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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : FloorRing K
v : K
n : ℕ
⊢ ∃ conts, (of v).contsAux (n + 1) = Pair.map Rat.cast conts

Tactic:
;

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : FloorRing K
v : K
n : ℕ
⊢ ∃ conts, (of v).contsAux (n + 1) = Pair.map Rat.cast conts
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
⊢ Stream'.map (Option.map (mapFr Rat.cast)) (IntFractPair.stream q) n = IntFractPair.stream v n

Tactic:
;

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
⊢ Stream'.map (Option.map (mapFr Rat.cast)) (IntFractPair.stream q) n = IntFractPair.stream v n
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n = some a✝ ↔ (of v).s.get? n = some a✝

Tactic:
;

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n = some a✝ ↔ (of v).s.get? n = some a✝
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n = some a✝ ↔
    Option.map (Pair.map Rat.cast) ((of q).s.get? n) = some a✝

Tactic:
;

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n = some a✝ ↔
    Option.map (Pair.map Rat.cast) ((of q).s.get? n) = some a✝
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
h : K
s : Stream'.Seq (Pair K)
gcf_v_eq : of v = { h := h, s := s }
⊢ { h := ↑(of q).h, s := Stream'.Seq.map (Pair.map Rat.cast) (of q).s } = { h := h, s := s }

Tactic:
;

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
inst✝ : IsStrictOrderedRing K
v : K
q : ℚ
v_eq_q : v = ↑q
h : K
s : Stream'.Seq (Pair K)
gcf_v_eq : of v = { h := h, s := s }
⊢ { h := ↑(of q).h, s := Stream'.Seq.map (Pair.map Rat.cast) (of q).s } = { h := h, s := s }
Module: Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat

Goal state before:
pp: q : ℚ
fract_q_num : ℤ := (Int.fract q).num
⊢ ∃ n, IntFractPair.stream q n = none

Tactic:
;

Goal state after:
q : ℚ
fract_q_num : ℤ := (Int.fract q).num
⊢ ∃ n, IntFractPair.stream q n = none
Module: Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries

Goal state before:
pp: case h₂
K : Type u_1
v : K
inst✝⁴ : Field K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing 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 : ℕ
n_ge_N : n ≥ N
g : GenContFract K := ⋯
not_terminatedAt_n : ¬g.TerminatedAt n
B : K := ⋯
nB : K := ⋯
abs_v_sub_conv_le : |v - g.convs n| ≤ 1 / (B * nB)
nB_ineq : ↑(fib (n + 2)) ≤ nB
B_ineq : ↑(fib (n + 1)) ≤ B
zero_lt_B : 0 < B
nB_pos : 0 < nB
zero_lt_mul_conts : 0 < B * nB
⊢ 1 ≤ 2

Tactic:
;

Goal state after:
case h₂
K : Type u_1
v : K
inst✝⁴ : Field K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing 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 : ℕ
n_ge_N : n ≥ N
g : GenContFract K := ⋯
not_terminatedAt_n : ¬g.TerminatedAt n
B : K := ⋯
nB : K := ⋯
abs_v_sub_conv_le : |v - g.convs n| ≤ 1 / (B * nB)
nB_ineq : ↑(fib (n + 2)) ≤ nB
B_ineq : ↑(fib (n + 1)) ≤ B
zero_lt_B : 0 < B
nB_pos : 0 < nB
zero_lt_mul_conts : 0 < B * nB
⊢ 1 ≤ 2
Module: Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
n : ℕ
IH :
  ∀ {g : GenContFract K},
    (∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) → g.convs n = g.convs' n
g : GenContFract K
s_pos : ∀ {gp : Pair K} {m : ℕ}, m < n + 1 → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b
g' : GenContFract K := g.squashGCF n
not_terminatedAt_n : ¬g.TerminatedAt n
this : g.convs (n + 1) = g'.convs n
gp' : Pair K
m : ℕ
m_lt_n : m < n
⊢ g'.s.get? m = some gp' → 0 < gp'.a ∧ 0 < gp'.b

