Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
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https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[291, 1]
[311, 87]
rw [hf i (Finset.mem_insert_self i s), mul_one]
ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ I + K = I + K * (I + J i)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[291, 1]
[311, 87]
ring
ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ I + K * (I + J i) = (1 + K) * I + K * J i
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[291, 1]
[311, 87]
gcongr
ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ (1 + K) * I + K * J i ≤ I + K ⊓ J i
case h₁ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ (1 + K) * I ≤ I case h₂ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ K * J i ≤ K ⊓ J i
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[291, 1]
[311, 87]
apply mul_le_left
case h₁ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ (1 + K) * I ≤ I case h₂ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ K * J i ≤ K ⊓ J i
case h₂ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ K * J i ≤ K ⊓ J i
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
isCoprime_Inf
[291, 1]
[311, 87]
apply mul_le_inf
case h₂ ι : Type u_1 R : Type u_2 inst✝ : CommRing R I : Ideal R J : ι → Ideal R i : ι s : Finset ι a✝ : i ∉ s hf : ∀ j ∈ insert i s, I + J j = 1 K : Ideal R := ⨅ j ∈ s, J j hs : (∀ j ∈ s, I + J j = 1) → I + K = 1 ⊢ K * J i ≤ K ⊓ J i
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
intro g
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) ⊢ Surjective ⇑(chineseMap I)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i ⊢ ∃ a, (chineseMap I) a = g
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i ⊢ ∃ a, (chineseMap I) a = g
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i ⊢ ∃ a, (chineseMap I) a = g
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
choose e he using key
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i key : ∀ (i : ι), ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0 ⊢ ∃ a, (chineseMap I) a = g
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 ⊢ ∃ a, (chineseMap I) a = g
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
use mk _ (∑ i, f i * e i)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 ⊢ ∃ a, (chineseMap I) a = g
case h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 ⊢ (chineseMap I) ((mk (⨅ i, I i)) (∑ i : ι, f i * e i)) = g
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
ext i
case h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 ⊢ (chineseMap I) ((mk (⨅ i, I i)) (∑ i : ι, f i * e i)) = g
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ (chineseMap I) ((mk (⨅ i, I i)) (∑ i : ι, f i * e i)) i = g i
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
rw [chineseMap_mk', map_sum, Fintype.sum_eq_single i]
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ (chineseMap I) ((mk (⨅ i, I i)) (∑ i : ι, f i * e i)) i = g i
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ (mk (I i)) (f i * e i) = g i case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ ∀ (x : ι), x ≠ i → (mk (I i)) (f x * e x) = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
intro i
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i ⊢ ∀ (i : ι), ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by intros j hj exact hI _ _ (by simpa [ne_comm, isCoprime_iff_add] using hj)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
rcases isCoprime_iff_exists.mp (isCoprime_Inf hI') with ⟨u, hu, e, he, hue⟩
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
case intro.intro.intro.intro ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R he : e ∈ ⨅ j ∈ {i}ᶜ, I j hue : u + e = 1 ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
replace he : ∀ j, j ≠ i → e ∈ I j := by simpa using he
case intro.intro.intro.intro ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R he : e ∈ ⨅ j ∈ {i}ᶜ, I j hue : u + e = 1 ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
case intro.intro.intro.intro ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
refine ⟨e, ?_, ?_⟩
case intro.intro.intro.intro ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ ∃ e, (mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
case intro.intro.intro.intro.refine_1 ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ (mk (I i)) e = 1 case intro.intro.intro.intro.refine_2 ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
intros j hj
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι ⊢ ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i j : ι hj : j ∈ {i}ᶜ ⊢ IsCoprime (I i) (I j)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
exact hI _ _ (by simpa [ne_comm, isCoprime_iff_add] using hj)
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i j : ι hj : j ∈ {i}ᶜ ⊢ IsCoprime (I i) (I j)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
simpa [ne_comm, isCoprime_iff_add] using hj
