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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture10Before.lean | frobeniusMorphism_injective | [343, 1] | [345, 8] | sorry | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
this : ∀ (x : K), x ^ p = 0 → x = 0
⊢ Injective ↑(frobeniusMorphism p K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
this : ∀ (x : K), x ^ p = 0 → x = 0
⊢ Injective ↑(frobeniusMorphism p K)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture10Before.lean | frobeniusMorphism_injective | [343, 1] | [345, 8] | exact? | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
⊢ ∀ (x : K), x ^ p = 0 → x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝¹ : Field K
inst✝ : CharP K p
x : K
⊢ ∀ (x : K), x ^ p = 0 → x = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture10Before.lean | frobeniusMorphism_bijective | [347, 1] | [348, 59] | sorry | R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝² : Field K
inst✝¹ : CharP K p
x : K
inst✝ : Finite K
⊢ Bijective ↑(frobeniusMorphism p K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
p : ℕ
hp : Fact (Nat.Prime p)
K : Type u_3
inst✝² : Field K
inst✝¹ : CharP K p
x : K
inst✝ : Finite K
⊢ Bijective ↑(frobeniusMorphism p K)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_antisymm | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_inf | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y) | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊓ (x ⊔ y)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_sup_left | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply inf_le_left | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ (x ⊔ y) ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb1 | [64, 1] | [69, 20] | apply le_refl | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_antisymm | α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_sup_left | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x ⊔ x ⊓ y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply sup_le | case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊔ x ⊓ y ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply inf_le_left | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ⊓ y ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | absorb2 | [71, 1] | [76, 20] | apply le_refl | case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
α : Type u_1
inst✝ : Lattice α
x y z : α
⊢ x ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux1 | [108, 1] | [110, 26] | rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ 0 ≤ b - a | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ 0 ≤ b - a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux1 | [108, 1] | [110, 26] | apply add_le_add_left h | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : a ≤ b
⊢ -a + a ≤ -a + b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux2 | [112, 1] | [114, 26] | rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a ≤ b | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a ≤ b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C02_Basics/solutions/Solutions_S05_Proving_Facts_about_Algebraic_Structures.lean | aux2 | [112, 1] | [114, 26] | apply add_le_add_left h | R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : StrictOrderedRing R
a b c : R
h : 0 ≤ b - a
⊢ a + 0 ≤ a + (b - a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
⊢ ConvergesTo (fun x => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | apply εpos | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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) | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | ring | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ c * s x✝ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ 0 * s x✝ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
⊢ ConvergesTo (fun n => s n * t n) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) | 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 | 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
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | have pos₀ : ε / B > 0 := div_pos εpos Bpos | 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
Bpos : 0 < B
⊢ ∃ 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
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | rcases ct _ pos₀ with ⟨N₁, h₁⟩ | 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
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.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
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | sorry | case intro.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
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | have h₁ : ConvergesTo (fun n ↦ s n * (t n + -b)) 0 := by
apply aux cs
convert convergesTo_add ct (convergesTo_const (-b))
ring | 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
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
⊢ ConvergesTo (fun n => s n * t n) (a * b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | have := convergesTo_add h₁ (convergesTo_mul_const b cs) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
⊢ ConvergesTo (fun n => s n * t n) (a * b) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ ConvergesTo (fun n => s n * t n) (a * b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
⊢ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ ConvergesTo (fun n => s n * t n) (a * b) | case h.e'_1
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n
case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ a * b = 0 + b * a | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | ring | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ a * b = 0 + b * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ a * b = 0 + b * a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | apply aux cs | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * (t n + -b)) 0 | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * (t n + -b)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | convert convergesTo_add ct (convergesTo_const (-b)) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0 | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | ring | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | ext | case h.e'_1
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n | case h.e'_1.h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
x✝ : ℕ
⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 7] | ring | case h.e'_1.h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
x✝ : ℕ
⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
x✝ : ℕ
⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | by_contra abne | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