Tactic:
s_mth_eq'

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
n : ℕ
IH :
  ∀ {g : GenContFract K},
    (∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) → g.convs n = g.convs' n
g : GenContFract K
s_pos : ∀ {gp : Pair K} {m : ℕ}, m < n + 1 → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b
g' : GenContFract K := g.squashGCF n
not_terminatedAt_n : ¬g.TerminatedAt n
this : g.convs (n + 1) = g'.convs n
gp' : Pair K
m : ℕ
m_lt_n : m < n
s_mth_eq' : g'.s.get? m = some gp'
⊢ 0 < gp'.a ∧ 0 < gp'.b
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
⊢ 0 ≤ ((of v).contsAux (n + 1)).b

Tactic:
;

Goal state after:
K : Type u_1
v : K
n : ℕ
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorRing K
⊢ 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:
· have : g.TerminatedAt n :=
      terminatedAt_iff_partDen_none.2 (by rwa [Stream'.Seq.TerminatedAt] at terminated)
    have : g.dens (n + 1) = g.dens n :=
      dens_stable_of_terminated n.le_succ this
    rw [this]

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.CubicDiscriminant

Goal state before:
pp: R : Type u_1
P : Cubic R
inst✝ : Semiring R
ha : P.a = 0
hb : P.b ≠ 0
⊢ P.toPoly.natDegree = 2

Tactic:
by
  rw [of_a_eq_zero ha, natDegree_quadratic hb]

Goal state after:
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}
⊢ C (φ P.a) * ((X - C x) ::ₘ (X - C y) ::ₘ {X - C z}).prod = C (φ P.a) * (X - C x) * (X - C y) * (X - C z)

Tactic:
prod_cons,

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}
⊢ C (φ P.a) * ((X - C x) * ((X - C y) ::ₘ {X - C z}).prod) = C (φ P.a) * (X - C x) * (X - C y) * (X - C z)
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}
⊢ C (φ P.a) * ((X - C x) * ((X - C y) ::ₘ {X - C z}).prod) = C (φ P.a) * (X - C x) * (X - C y) * (X - C z)

Tactic:
prod_cons,

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}
⊢ C (φ P.a) * ((X - C x) * ((X - C y) * {X - C z}.prod)) = C (φ P.a) * (X - C x) * (X - C y) * (X - C z)
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: 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
n_eq_zero_or_not_terminatedAt_pred_n : 0 = 0 ∨ ¬g.TerminatedAt (0 - 1)
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)
⊢ pA * B - pB * A = (-1) ^ 0

Tactic:
;

Goal state after:
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
n_eq_zero_or_not_terminatedAt_pred_n : 0 = 0 ∨ ¬g.TerminatedAt (0 - 1)
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)
⊢ pA * B - pB * A = (-1) ^ 0
Module: Mathlib.Algebra.ContinuedFractions.Computation.Approximations

Goal state before:
pp: 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 := of v
nextConts : Pair K := g.contsAux (n + 2)
conts : Pair K := g.contsAux (n + 1)
conts_eq : conts = g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
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 := conts.b * (pred_conts.b + gp.b * conts.b)
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 := conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)
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
this : gp.b * conts.b ≤ ifp_n.fr⁻¹ * conts.b
⊢ conts.b * (pred_conts.b + gp.b * conts.b) ≤ conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)

Tactic:
;

Goal state after:
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 := of v
nextConts : Pair K := g.contsAux (n + 2)
conts : Pair K := g.contsAux (n + 1)
conts_eq : conts = g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
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 := conts.b * (pred_conts.b + gp.b * conts.b)
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 := conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)
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
this : gp.b * conts.b ≤ ifp_n.fr⁻¹ * conts.b
⊢ conts.b * (pred_conts.b + gp.b * conts.b) ≤ conts.b * (pred_conts.b + ifp_n.fr⁻¹ * conts.b)
Module: Mathlib.RingTheory.TensorProduct.Free

Goal state before:
pp: R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
ι : Type u_4
ι₂ : Type u_5
A : Type u_6
inst✝⁹ : Fintype ι
inst✝⁸ : Finite ι₂
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
f : M₁ →ₗ[R] M₂
b₁ : Basis ι R M₁
b₂ : Basis ι₂ R M₂
i✝ : ι₂
j✝ : ι
⊢ (toMatrix (basis A b₁) (basis A b₂)) (baseChange A f) i✝ j✝ = ((toMatrix b₁ b₂) f).map (⇑(algebraMap R A)) i✝ j✝

Tactic:
;

Goal state after:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
ι : Type u_4
ι₂ : Type u_5
A : Type u_6
inst✝⁹ : Fintype ι
inst✝⁸ : Finite ι₂
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
f : M₁ →ₗ[R] M₂
b₁ : Basis ι R M₁
b₂ : Basis ι₂ R M₂
i✝ : ι₂
j✝ : ι
⊢ (toMatrix (basis A b₁) (basis A b₂)) (baseChange A f) i✝ j✝ = ((toMatrix b₁ b₂) f).map (⇑(algebraMap R A)) i✝ j✝
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_4
inst✝ : CommSemiring R
x y : R
⊢ x * y = 0 ∧ x + y = 1 → y * x = 0

Tactic:
;

Goal state after:
R : Type u_4
inst✝ : CommSemiring R
x y : R
⊢ x * y = 0 ∧ x + y = 1 → y * x = 0
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case pos
I : Type u_4
inst✝² : Fintype I
inst✝¹ : DecidableEq I
R : I → Type u_5
inst✝ : (i : I) → Semiring (R i)
i j : I
hij : i ≠ j
⊢ (Pi.single i 1 * Pi.single j 1) i = 0 i

Tactic:
;

Goal state after:
case pos
I : Type u_4
inst✝² : Fintype I
inst✝¹ : DecidableEq I
R : I → Type u_5
inst✝ : (i : I) → Semiring (R i)
i j : I
hij : i ≠ j
⊢ (Pi.single i 1 * Pi.single j 1) i = 0 i
Module: Mathlib.LinearAlgebra.GeneralLinearGroup.AlgEquiv

Goal state before:
pp: case e_6
K : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : Semifield K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : Projective K V
inst✝² : AddCommMonoid W
inst✝¹ : Module K W
inst✝ : Projective K W
f : End K V ≃ₐ[K] End K W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : Dual K V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →ₗ[K] W := ⋯
hT : ∀ (x : V), T x = (f (smulRight v x)) z
A : End K V
x : V
⊢ smulRight v (A x) = A * smulRight v x

Tactic:
;

Goal state after:
case e_6
K : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : Semifield K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : Projective K V
inst✝² : AddCommMonoid W
inst✝¹ : Module K W
inst✝ : Projective K W
f : End K V ≃ₐ[K] End K W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : Dual K V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →ₗ[K] W := ⋯
hT : ∀ (x : V), T x = (f (smulRight v x)) z
A : End K V
x : V
⊢ smulRight v (A x) = A * smulRight v x
Module: Mathlib.LinearAlgebra.GeneralLinearGroup.AlgEquiv

Goal state before:
pp: case e_6
K : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : Semifield K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : Projective K V
inst✝² : AddCommMonoid W
inst✝¹ : Module K W
inst✝ : Projective K W
f : End K V ≃ₐ[K] End K W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : Dual K V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →ₗ[K] W := applyₗ z ∘ₗ f.toLinearMap ∘ₗ smulRightₗ v
hT : ∀ (x : V), T x = (f (smulRight v x)) z
A : End K V
x x✝ : V
⊢ (smulRight v (A x)) x✝ = (A * smulRight v x) x✝

Tactic:
;

