2, b)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
have := nestedInterval_le n
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ (nestedInterval U (n + 1)).1 < (nestedInterval U (n + 1)).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ (nestedInterval U (n + 1)).1 < (nestedInterval U (n + 1)).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
cases nestedIntervalSucc_eq_or_eq U (α n) (β n) with
| inl h => rw [nestedInterval, h]; dsimp only; linarith
| inr h => rw [nestedInterval, h]; dsimp only; linarith
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ (nestedInterval U (n + 1)).1 < (nestedInterval U (n + 1)).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
rw [nestedInterval, h]
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ (nestedInterval U (n + 1)).1 < (nestedInterval U (n + 1)).2
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).1 <
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
dsimp only
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).1 <
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ (nestedInterval U n).1 < ((nestedInterval U n).1 + (nestedInterval U n).2) / 2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
linarith
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ (nestedInterval U n).1 < ((nestedInterval U n).1 + (nestedInterval U n).2) / 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
rw [nestedInterval, h]
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U (n + 1)).1 < (nestedInterval U (n + 1)).2
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1 <
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
dsimp only
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1 <
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 < (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_le
|
[138, 1]
|
[144, 60]
|
linarith
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 < (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
have := nestedInterval_le U n
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2 ⊆ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2 ⊆ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
cases nestedIntervalSucc_eq_or_eq U (α n) (β n) with
| inl h =>
apply Icc_subset_Icc (by rw [nestedInterval, h]) (by rw [nestedInterval, h]; dsimp only; linarith)
| inr h =>
apply Icc_subset_Icc (by rw [nestedInterval, h]; dsimp only; linarith) (by rw [nestedInterval, h])
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2 ⊆ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
apply Icc_subset_Icc (by rw [nestedInterval, h]) (by rw [nestedInterval, h]; dsimp only; linarith)
|
case inl
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2 ⊆ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
rw [nestedInterval, h]
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ (nestedInterval U n).1 ≤ (nestedInterval U (n + 1)).1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
rw [nestedInterval, h]
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ (nestedInterval U (n + 1)).2 ≤ (nestedInterval U n).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2 ≤ (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
dsimp only
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2 ≤ (nestedInterval U n).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
linarith
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2)
⊢ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
apply Icc_subset_Icc (by rw [nestedInterval, h]; dsimp only; linarith) (by rw [nestedInterval, h])
|
case inr
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2 ⊆ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
rw [nestedInterval, h]
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U n).1 ≤ (nestedInterval U (n + 1)).1
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U n).1 ≤ (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
dsimp only
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U n).1 ≤ (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
linarith
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested_succ
|
[146, 1]
|
[152, 103]
|
rw [nestedInterval, h]
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
h :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
⊢ (nestedInterval U (n + 1)).2 ≤ (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
rw [(Nat.add_sub_of_le hij).symm]
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
⊢ Icc (nestedInterval U j).1 (nestedInterval U j).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
⊢ Icc (nestedInterval U (i + (j - i))).1 (nestedInterval U (i + (j - i))).2 ⊆
Icc (nestedInterval U i).1 (nestedInterval U i).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
set k := j - i
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
⊢ Icc (nestedInterval U (i + (j - i))).1 (nestedInterval U (i + (j - i))).2 ⊆
Icc (nestedInterval U i).1 (nestedInterval U i).2
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k : ℕ := j - i
⊢ Icc (nestedInterval U (i + k)).1 (nestedInterval U (i + k)).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
induction k with
| zero => apply rfl.subset
| succ k ih => intro x hx; apply ih (nestedIntervalSeq_is_nested_succ U (i + k) hx)
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k : ℕ := j - i
⊢ Icc (nestedInterval U (i + k)).1 (nestedInterval U (i + k)).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
apply rfl.subset
|
case zero
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k : ℕ := j - i
⊢ Icc (nestedInterval U (i + Nat.zero)).1 (nestedInterval U (i + Nat.zero)).2 ⊆
Icc (nestedInterval U i).1 (nestedInterval U i).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
intro x hx
|
case succ
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k✝ : ℕ := j - i
k : ℕ
ih : Icc (nestedInterval U (i + k)).1 (nestedInterval U (i + k)).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
⊢ Icc (nestedInterval U (i + Nat.succ k)).1 (nestedInterval U (i + Nat.succ k)).2 ⊆
Icc (nestedInterval U i).1 (nestedInterval U i).2
|
case succ
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k✝ : ℕ := j - i
k : ℕ
ih : Icc (nestedInterval U (i + k)).1 (nestedInterval U (i + k)).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
x : ℝ
hx : x ∈ Icc (nestedInterval U (i + Nat.succ k)).1 (nestedInterval U (i + Nat.succ k)).2
⊢ x ∈ Icc (nestedInterval U i).1 (nestedInterval U i).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_is_nested
|
[154, 1]
|
[159, 86]
|
apply ih (nestedIntervalSeq_is_nested_succ U (i + k) hx)
|
case succ
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
k✝ : ℕ := j - i
k : ℕ
ih : Icc (nestedInterval U (i + k)).1 (nestedInterval U (i + k)).2 ⊆ Icc (nestedInterval U i).1 (nestedInterval U i).2
x : ℝ
hx : x ∈ Icc (nestedInterval U (i + Nat.succ k)).1 (nestedInterval U (i + Nat.succ k)).2
⊢ x ∈ Icc (nestedInterval U i).1 (nestedInterval U i).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_mem
|
[161, 1]
|
[164, 26]
|
simp only [mem_Icc, nestedIntervalSeq]
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ nestedIntervalSeq U n ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ∧
((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_mem
|
[161, 1]
|
[164, 26]
|
have := nestedInterval_le U n
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ∧
((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ∧
((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_mem
|
[161, 1]
|
[164, 26]
|
split_ands <;> linarith
|
ι : Type
U : ι → Set ℝ
n : ℕ
this : (nestedInterval U n).1 < (nestedInterval U n).2
⊢ (nestedInterval U n).1 ≤ ((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ∧
((nestedInterval U n).1 + (nestedInterval U n).2) / 2 ≤ (nestedInterval U n).2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
by_cases H : HasFinSubCover U (Icc (α n) ((α n + β n) / 2))
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
case pos
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
case neg
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : ¬HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
case pos =>
rw [nestedInterval, nestedIntervalSucc_right H]
apply not_HasFinSubCover_concat ?_ H
apply nestedInterval_not_HasFinSubCover h n
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
case neg =>
rw [nestedInterval, nestedIntervalSucc_left H]
apply H
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : ¬HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
rw [nestedInterval, nestedIntervalSucc_right H]
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
apply not_HasFinSubCover_concat ?_ H
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).1
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2)
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
apply nestedInterval_not_HasFinSubCover h n
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2).2)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
rw [nestedInterval, nestedIntervalSucc_left H]
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : ¬HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U (Icc (nestedInterval U (n + 1)).1 (nestedInterval U (n + 1)).2)
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : ¬HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).1
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_not_HasFinSubCover
|
[173, 1]
|
[183, 14]
|
apply H
|
ι : Type
U : ι → Set ℝ
h : ¬HasFinSubCover U (Icc (nestedInterval U 0).1 (nestedInterval U 0).2)
n : ℕ
H : ¬HasFinSubCover U (Icc (nestedInterval U n).1 (((nestedInterval U n).1 + (nestedInterval U n).2) / 2))
⊢ ¬HasFinSubCover U
(Icc ((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).1
((nestedInterval U n).1, ((nestedInterval U n).1 + (nestedInterval U n).2) / 2).2)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
simp [nestedInterval]
|
ι : Type
U : ι → Set ℝ
