Theorem: GenContFract.of_convergence_epsilon
File: Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean

Proof steps:
state_before: K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
state_after: K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
tactic: intro ε ε_pos
state_before: K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
state_after: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
tactic: rcases (exists_nat_gt (1 / ε) : ∃ N' : ℕ, 1 / ε < N') with ⟨N', one_div_ε_lt_N'⟩
state_before: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
state_after: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
tactic: let N := max N' 5
state_before: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
⊢ ∃ N, ∀ n ≥ N, |v - (of v).convs n| < ε
state_after: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
⊢ ∀ n ≥ N, |v - (of v).convs n| < ε
tactic: exists N
state_before: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
⊢ ∀ n ≥ N, |v - (of v).convs n| < ε
state_after: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
⊢ |v - (of v).convs n| < ε
tactic: intro n n_ge_N
state_before: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
⊢ |v - (of v).convs n| < ε
state_after: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
g : GenContFract K := of v
⊢ |v - (of v).convs n| < ε
tactic: let g := of v
state_before: case intro
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
g : GenContFract K := of v
⊢ |v - (of v).convs n| < ε
state_after: case intro.inl
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
g : GenContFract K := of v
terminatedAt_n : g.TerminatedAt n
⊢ |v - (of v).convs n| < ε

case intro.inr
K : Type u_1
v : K
inst✝² : LinearOrderedField K
inst✝¹ : FloorRing K
inst✝ : Archimedean K
ε : K
ε_pos : ε > 0
N' : ℕ
one_div_ε_lt_N' : 1 / ε < ↑N'
N : ℕ := max N' 5
n : ℕ
n_ge_N : n ≥ N
g : GenContFract K := of v
not_terminatedAt_n : ¬g.TerminatedAt n
⊢ |v - (of v).convs n| < ε
tactic: cases' Decidable.em (g.TerminatedAt n) with terminatedAt_n not_terminatedAt_n