Goal state after:
case e_6
K : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : Semifield K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : Projective K V
inst✝² : AddCommMonoid W
inst✝¹ : Module K W
inst✝ : Projective K W
f : End K V ≃ₐ[K] End K W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : Dual K V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →ₗ[K] W := applyₗ z ∘ₗ f.toLinearMap ∘ₗ smulRightₗ v
hT : ∀ (x : V), T x = (f (smulRight v x)) z
A : End K V
x x✝ : V
⊢ (smulRight v (A x)) x✝ = (A * smulRight v x) x✝
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
inst✝ : Finite I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e : I → S
he : OrthogonalIdempotents e
he' : ∀ (i : I), e i ∈ f.range
val✝ : Fintype I
e' : Fin (Fintype.card I) → R
h₁ : OrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = e ∘ ⇑(Fintype.equivFin I).symm.toEmbedding
x : I
⊢ (⇑f ∘ e' ∘ ⇑(Fintype.equivFin I).toEmbedding) x = e x

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
inst✝ : Finite I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e : I → S
he : OrthogonalIdempotents e
he' : ∀ (i : I), e i ∈ f.range
val✝ : Fintype I
e' : Fin (Fintype.card I) → R
h₁ : OrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = e ∘ ⇑(Fintype.equivFin I).symm.toEmbedding
x : I
⊢ (⇑f ∘ e' ∘ ⇑(Fintype.equivFin I).toEmbedding) x = e x
Module: Mathlib.GroupTheory.Congruence.BigOperators

Goal state before:
pp: case mk
ι : Type u_1
M : Type u_2
inst✝ : CommMonoid M
c : Con M
s : Multiset ι
f g : ι → M
a✝ : List ι
h : ∀ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, c (f x) (g x)
⊢ c (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod (Multiset.map g (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod

Tactic:
;

Goal state after:
case mk
ι : Type u_1
M : Type u_2
inst✝ : CommMonoid M
c : Con M
s : Multiset ι
f g : ι → M
a✝ : List ι
h : ∀ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, c (f x) (g x)
⊢ c (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod (Multiset.map g (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case inr.inr.zero
R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
h✝¹ : Nontrivial R
h✝ : Nontrivial S
e : Fin 0 → S
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : Fin 0), e i ∈ f.range
⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e

Tactic:
simpa using he.complete

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e : I → S
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : I), e i ∈ f.range
e' : Fin (Fintype.card I) → R
h₁ : CompleteOrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = e ∘ ⇑(Fintype.equivFin I).symm
x : I
⊢ (⇑f ∘ e' ∘ ⇑(Fintype.equivFin I)) x = e x

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e : I → S
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : I), e i ∈ f.range
e' : Fin (Fintype.card I) → R
h₁ : CompleteOrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = e ∘ ⇑(Fintype.equivFin I).symm
x : I
⊢ (⇑f ∘ e' ∘ ⇑(Fintype.equivFin I)) x = e x
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
e₁ e₂ : R
he₁ : IsIdempotentElem e₁
he₂ : IsIdempotentElem e₂
H' : Commute e₁ e₂
this : (e₁ - e₂) ^ 3 = e₁ - e₂
n : ℕ
⊢ (e₁ - e₂) ^ (2 * n + 1) = e₁ - e₂

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : Ring R
e₁ e₂ : R
he₁ : IsIdempotentElem e₁
he₂ : IsIdempotentElem e₂
H' : Commute e₁ e₂
this : (e₁ - e₂) ^ 3 = e₁ - e₂
n : ℕ
⊢ (e₁ - e₂) ^ (2 * n + 1) = e₁ - e₂
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: case mp
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : ∀ (i j : n), M i j = 0
⊢ M = 0

Tactic:
;

Goal state after:
case mp
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : ∀ (i j : n), M i j = 0
⊢ M = 0
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: case mp
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : ∀ (i j : n), M i j = 0
i✝ j✝ : n
⊢ M i✝ j✝ = 0 i✝ j✝

