-- Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
theorem succ_nth_conv_eq_squashGCF_nth_conv [Field K]
    (nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? n = some b → b ≠ 0) :
    g.convs (n + 1) = (squashGCF g n).convs n

Proof state:
case inr.intro.mk.succ.intro.mk
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
⊢ g.convs (n' + 1 + 1) = (g.squashGCF (n' + 1)).convs (n' + 1)

Tactic:
set ppred_conts := g.contsAux n' with n'th_contsAux_eq

Result:
case inr.intro.mk.succ.intro.mk
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
ppred_conts : Pair K := g.contsAux n'
n'th_contsAux_eq : ppred_conts = g.contsAux n'
⊢ g.convs (n' + 1 + 1) = (g.squashGCF (n' + 1)).convs (n' + 1)

Proof state:
case intro.intro.intro
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
ppred_conts : Pair K := g.contsAux n'
n'th_contsAux_eq : ppred_conts = g.contsAux n'
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
pred_conts' : Pair K := g'.contsAux (n' + 1)
succ_n'th_contsAux_eq' : pred_conts' = g'.contsAux (n' + 1)
ppred_conts' : Pair K := g'.contsAux n'
n'th_contsAux_eq' : ppred_conts' = g'.contsAux n'
pA' : K := pred_conts'.a
pB' : K := pred_conts'.b
ppA' : K := ppred_conts'.a
ppB' : K := ppred_conts'.b
this✝¹ : g'.convs (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')
this✝ : g.convs (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
this :
  ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
eq1 : pA' = pA
eq2 : pB' = pB
eq3 : ppA' = ppA
eq4 : ppB' = ppB
⊢ (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB) =
    ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')

Tactic:
symm

Result:
case intro.intro.intro
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
ppred_conts : Pair K := g.contsAux n'
n'th_contsAux_eq : ppred_conts = g.contsAux n'
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
pred_conts' : Pair K := g'.contsAux (n' + 1)
succ_n'th_contsAux_eq' : pred_conts' = g'.contsAux (n' + 1)
ppred_conts' : Pair K := g'.contsAux n'
n'th_contsAux_eq' : ppred_conts' = g'.contsAux n'
pA' : K := pred_conts'.a
pB' : K := pred_conts'.b
ppA' : K := ppred_conts'.a
ppB' : K := ppred_conts'.b
this✝¹ : g'.convs (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')
this✝ : g.convs (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
this :
  ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
eq1 : pA' = pA
eq2 : pB' = pB
eq3 : ppA' = ppA
eq4 : ppB' = ppB
⊢ ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)

Proof state:
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
ppred_conts : Pair K := g.contsAux n'
n'th_contsAux_eq : ppred_conts = g.contsAux n'
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
pred_conts' : Pair K := g'.contsAux (n' + 1)
succ_n'th_contsAux_eq' : pred_conts' = g'.contsAux (n' + 1)
ppred_conts' : Pair K := g'.contsAux n'
n'th_contsAux_eq' : ppred_conts' = g'.contsAux n'
pA' : K := pred_conts'.a
pB' : K := pred_conts'.b
ppA' : K := ppred_conts'.a
ppB' : K := ppred_conts'.b
this : g'.s.get? n' = some { a := pa, b := pb + a / b }
⊢ g'.convs (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')

Tactic:
rw [conv_eq_conts_a_div_conts_b,
  conts_recurrenceAux this n'th_contsAux_eq'.symm succ_n'th_contsAux_eq'.symm]

Result:
no goals

Proof state:
case intro.intro.intro
K : Type u_1
g : GenContFract K
inst✝ : Field K
a b : K
b_ne_zero : b ≠ 0
n' : ℕ
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt (n' + 1)
s_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }
pa pb : K
s_n'th_eq : g.s.get? n' = some { a := pa, b := pb }
g' : GenContFract K := g.squashGCF (n' + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
succ_n'th_contsAux_eq : pred_conts = g.contsAux (n' + 1)
ppred_conts : Pair K := g.contsAux n'
n'th_contsAux_eq : ppred_conts = g.contsAux n'
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
pred_conts' : Pair K := g'.contsAux (n' + 1)
succ_n'th_contsAux_eq' : pred_conts' = g'.contsAux (n' + 1)
ppred_conts' : Pair K := g'.contsAux n'
n'th_contsAux_eq' : ppred_conts' = g'.contsAux n'
pA' : K := pred_conts'.a
pB' : K := pred_conts'.b
ppA' : K := ppred_conts'.a
ppB' : K := ppred_conts'.b
this✝¹ : g'.convs (n' + 1) = ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB')
this✝ : g.convs (n' + 2) = (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
this :
  ((pb + a / b) * pA + pa * ppA) / ((pb + a / b) * pB + pa * ppB) =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)
eq1 : pA' = pA
eq2 : pB' = pB
eq3 : ppA' = ppA
eq4 : ppB' = ppB
⊢ ((pb + a / b) * pA' + pa * ppA') / ((pb + a / b) * pB' + pa * ppB') =
    (b * (pb * pA + pa * ppA) + a * pA) / (b * (pb * pB + pa * ppB) + a * pB)

Tactic:
simpa only [eq1, eq2, eq3, eq4, mul_div_cancel_right₀ _ b_ne_zero]

Result:
no goals