mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case succ.inl.zero
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
IH : 0 ≤ ((of v).contsAux 0).b
terminated : g.TerminatedAt (0 - 1)
⊢ 0 ≤ ((of v).contsAux (0 + 1)).b

pp: case succ.inl.succ
K : Type u_1
v : K
n✝ : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux (n + 1)).b
terminated : g.TerminatedAt (n + 1 - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1 + 1)).b

Tactic:
·

Goal state after:
case succ.inl.zero
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
IH : 0 ≤ ((of v).contsAux 0).b
terminated : g.TerminatedAt (0 - 1)
⊢ 0 ≤ ((of v).contsAux (0 + 1)).b
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: case inl
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
f : { p // p.degree ≤ 3 }
n : ℕ
hn : n ≤ 3
⊢ (↑((fun P ↦ ⟨P.toPoly, ⋯⟩)
            ((fun f ↦ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f))).coeff
      n =
    (↑f).coeff n

pp: case inr
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
f : { p // p.degree ≤ 3 }
n : ℕ
hn : 3 < n
⊢ (↑((fun P ↦ ⟨P.toPoly, ⋯⟩)
            ((fun f ↦ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f))).coeff
      n =
    (↑f).coeff n

Tactic:
·

Goal state after:
case inl
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
f : { p // p.degree ≤ 3 }
n : ℕ
hn : n ≤ 3
⊢ (↑((fun P ↦ ⟨P.toPoly, ⋯⟩)
            ((fun f ↦ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f))).coeff
      n =
    (↑f).coeff n
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case succ.inl.succ
K : Type u_1
v : K
n✝ : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux (n + 1)).b
terminated : g.TerminatedAt (n + 1 - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1 + 1)).b

Tactic:
·

Goal state after:
case succ.inl.succ
K : Type u_1
v : K
n✝ : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux (n + 1)).b
terminated : g.TerminatedAt (n + 1 - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1 + 1)).b
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case succ.inr
K : Type u_1
v : K
n✝ : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
not_terminated : ¬g.TerminatedAt (n - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1)).b

Tactic:
·

Goal state after:
case succ.inr
K : Type u_1
v : K
n✝ : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
IH : 0 ≤ ((of v).contsAux n).b
not_terminated : ¬g.TerminatedAt (n - 1)
⊢ 0 ≤ ((of v).contsAux (n + 1)).b
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)

pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)

Tactic:
·

Goal state after:
case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
terminated : g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
f : { p // p.degree ≤ 3 }
n : ℕ
hn : 3 < n
⊢ (↑((fun P ↦ ⟨P.toPoly, ⋯⟩)
            ((fun f ↦ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f))).coeff
      n =
    (↑f).coeff n

Tactic:
·

Goal state after:
case inr
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
f : { p // p.degree ≤ 3 }
n : ℕ
hn : 3 < n
⊢ (↑((fun P ↦ ⟨P.toPoly, ⋯⟩)
            ((fun f ↦ { a := (↑f).coeff 3, b := (↑f).coeff 2, c := (↑f).coeff 1, d := (↑f).coeff 0 }) f))).coeff
      n =
    (↑f).coeff n
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)

Tactic:
·

Goal state after:
case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
⊢ (of v).dens n ≤ (of v).dens (n + 1)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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))

pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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:
·

Goal state after:
case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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))
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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:
·

Goal state after:
case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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))
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: case pos
R : Type u_1
P : Cubic R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
hP : P.toPoly = 0
⊢ P.toPoly.roots.card ≤ 3

pp: case neg
R : Type u_1
P : Cubic R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
hP : ¬P.toPoly = 0
⊢ P.toPoly.roots.card ≤ 3

Tactic:
·

Goal state after:
case pos
R : Type u_1
P : Cubic R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
hP : P.toPoly = 0
⊢ P.toPoly.roots.card ≤ 3
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: case neg
R : Type u_1
P : Cubic R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
hP : ¬P.toPoly = 0
⊢ P.toPoly.roots.card ≤ 3