Tactic:
;

Goal state after:
case mp
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : ∀ (i j : n), M i j = 0
i✝ j✝ : n
⊢ M i✝ j✝ = 0 i✝ j✝
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: case mpr
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : M = 0
⊢ ∀ (i j : n), M i j = 0

Tactic:
;

Goal state after:
case mpr
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n R
H : M = 0
⊢ ∀ (i j : n), M i j = 0
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
x✝ : Matrix n n R
⊢ x✝ ∈ matrix n ⊤ ↔ x✝ ∈ ⊤

Tactic:
;

Goal state after:
R : Type u_1
inst✝² : Semiring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
x✝ : Matrix n n R
⊢ x✝ ∈ matrix n ⊤ ↔ x✝ ∈ ⊤
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Type u_3
inst✝ : Finite I
e : I → R
he : OrthogonalIdempotents e
this : Pairwise fun i j ↦ IsCoprime (Ideal.span {1 - e i}) (Ideal.span {1 - e j})
x : R
i : I
⊢ (RingHom.pi fun i ↦ Ideal.Quotient.mk (Ideal.span {1 - e i})) x i =
    (Ideal.quotientInfToPiQuotient fun i ↦ Ideal.span {1 - e i}) ((Ideal.Quotient.mk (⨅ i, Ideal.span {1 - e i})) x) i

Tactic:
;

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
I : Type u_3
inst✝ : Finite I
e : I → R
he : OrthogonalIdempotents e
this : Pairwise fun i j ↦ IsCoprime (Ideal.span {1 - e i}) (Ideal.span {1 - e j})
x : R
i : I
⊢ (RingHom.pi fun i ↦ Ideal.Quotient.mk (Ideal.span {1 - e i})) x i =
    (Ideal.quotientInfToPiQuotient fun i ↦ Ideal.span {1 - e i}) ((Ideal.Quotient.mk (⨅ i, Ideal.span {1 - e i})) x) i
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
k : ℕ
x✝ : M
⊢ x✝ ∈ (f.genEigenspace μ) ↑k ↔ x✝ ∈ ((f - μ • 1) ^ k).ker

Tactic:
;

Goal state after:
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
k : ℕ
x✝ : M
⊢ x✝ ∈ (f.genEigenspace μ) ↑k ↔ x✝ ∈ ((f - μ • 1) ^ k).ker
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
x✝ : M
⊢ x✝ ∈ (f.genEigenspace μ) 0 ↔ x✝ ∈ ⊥

Tactic:
;

Goal state after:
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
x✝ : M
⊢ x✝ ∈ (f.genEigenspace μ) 0 ↔ x✝ ∈ ⊥
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
k : ℕ
x✝ : M
⊢ x✝ ∈ (f.genEigenspace 0) ↑k ↔ x✝ ∈ LinearMap.ker (f ^ k)

Tactic:
;

Goal state after:
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
k : ℕ
x✝ : M
⊢ x✝ ∈ (f.genEigenspace 0) ↑k ↔ x✝ ∈ LinearMap.ker (f ^ k)
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: case h
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
k : ℕ∞
x : M
h : f.HasUnifEigenvector μ k x
⊢ x ∈ (f.genEigenspace μ) k ∧ x ≠ 0

Tactic:
;

Goal state after:
case h
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
k : ℕ∞
x : M
h : f.HasUnifEigenvector μ k x
⊢ x ∈ (f.genEigenspace μ) k ∧ x ≠ 0
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝² : Ring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
x✝ : Matrix n n R
⊢ x✝ ∈ asIdeal (matrix n I) ↔ x✝ ∈ Ideal.matrix n (asIdeal I)

Tactic:
;

Goal state after:
R : Type u_1
n : Type u_2
inst✝² : Ring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
x✝ : Matrix n n R
⊢ x✝ ∈ asIdeal (matrix n I) ↔ x✝ ∈ Ideal.matrix n (asIdeal I)