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i j : ι hj : j ∈ {i}ᶜ ⊢ i ≠ j
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
simpa using he
ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R he : e ∈ ⨅ j ∈ {i}ᶜ, I j hue : u + e = 1 ⊢ ∀ (j : ι), j ≠ i → e ∈ I j
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
simp [eq_sub_of_add_eq' hue, map_sub, eq_zero_iff_mem.mpr hu]
case intro.intro.intro.intro.refine_1 ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ (mk (I i)) e = 1
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj)
case intro.intro.intro.intro.refine_2 ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i i : ι hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j) u : R hu : u ∈ I i e : R hue : u + e = 1 he : ∀ (j : ι), j ≠ i → e ∈ I j ⊢ ∀ (j : ι), j ≠ i → (mk (I j)) e = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
simp [(he i).1, hf]
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ (mk (I i)) (f i * e i) = g i
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
intros j hj
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i : ι ⊢ ∀ (x : ι), x ≠ i → (mk (I i)) (f x * e x) = 0
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i j : ι hj : j ≠ i ⊢ (mk (I i)) (f j * e j) = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C08_Groups_and_Rings/S02_Rings.lean
chineseMap_surj
[319, 1]
[350, 30]
simp [(he j).2 i hj.symm]
case h.h ι : Type u_1 R : Type u_2 inst✝¹ : CommRing R inst✝ : Fintype ι I : ι → Ideal R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) g : Π (i : ι), R ⧸ I i f : ι → R hf : ∀ (i : ι), (mk (I i)) (f i) = g i e : ι → R he : ∀ (i : ι), (mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (mk (I j)) (e i) = 0 i j : ι hj : j ≠ i ⊢ (mk (I i)) (f j * e j) = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.add_comm
[137, 11]
[140, 25]
rw [add, add]
a b : Point ⊢ a.add b = b.add a
a b : Point ⊢ { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } = { x := b.x + a.x, y := b.y + a.y, z := b.z + a.z }
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.add_comm
[137, 11]
[140, 25]
ext <;> dsimp
a b : Point ⊢ { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } = { x := b.x + a.x, y := b.y + a.y, z := b.z + a.z }
case x a b : Point ⊢ a.x + b.x = b.x + a.x case y a b : Point ⊢ a.y + b.y = b.y + a.y case z a b : Point ⊢ a.z + b.z = b.z + a.z
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.add_comm
[137, 11]
[140, 25]
repeat' apply add_comm
case x a b : Point ⊢ a.x + b.x = b.x + a.x case y a b : Point ⊢ a.y + b.y = b.y + a.y case z a b : Point ⊢ a.z + b.z = b.z + a.z
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.add_comm
[137, 11]
[140, 25]
apply add_comm
case z a b : Point ⊢ a.z + b.z = b.z + a.z
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.addAlt_x
[173, 1]
[174, 6]
rfl
a b : Point ⊢ (a.addAlt b).x = a.x + b.x
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.addAlt_comm
[176, 1]
[180, 25]
rw [addAlt, addAlt]
a b : Point ⊢ a.addAlt b = b.addAlt a
a b : Point ⊢ (match a, b with | { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }) = match b, a with | { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.addAlt_comm
[176, 1]
[180, 25]
ext <;> dsimp
a b : Point ⊢ (match a, b with | { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }) = match b, a with | { x := x₁, y := y₁, z := z₁ }, { x := x₂, y := y₂, z := z₂ } => { x := x₁ + x₂, y := y₁ + y₂, z := z₁ + z₂ }
case x a b : Point ⊢ a.x + b.x = b.x + a.x case y a b : Point ⊢ a.y + b.y = b.y + a.y case z a b : Point ⊢ a.z + b.z = b.z + a.z
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.addAlt_comm
[176, 1]
[180, 25]
repeat' apply add_comm
case x a b : Point ⊢ a.x + b.x = b.x + a.x case y a b : Point ⊢ a.y + b.y = b.y + a.y case z a b : Point ⊢ a.z + b.z = b.z + a.z
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.addAlt_comm
[176, 1]
[180, 25]
apply add_comm
case z a b : Point ⊢ a.z + b.z = b.z + a.z
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.add_assoc
[193, 11]
[197, 24]
simp [add, add_assoc]
a b c : Point ⊢ (a.add b).add c = a.add (b.add c)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C06_Structures/S01_Structures.lean
C06S01.Point.smul_distrib
[207, 1]
[212, 28]
simp [add, smul, mul_add]
r : ℝ a b : Point ⊢ (smul r a).add (smul r b) = smul r (a.add b)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_const
[108, 1]
[113, 13]
intro ε εpos
a : ℝ ⊢ ConvergesTo (fun x => a) a
a ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_const
[108, 1]
[113, 13]
use 0
a ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
case h a ε : ℝ εpos : ε > 0 ⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_const
[108, 1]
[113, 13]
intro n nge
case h a ε : ℝ εpos : ε > 0 ⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
case h a ε : ℝ εpos : ε > 0 n : ℕ nge : n ≥ 0 ⊢ |(fun x => a) n - a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_const
[108, 1]
[113, 13]
rw [sub_self, abs_zero]
