internlm/Lean-Github · Datasets at Hugging Face

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	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.comp	[35, 1]	[43, 8]	rw [Injective]	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ Injective (g ∘ f)	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.comp	[35, 1]	[43, 8]	intro x₁ x₂ hgf	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ ⊢ x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.comp	[35, 1]	[43, 8]	have hf := hginj hgf	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ ⊢ x₁ = x₂	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ hf : f x₁ = f x₂ ⊢ x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.comp	[35, 1]	[43, 8]	sorry	X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ hf : f x₁ = f x₂ ⊢ x₁ = x₂	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.of_comp	[69, 1]	[77, 8]	rw [Injective]	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ Injective f	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.of_comp	[69, 1]	[77, 8]	intro x₁ x₂ hf	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.of_comp	[69, 1]	[77, 8]	have h : g (f x₁) = g (f x₂) := by sorry	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ x₁ = x₂	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ h : g (f x₁) = g (f x₂) ⊢ x₁ = x₂
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.of_comp	[69, 1]	[77, 8]	sorry	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ h : g (f x₁) = g (f x₂) ⊢ x₁ = x₂	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Injective.of_comp	[69, 1]	[77, 8]	sorry	X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ g (f x₁) = g (f x₂)	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Surjective.comp	[119, 1]	[126, 8]	rw [Surjective]	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ Surjective (g ∘ f)	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Surjective.comp	[119, 1]	[126, 8]	intro z	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z ⊢ ∃ x, (g ∘ f) x = z
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Surjective.comp	[119, 1]	[126, 8]	have ⟨y, hy⟩ := hgsurj z	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z ⊢ ∃ x, (g ∘ f) x = z	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z y : Y hy : g y = z ⊢ ∃ x, (g ∘ f) x = z
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Surjective.comp	[119, 1]	[126, 8]	sorry	X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z y : Y hy : g y = z ⊢ ∃ x, (g ∘ f) x = z	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Basic/Lecture4.lean	Tutorial.Surjective.of_comp	[137, 1]	[138, 8]	sorry	X Y Z : Type f : X → Y g : Y → Z h : Surjective (g ∘ f) ⊢ Surjective g	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.Subgroup.mem_comm	[31, 1]	[42, 8]	intro hab	G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G ⊢ a * b ∈ N → b * a ∈ N	G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G hab : a * b ∈ N ⊢ b * a ∈ N
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.Subgroup.mem_comm	[31, 1]	[42, 8]	sorry	G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G hab : a * b ∈ N ⊢ b * a ∈ N	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.mem_of_eq_one	[82, 1]	[83, 8]	sorry	G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Subgroup.Normal N a : G ⊢ LeftQuotient.mk a = 1 ↔ a ∈ N	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.Subgroup.coe_one	[170, 1]	[170, 48]	simp	G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H K : Subgroup G ⊢ 1 ∈ K	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.GroupHom.rangeKerLift_injective	[201, 1]	[202, 8]	sorry	G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H ⊢ Function.Injective ⇑(rangeKerLift f)	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.GroupHom.rangeKerLift_surjective	[205, 1]	[208, 8]	intro ⟨y, hy⟩	G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H ⊢ Function.Surjective ⇑(rangeKerLift f)	G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H y : H hy : y ∈ range f ⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Algebra/Lecture5.lean	Tutorial.GroupHom.rangeKerLift_surjective	[205, 1]	[208, 8]	sorry	G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H y : H hy : y ∈ range f ⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	have ⟨n, hn⟩ : ∃ n : ℕ, ε⁻¹ < 10 ^ n := pow_unbounded_of_one_lt ε⁻¹ rfl	case e_cauchy.a ε : ℚ ε0 : ε > 0 ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	have : (10 ^ n : ℚ)⁻¹ < ε := inv_lt_of_inv_lt ε0 hn	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	exists n	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∀ j ≥ n, (10 ^ j)⁻¹ < ε
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	intro h hj	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∀ j ≥ n, (10 ^ j)⁻¹ < ε	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ (10 ^ h)⁻¹ < ε
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	calc (10 ^ h : ℚ )⁻¹ ≤ (10 ^ n : ℚ)⁻¹ := inv_pow_le_inv_pow_of_le (by linarith) hj _ < ε := this	case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ (10 ^ h)⁻¹ < ε	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.«0.9999999 = 1»	[39, 1]	[54, 18]	linarith	ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ 1 ≤ 10	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.hasFinSubCover_concat	[109, 1]	[123, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.not_HasFinSubCover_concat	[125, 1]	[128, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.not_HasFinSubCover_concat	[125, 1]	[128, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSucc_eq_or_eq	[140, 1]	[143, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_le	[145, 1]	[151, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested_succ	[153, 1]	[159, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_is_nested	[161, 1]	[166, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_mem	[168, 1]	[171, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_mem	[168, 1]	[171, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_mem	[168, 1]	[171, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_not_HasFinSubCover	[180, 1]	[190, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedInterval_len	[195, 1]	[202, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_isCauSeq_aux	[204, 1]	[207, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_isCauSeq_aux	[204, 1]	[207, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_isCauSeq_aux'	[209, 1]	[212, 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	Solution/Advanced/Analysis/Lecture3.lean	Tutorial.nestedIntervalSeq_isCauSeq_aux'	[209, 1]	[212, 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/Basic/Lecture4.lean

