Theorem: GenContFract.sub_convs_eq
File: Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean

Proof steps:
state_before: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
state_after: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
tactic: let g := of v
state_before: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
state_after: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
tactic: let conts := g.contsAux (n + 1)
state_before: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
state_after: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
tactic: let pred_conts := g.contsAux n
state_before: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
state_after: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_conts conts ifp.fr
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
tactic: have g_finite_correctness :
    v = GenContFract.compExactValue pred_conts conts ifp.fr :=
    compExactValue_correctness_of_stream_eq_some stream_nth_eq
state_before: K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_conts conts ifp.fr
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
state_after: case inl
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_conts conts ifp.fr
ifp_fr_eq_zero : ifp.fr = 0
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))

case inr
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_conts conts ifp.fr
ifp_fr_ne_zero : ifp.fr ≠ 0
⊢ let g := of v;
  let B := (g.contsAux (n + 1)).b;
  let pB := (g.contsAux n).b;
  v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))
tactic: obtain (ifp_fr_eq_zero | ifp_fr_ne_zero) := eq_or_ne ifp.fr 0