case h a ε : ℝ εpos : ε > 0 n : ℕ nge : n ≥ 0 ⊢ |(fun x => a) n - a| < ε
case h a ε : ℝ εpos : ε > 0 n : ℕ nge : n ≥ 0 ⊢ 0 < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_const
[108, 1]
[113, 13]
apply εpos
case h a ε : ℝ εpos : ε > 0 n : ℕ nge : n ≥ 0 ⊢ 0 < ε
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
intro ε εpos
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n + t n) (a + b)
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
dsimp
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
have ε2pos : 0 < ε / 2 := by linarith
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩
case intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case intro.intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
use max Ns Nt
case intro.intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
sorry
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_add
[143, 1]
[152, 8]
linarith
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ 0 < ε / 2
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
intro ε εpos
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ⊢ ConvergesTo (fun n => s n + t n) (a + b)
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
dsimp
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
have ε2pos : 0 < ε / 2 := by linarith
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩
case intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case intro.intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
use max Ns Nt
case intro.intro s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
intro n hn
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 ⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ⊢ |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
have ngeNs : n ≥ Ns := le_of_max_le_left hn
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ⊢ |s n + t n - (a + b)| < ε
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ⊢ |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
have ngeNt : n ≥ Nt := le_of_max_le_right hn
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ⊢ |s n + t n - (a + b)| < ε
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ |s n + t n - (a + b)| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
calc |s n + t n - (a + b)| = |s n - a + (t n - b)| := by congr ring _ ≤ |s n - a| + |t n - b| := (abs_add _ _) _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) _ = ε := by norm_num
case h s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ |s n + t n - (a + b)| < ε
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
linarith
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ⊢ 0 < ε / 2
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
congr
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ |s n + t n - (a + b)| = |s n - a + (t n - b)|
case e_a s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ s n + t n - (a + b) = s n - a + (t n - b)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
ring
case e_a s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ s n + t n - (a + b) = s n - a + (t n - b)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_addαα
[156, 1]
[174, 25]
norm_num
s t : ℕ → ℝ a b : ℝ cs : ConvergesTo s a ct : ConvergesTo t b ε : ℝ εpos : ε > 0 ε2pos : 0 < ε / 2 Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / 2 Nt : ℕ ht : ∀ n ≥ Nt, |t n - b| < ε / 2 n : ℕ hn : n ≥ max Ns Nt ngeNs : n ≥ Ns ngeNt : n ≥ Nt ⊢ ε / 2 + ε / 2 = ε
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
by_cases h : c = 0
s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a ⊢ ConvergesTo (fun n => c * s n) (c * a)
case pos s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a) case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
have acpos : 0 < |c| := abs_pos.mpr h
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ⊢ ConvergesTo (fun n => c * s n) (c * a)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
sorry
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ⊢ ConvergesTo (fun n => c * s n) (c * a)
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
convert convergesTo_const 0
case pos s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0 case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ c * a = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
rw [h]
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ c * a = 0
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
ring
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
rw [h]
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_const
[199, 1]
[208, 8]
ring
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
by_cases h : c = 0
s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a ⊢ ConvergesTo (fun n => c * s n) (c * a)
case pos s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a) case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
have acpos : 0 < |c| := abs_pos.mpr h
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ⊢ ConvergesTo (fun n => c * s n) (c * a)