Tutorial.Injective.comp

[35, 1]

[43, 8]

rw [Injective]

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ Injective (g ∘ f)

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.comp

[35, 1]

[43, 8]

intro x₁ x₂ hgf

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g ⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ ⊢ x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.comp

[35, 1]

[43, 8]

have hf := hginj hgf

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ ⊢ x₁ = x₂

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ hf : f x₁ = f x₂ ⊢ x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.comp

[35, 1]

[43, 8]

sorry

X Y Z : Type f : X → Y g : Y → Z hfinj : Injective f hginj : Injective g x₁ x₂ : X hgf : (g ∘ f) x₁ = (g ∘ f) x₂ hf : f x₁ = f x₂ ⊢ x₁ = x₂

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.of_comp

[69, 1]

[77, 8]

rw [Injective]

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ Injective f

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.of_comp

[69, 1]

[77, 8]

intro x₁ x₂ hf

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) ⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.of_comp

[69, 1]

[77, 8]

have h : g (f x₁) = g (f x₂) := by sorry

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ x₁ = x₂

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ h : g (f x₁) = g (f x₂) ⊢ x₁ = x₂

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.of_comp

[69, 1]

[77, 8]

sorry

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ h : g (f x₁) = g (f x₂) ⊢ x₁ = x₂

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Injective.of_comp

[69, 1]

[77, 8]

sorry

X Y Z : Type f : X → Y g : Y → Z hgfinj : Injective (g ∘ f) x₁ x₂ : X hf : f x₁ = f x₂ ⊢ g (f x₁) = g (f x₂)

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Surjective.comp

[119, 1]

[126, 8]

rw [Surjective]

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ Surjective (g ∘ f)

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Surjective.comp

[119, 1]

[126, 8]

intro z

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g ⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z ⊢ ∃ x, (g ∘ f) x = z

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Surjective.comp

[119, 1]

[126, 8]

have ⟨y, hy⟩ := hgsurj z

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z ⊢ ∃ x, (g ∘ f) x = z

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z y : Y hy : g y = z ⊢ ∃ x, (g ∘ f) x = z

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Surjective.comp

[119, 1]

[126, 8]

sorry

X Y Z : Type f : X → Y g : Y → Z hfsurj : Surjective f hgsurj : Surjective g z : Z y : Y hy : g y = z ⊢ ∃ x, (g ∘ f) x = z