Tactic:
·

Goal state after:
case neg
R : Type u_1
P : Cubic R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
hP : ¬P.toPoly = 0
⊢ P.toPoly.roots.card ≤ 3
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ (map φ P).toPoly =
    { a := φ P.a, b := φ P.a * -(x + y + z), c := φ P.a * (x * y + x * z + y * z), d := φ P.a * -(x * y * z) }.toPoly

pp: case inj
F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ Function.Injective toPoly

Tactic:
·

Goal state after:
F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ (map φ P).toPoly =
    { a := φ P.a, b := φ P.a * -(x + y + z), c := φ P.a * (x * y + x * z + y * z), d := φ P.a * -(x * y * z) }.toPoly
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case zero
K : Type u_1
v : K
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
ifp : IntFractPair K
g : GenContFract K := of v
ifp_fr_ne_zero : ifp.fr ≠ 0
stream_nth_eq : IntFractPair.stream v 0 = some ifp
conts : Pair K := g.contsAux (0 + 1)
pred_conts : Pair K := g.contsAux 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
⊢ 0 = 0 ∨ ¬g.TerminatedAt (0 - 1)

pp: case succ
K : Type u_1
v : K
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
ifp : IntFractPair K
g : GenContFract K := of v
ifp_fr_ne_zero : ifp.fr ≠ 0
n' : ℕ
stream_nth_eq : IntFractPair.stream v (n' + 1) = some ifp
conts : Pair K := g.contsAux (n' + 1 + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
⊢ n' + 1 = 0 ∨ ¬g.TerminatedAt (n' + 1 - 1)

Tactic:
·

Goal state after:
case zero
K : Type u_1
v : K
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
ifp : IntFractPair K
g : GenContFract K := of v
ifp_fr_ne_zero : ifp.fr ≠ 0
stream_nth_eq : IntFractPair.stream v 0 = some ifp
conts : Pair K := g.contsAux (0 + 1)
pred_conts : Pair K := g.contsAux 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
⊢ 0 = 0 ∨ ¬g.TerminatedAt (0 - 1)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case succ
K : Type u_1
v : K
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
ifp : IntFractPair K
g : GenContFract K := of v
ifp_fr_ne_zero : ifp.fr ≠ 0
n' : ℕ
stream_nth_eq : IntFractPair.stream v (n' + 1) = some ifp
conts : Pair K := g.contsAux (n' + 1 + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
⊢ n' + 1 = 0 ∨ ¬g.TerminatedAt (n' + 1 - 1)

Tactic:
·

Goal state after:
case succ
K : Type u_1
v : K
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
ifp : IntFractPair K
g : GenContFract K := of v
ifp_fr_ne_zero : ifp.fr ≠ 0
n' : ℕ
stream_nth_eq : IntFractPair.stream v (n' + 1) = some ifp
conts : Pair K := g.contsAux (n' + 1 + 1)
pred_conts : Pair K := g.contsAux (n' + 1)
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
⊢ n' + 1 = 0 ∨ ¬g.TerminatedAt (n' + 1 - 1)
Module: Mathlib.Algebra.CubicDiscriminant

Goal state before:
pp: case inj
F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ Function.Injective toPoly

Tactic:
·

Goal state after:
case inj
F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ Function.Injective toPoly
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
n_eq_zero : n = 0
⊢ n ≤ 1 ∨ ¬g.TerminatedAt (n - 2)

pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n ≤ 1 ∨ ¬g.TerminatedAt (n - 2)

Tactic:
·

Goal state after:
case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
n_eq_zero : n = 0
⊢ n ≤ 1 ∨ ¬g.TerminatedAt (n - 2)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n ≤ 1 ∨ ¬g.TerminatedAt (n - 2)

Tactic:
·

Goal state after:
case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n ≤ 1 ∨ ¬g.TerminatedAt (n - 2)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
n_eq_zero : n = 0
⊢ n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2)

pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2)

Tactic:
·

Goal state after:
case inl
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
n_eq_zero : n = 0
⊢ n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2)

Tactic:
·

Goal state after:
case inr
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing 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
ifp_fr_ne_zero : ifp.fr ≠ 0
A : K := conts.a
B : K := conts.b
pA : K := pred_conts.a
pB : K := pred_conts.b
g_finite_correctness : v = (pA + ifp.fr⁻¹ * A) / (pB + ifp.fr⁻¹ * B)
determinant_eq : pA * B - pB * A = (-1) ^ n
pB_ineq : ↑(fib n) ≤ pB
not_terminatedAt_pred_n : ¬g.TerminatedAt (n - 1)
⊢ n + 1 ≤ 1 ∨ ¬g.TerminatedAt (n + 1 - 2)
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case isTrue
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : DecidableEq I
he : OrthogonalIdempotents e
i j : I
h✝ : i = j
⊢ e i * e j = e i

pp: case isFalse
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : DecidableEq I
he : OrthogonalIdempotents e
i j : I
h✝ : ¬i = j
⊢ e i * e j = 0

Tactic:
·

Goal state after:
case isTrue
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : DecidableEq I
he : OrthogonalIdempotents e
i j : I
h✝ : i = j
⊢ e i * e j = e i
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case isFalse
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : DecidableEq I
he : OrthogonalIdempotents e
i j : I
h✝ : ¬i = j
⊢ e i * e j = 0

Tactic:
·

Goal state after:
case isFalse
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : DecidableEq I
he : OrthogonalIdempotents e
i j : I
h✝ : ¬i = j
⊢ e i * e j = 0
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: case left
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := ⋯
nextConts : Pair K := ⋯
conts : Pair K := ⋯
conts_eq : conts = g.contsAux (n + 1)
pred_conts : Pair K := ⋯
pred_conts_eq : pred_conts = g.contsAux n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
gp_a_eq_one : gp.a = 1
nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b
den : K := ⋯
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_succ_n_b_eq_gp_b : ↑ifp_succ_n.b = gp.b
ifp_n : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp_n
stream_nth_fr_ne_zero : ifp_n.fr ≠ 0
if_of_eq_ifp_succ_n : IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
den' : K := ⋯
nextConts_b_ineq : ↑(fib (n + 2)) ≤ pred_conts.b + gp.b * conts.b
conts_b_ineq : ↑(fib (n + 1)) ≤ conts.b
zero_lt_conts_b : 0 < conts.b
⊢ 0 < den

pp: case right
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := ⋯
nextConts : Pair K := ⋯
conts : Pair K := ⋯
conts_eq : conts = g.contsAux (n + 1)
pred_conts : Pair K := ⋯
pred_conts_eq : pred_conts = g.contsAux n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
gp_a_eq_one : gp.a = 1
nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b
den : K := ⋯
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_succ_n_b_eq_gp_b : ↑ifp_succ_n.b = gp.b
ifp_n : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp_n
stream_nth_fr_ne_zero : ifp_n.fr ≠ 0
if_of_eq_ifp_succ_n : IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
den' : K := ⋯
nextConts_b_ineq : ↑(fib (n + 2)) ≤ pred_conts.b + gp.b * conts.b
conts_b_ineq : ↑(fib (n + 1)) ≤ conts.b
zero_lt_conts_b : 0 < conts.b
⊢ den ≤ den'

Tactic:
·

Goal state after:
case left
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := ⋯
nextConts : Pair K := ⋯
conts : Pair K := ⋯
conts_eq : conts = g.contsAux (n + 1)
pred_conts : Pair K := ⋯
pred_conts_eq : pred_conts = g.contsAux n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
gp_a_eq_one : gp.a = 1
nextConts_b_eq : nextConts.b = pred_conts.b + gp.b * conts.b
den : K := ⋯
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_succ_n_b_eq_gp_b : ↑ifp_succ_n.b = gp.b
ifp_n : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp_n
stream_nth_fr_ne_zero : ifp_n.fr ≠ 0
if_of_eq_ifp_succ_n : IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n
den' : K := ⋯
nextConts_b_ineq : ↑(fib (n + 2)) ≤ pred_conts.b + gp.b * conts.b
conts_b_ineq : ↑(fib (n + 1)) ≤ conts.b
zero_lt_conts_b : 0 < conts.b
⊢ 0 < den