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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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