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Basic/Lecture4.lean

Tutorial.Surjective.of_comp

[137, 1]

[138, 8]

sorry

X Y Z : Type f : X → Y g : Y → Z h : Surjective (g ∘ f) ⊢ Surjective g

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.Subgroup.mem_comm

[31, 1]

[42, 8]

intro hab

G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G ⊢ a * b ∈ N → b * a ∈ N

G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G hab : a * b ∈ N ⊢ b * a ∈ N

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.Subgroup.mem_comm

[31, 1]

[42, 8]

sorry

G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Normal N a b : G hab : a * b ∈ N ⊢ b * a ∈ N

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.mem_of_eq_one

[82, 1]

[83, 8]

sorry

G : Type inst✝¹ : Group G N : Subgroup G inst✝ : Subgroup.Normal N a : G ⊢ LeftQuotient.mk a = 1 ↔ a ∈ N

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.Subgroup.coe_one

[170, 1]

[170, 48]

simp

G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H K : Subgroup G ⊢ 1 ∈ K

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.GroupHom.rangeKerLift_injective

[201, 1]

[202, 8]

sorry

G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H ⊢ Function.Injective ⇑(rangeKerLift f)

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.GroupHom.rangeKerLift_surjective

[205, 1]

[208, 8]

intro ⟨y, hy⟩

G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H ⊢ Function.Surjective ⇑(rangeKerLift f)

G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H y : H hy : y ∈ range f ⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Algebra/Lecture5.lean

Tutorial.GroupHom.rangeKerLift_surjective

[205, 1]

[208, 8]

sorry

G H : Type inst✝¹ : Group G inst✝ : Group H f : G →* H y : H hy : y ∈ range f ⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

have ⟨n, hn⟩ : ∃ n : ℕ, ε⁻¹ < 10 ^ n := pow_unbounded_of_one_lt ε⁻¹ rfl

case e_cauchy.a ε : ℚ ε0 : ε > 0 ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

have : (10 ^ n : ℚ)⁻¹ < ε := inv_lt_of_inv_lt ε0 hn

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

exists n

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∃ i, ∀ j ≥ i, (10 ^ j)⁻¹ < ε

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∀ j ≥ n, (10 ^ j)⁻¹ < ε

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

intro h hj

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε ⊢ ∀ j ≥ n, (10 ^ j)⁻¹ < ε

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ (10 ^ h)⁻¹ < ε

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

calc (10 ^ h : ℚ )⁻¹ ≤ (10 ^ n : ℚ)⁻¹ := inv_pow_le_inv_pow_of_le (by linarith) hj _ < ε := this

case e_cauchy.a ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ (10 ^ h)⁻¹ < ε

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.«0.9999999 = 1»

[39, 1]

[54, 18]

linarith

ε : ℚ ε0 : ε > 0 n : ℕ hn : ε⁻¹ < 10 ^ n this : (10 ^ n)⁻¹ < ε h : ℕ hj : h ≥ n ⊢ 1 ≤ 10

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.hasFinSubCover_concat

[109, 1]

[123, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.not_HasFinSubCover_concat

[125, 1]

[128, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.not_HasFinSubCover_concat

[125, 1]

[128, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSucc_eq_or_eq

[140, 1]

[143, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_le

[145, 1]

[151, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested_succ

[153, 1]

[159, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_is_nested

[161, 1]

[166, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_mem

[168, 1]

[171, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_mem

[168, 1]

[171, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_mem

[168, 1]

[171, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_not_HasFinSubCover

[180, 1]

[190, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedInterval_len

[195, 1]

[202, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_isCauSeq_aux

[204, 1]

[207, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_isCauSeq_aux

[204, 1]

[207, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_isCauSeq_aux'

[209, 1]

[212, 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

Solution/Advanced/Analysis/Lecture3.lean

Tutorial.nestedIntervalSeq_isCauSeq_aux'

[209, 1]

[212, 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