⊢ a = b | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
⊢ a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | have : |a - b| > 0 := by sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | let ε := |a - b| / 2 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | have εpos : ε > 0 := by
change |a - b| / 2 > 0
linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | rcases sa ε εpos with ⟨Na, hNa⟩ | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | rcases sb ε εpos with ⟨Nb, hNb⟩ | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | let N := max Na Nb | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | have absa : |s N - a| < ε := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | have absb : |s N - b| < ε := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | have : |a - b| < |a - b| := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | exact lt_irrefl _ this | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | change |a - b| / 2 > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S01_Basics.lean | left_inv_eq_right_inv₁ | [103, 1] | [104, 90] | rw [← DiaOneClass₁.one_dia c, ← hba, Semigroup₁.dia_assoc, hac, DiaOneClass₁.dia_one b] | M : Type
inst✝ : Monoid₁ M
a b c : M
hba : b ⋄ a = 𝟙
hac : a ⋄ c = 𝟙
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type
inst✝ : Monoid₁ M
a b c : M
hba : b ⋄ a = 𝟙
hac : a ⋄ c = 𝟙
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S01_Basics.lean | left_inv_eq_right_inv' | [145, 1] | [147, 54] | rw [← one_mul c, ← hba, mul_assoc₃, hac, mul_one b] | M : Type
inst✝ : Monoid₃ M
a b c : M
hba : b * a = 1
hac : a * c = 1
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type
inst✝ : Monoid₃ M
a b c : M
hba : b * a = 1
hac : a * c = 1
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S01_Basics.lean | Group₃.mul_inv | [180, 1] | [182, 8] | sorry | G : Type
inst✝ : Group₃ G
a : G
⊢ a * a⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group₃ G
a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S01_Basics.lean | mul_left_cancel₃ | [184, 1] | [186, 8] | sorry | G : Type
inst✝ : Group₃ G
a b c : G
h : a * b = a * c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group₃ G
a b c : G
h : a * b = a * c
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S01_Basics.lean | mul_right_cancel₃ | [188, 1] | [190, 8] | sorry | G : Type
inst✝ : Group₃ G
a b c : G
h : b * a = c * a
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group₃ G
a b c : G
h : b * a = c * a
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | intro x y ε epos ele1 xlt ylt | ⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | calc
|x * y| = |x| * |y| := by apply abs_mul
_ ≤ |x| * ε := by apply mul_le_mul; linarith; linarith; apply abs_nonneg; apply abs_nonneg;
_ < 1 * ε := by rw [mul_lt_mul_right epos]; linarith
_ = ε := by apply one_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| = |x| * |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| = |x| * |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply mul_le_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * |y| ≤ |x| * ε | case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * |y| ≤ |x| * ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_nonneg | case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_nonneg | case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | rw [mul_lt_mul_right epos] | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * ε < 1 * ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * ε < 1 * ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply one_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 1 * ε = ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 1 * ε = ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | have : ∀ n, (∑ i in range (n + 1), i : ℚ) = n * (n + 1) / 2 | n : ℕ
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (∑ i in range (n + 1), ↑i) ^ 2 | case this
n : ℕ
⊢ ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (∑ i in range (n + 1), ↑i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (∑ i in range (n + 1), ↑i) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | rw [this] | n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (∑ i in range (n + 1), ↑i) ^ 2 | n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (∑ i in range (n + 1), ↑i) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | induction n with
| zero => simp
| succ n ih =>
rw [sum_range_succ, ih]
push_cast
ring | n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1), ↑(i ^ 3) = (↑n * (↑n + 1) / 2) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | intro n | case this
n : ℕ
⊢ ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2 | case this
n✝ n : ℕ
⊢ ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case this
n : ℕ
⊢ ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | induction n with
| zero => simp
| succ n ih =>
rw [sum_range_succ, ih]
push_cast
ring | case this
n✝ n : ℕ
⊢ ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
n✝ n : ℕ
⊢ ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | simp | case this.zero
n : ℕ
⊢ ∑ i in range (0 + 1), ↑i = ↑0 * (↑0 + 1) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.zero
n : ℕ
⊢ ∑ i in range (0 + 1), ↑i = ↑0 * (↑0 + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | rw [sum_range_succ, ih] | case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1 + 1), ↑i = ↑(n + 1) * (↑(n + 1) + 1) / 2 | case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + ↑(n + 1) = ↑(n + 1) * (↑(n + 1) + 1) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (n + 1 + 1), ↑i = ↑(n + 1) * (↑(n + 1) + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | push_cast | case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + ↑(n + 1) = ↑(n + 1) * (↑(n + 1) + 1) / 2 | case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + (↑n + 1) = (↑n + 1) * (↑n + 1 + 1) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + ↑(n + 1) = ↑(n + 1) * (↑(n + 1) + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | ring | case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + (↑n + 1) = (↑n + 1) * (↑n + 1 + 1) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.succ
n✝ n : ℕ
ih : ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ↑n * (↑n + 1) / 2 + (↑n + 1) = (↑n + 1) * (↑n + 1 + 1) / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_1 | [24, 1] | [39, 9] | simp | case zero
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (0 + 1), ↑(i ^ 3) = (↑0 * (↑0 + 1) / 2) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
this : ∀ (n : ℕ), ∑ i in range (n + 1), ↑i = ↑n * (↑n + 1) / 2
⊢ ∑ i in range (0 + 1), ↑(i ^ 3) = (↑0 * (↑0 + 1) / 2) ^ 2
TACTIC:
|
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