⊢ (nestedInterval U 0).2 - (nestedInterval U 0).1 = (2 ^ 0)⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
have ih := nestedInterval_len n
|
ι : Type
U : ι → Set ℝ
n : ℕ
⊢ (nestedInterval U (n + 1)).2 - (nestedInterval U (n + 1)).1 = (2 ^ (n + 1))⁻¹
|
ι : Type
U : ι → Set ℝ
n : ℕ
ih : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ (nestedInterval U (n + 1)).2 - (nestedInterval U (n + 1)).1 = (2 ^ (n + 1))⁻¹
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
rcases nestedIntervalSucc_eq_or_eq U (α n) (β n) with H | H <;>
rw [nestedInterval, H] <;> field_simp at ih ⊢ <;>
calc _ = (β n - α n) * 2 ^ n * 2 := by ring
_ = 2 := by rw [ih]; ring
|
ι : Type
U : ι → Set ℝ
n : ℕ
ih : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ (nestedInterval U (n + 1)).2 - (nestedInterval U (n + 1)).1 = (2 ^ (n + 1))⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
ring
|
ι : Type
U : ι → Set ℝ
n : ℕ
H :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
ih : ((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n = 1
⊢ ((nestedInterval U n).2 * 2 - ((nestedInterval U n).1 + (nestedInterval U n).2)) * 2 ^ (n + 1) =
((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n * 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
rw [ih]
|
ι : Type
U : ι → Set ℝ
n : ℕ
H :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
ih : ((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n = 1
⊢ ((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n * 2 = 2
|
ι : Type
U : ι → Set ℝ
n : ℕ
H :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
ih : ((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n = 1
⊢ 1 * 2 = 2
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedInterval_len
|
[188, 1]
|
[195, 61]
|
ring
|
ι : Type
U : ι → Set ℝ
n : ℕ
H :
nestedIntervalSucc U (nestedInterval U n).1 (nestedInterval U n).2 =
(((nestedInterval U n).1 + (nestedInterval U n).2) / 2, (nestedInterval U n).2)
ih : ((nestedInterval U n).2 - (nestedInterval U n).1) * 2 ^ n = 1
⊢ 1 * 2 = 2
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq_aux
|
[197, 1]
|
[200, 47]
|
dsimp [Icc] at hx hy
|
ι : Type
U : ι → Set ℝ
a b x y : ℝ
hx : x ∈ Icc a b
hy : y ∈ Icc a b
⊢ |y - x| ≤ b - a
|
ι : Type
U : ι → Set ℝ
a b x y : ℝ
hx : a ≤ x ∧ x ≤ b
hy : a ≤ y ∧ y ≤ b
⊢ |y - x| ≤ b - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq_aux
|
[197, 1]
|
[200, 47]
|
apply (abs_sub_le_iff.2 ⟨_, _⟩) <;> linarith
|
ι : Type
U : ι → Set ℝ
a b x y : ℝ
hx : a ≤ x ∧ x ≤ b
hy : a ≤ y ∧ y ≤ b
⊢ |y - x| ≤ b - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq_aux'
|
[202, 1]
|
[205, 40]
|
have := nestedIntervalSeq_isCauSeq_aux (nestedIntervalSeq_mem U i) (nestedIntervalSeq_mem_of_le U hij)
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| ≤ (2 ^ i)⁻¹
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
this : |nestedIntervalSeq U j - nestedIntervalSeq U i| ≤ (nestedInterval U i).2 - (nestedInterval U i).1
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| ≤ (2 ^ i)⁻¹
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq_aux'
|
[202, 1]
|
[205, 40]
|
simpa [nestedInterval_len] using this
|
ι : Type
U : ι → Set ℝ
i j : ℕ
hij : i ≤ j
this : |nestedIntervalSeq U j - nestedIntervalSeq U i| ≤ (nestedInterval U i).2 - (nestedInterval U i).1
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| ≤ (2 ^ i)⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_isCauSeq
|
[207, 1]
|
[215, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.nestedIntervalSeq_tendsto
|
[225, 1]
|
[227, 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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
have ⟨n, hn⟩ : ∃ n : ℕ, (ε / 2)⁻¹ < 2 ^ n := by
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
suffices HasFinSubCover U I(n) by
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
suffices I(n) ⊆ U i by
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
suffices ∀ x, x ∈ I(n) → |x - c| < ε by
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
sorry
|
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
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
sorry
|
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
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
sorry
|
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)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture3.lean
|
Tutorial.HasFinSubCover_of_Icc
|
[247, 1]
|
[261, 8]
|
sorry
|
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
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Category/Lecture1.lean
|
Tutorial.comp_app
|
[115, 1]
|
[116, 6]
|
rfl
|
C : Type u
inst✝ : Category C
a b c d e : C
X Y Z : Type
f : Hom X Y
g : Hom Y Z
x : X
⊢ (f ≫ g) x = g (f x)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Category/Lecture1.lean
|
Tutorial.id_app
|
[119, 1]
|
[120, 6]
|
rfl
|
C : Type u
inst✝ : Category C
a b c d e : C
X : Type
x : X
⊢ 𝟙 X x = x
|
no goals
|
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