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
intro ε εpos
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
dsimp
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩
case neg s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
case neg.intro s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
use Ns
case neg.intro s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| ⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
case h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| ⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
intro n ngt
case h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| ⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε
case h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| n : ℕ ngt : n ≥ Ns ⊢ |c * s n - c * a| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
calc |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm
case h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| n : ℕ ngt : n ≥ Ns ⊢ |c * s n - c * a| < ε
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
convert convergesTo_const 0
case pos s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ ConvergesTo (fun n => c * s n) (c * a)
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0 case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ c * a = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
rw [h]
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ c * a = 0
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
ring
case h.e'_2 s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 ⊢ 0 * a = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
rw [h]
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ c * s x✝ = 0
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
ring
case h.e'_1.h s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : c = 0 x✝ : ℕ ⊢ 0 * s x✝ = 0
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
apply div_pos εpos acpos
s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 ⊢ 0 < ε / |c|
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.convergesTo_mul_constαα
[212, 1]
[230, 53]
rw [← abs_mul, mul_sub]
s : ℕ → ℝ a c : ℝ cs : ConvergesTo s a h : ¬c = 0 acpos : 0 < |c| ε : ℝ εpos : ε > 0 εcpos : 0 < ε / |c| Ns : ℕ hs : ∀ n ≥ Ns, |s n - a| < ε / |c| n : ℕ ngt : n ≥ Ns ⊢ |c * s n - c * a| = |c| * |s n - a|
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesTo
[239, 1]
[243, 8]
rcases cs 1 zero_lt_one with ⟨N, h⟩
s : ℕ → ℝ a : ℝ cs : ConvergesTo s a ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesTo
[239, 1]
[243, 8]
use N, |a| + 1
case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesTo
[239, 1]
[243, 8]
sorry
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
rcases cs 1 zero_lt_one with ⟨N, h⟩
s : ℕ → ℝ a : ℝ cs : ConvergesTo s a ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
use N, |a| + 1
case intro s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
intro n ngt
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 ⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ |s n| < |a| + 1
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
calc |s n| = |s n - a + a| := by congr abel _ ≤ |s n - a| + |a| := (abs_add _ _) _ < |a| + 1 := by linarith [h n ngt]
case h s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ |s n| < |a| + 1
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
congr
s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ |s n| = |s n - a + a|
case e_a s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ s n = s n - a + a
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
abel
case e_a s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ s n = s n - a + a
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.exists_abs_le_of_convergesToαα
[247, 1]
[257, 41]
linarith [h n ngt]
s : ℕ → ℝ a : ℝ cs : ConvergesTo s a N : ℕ h : ∀ n ≥ N, |s n - a| < 1 n : ℕ ngt : N ≤ n ⊢ |s n - a| + |a| < |a| + 1
no goals
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.aux
[275, 1]
[283, 8]
intro ε εpos
s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ⊢ ConvergesTo (fun n => s n * t n) 0
s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n * t n) n - 0| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.aux
[275, 1]
[283, 8]
dsimp
s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |(fun n => s n * t n) n - 0| < ε
s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε
https://github.com/avigad/mathematics_in_lean_source.git
3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3
MIL/C03_Logic/S06_Sequences_and_Convergence.lean
C03S06.aux
[275, 1]
[283, 8]
rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩
s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 ⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε
case intro.intro s t : ℕ → ℝ a : ℝ cs : ConvergesTo s a ct : ConvergesTo t 0 ε : ℝ εpos : ε > 0 N₀ : ℕ B : ℝ h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B ⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε