mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Colimit.Ring

Goal state before:
pp: case e_a.e_f
ι : Type u_1
inst✝³ : Preorder ι
G : ι → Type u_2
inst✝² : (i : ι) → CommRing (G i)
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
q : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
q₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
x✝¹ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₁
x✝ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₂
i₁ : ι
p₁ : (G i₁)[X]
ih₁ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₁) p₁ = q₁
i₂ : ι
p₂ : (G i₂)[X]
ih₂ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₂) p₂ = q₂
i : ι
h1 : i₁ ≤ i
h2 : i₂ ≤ i
⊢ (of G (fun i j h ↦ ⇑(f' i j h)) i).comp (f' i₁ i h1) = of G (fun i j h ↦ ⇑(f' i j h)) i₁

Tactic:
ext x

Goal state after:
case e_a.e_f
ι : Type u_1
inst✝³ : Preorder ι
G : ι → Type u_2
inst✝² : (i : ι) → CommRing (G i)
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
q : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
q₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
x✝¹ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₁
x✝ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₂
i₁ : ι
p₁ : (G i₁)[X]
ih₁ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₁) p₁ = q₁
i₂ : ι
p₂ : (G i₂)[X]
ih₂ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₂) p₂ = q₂
i : ι
h1 : i₁ ≤ i
h2 : i₂ ≤ i
x : G i₁
⊢ ((of G (fun i j h ↦ ⇑(f' i j h)) i).comp (f' i₁ i h1)) x = (of G (fun i j h ↦ ⇑(f' i j h)) i₁) x
Module: Mathlib.Algebra.Colimit.Ring

Goal state before:
pp: case e_a.e_f
ι : Type u_1
inst✝³ : Preorder ι
G : ι → Type u_2
inst✝² : (i : ι) → CommRing (G i)
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
q : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
q₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
x✝¹ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₁
x✝ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₂
i₁ : ι
p₁ : (G i₁)[X]
ih₁ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₁) p₁ = q₁
i₂ : ι
p₂ : (G i₂)[X]
ih₂ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₂) p₂ = q₂
i : ι
h1 : i₁ ≤ i
h2 : i₂ ≤ i
⊢ (of G (fun i j h ↦ ⇑(f' i j h)) i).comp (f' i₂ i h2) = of G (fun i j h ↦ ⇑(f' i j h)) i₂

Tactic:
ext x

Goal state after:
case e_a.e_f
ι : Type u_1
inst✝³ : Preorder ι
G : ι → Type u_2
inst✝² : (i : ι) → CommRing (G i)
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
q : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
q₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]
x✝¹ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₁
x✝ : ∃ i p, Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i) p = q₂
i₁ : ι
p₁ : (G i₁)[X]
ih₁ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₁) p₁ = q₁
i₂ : ι
p₂ : (G i₂)[X]
ih₂ : Polynomial.map (of G (fun i j h ↦ ⇑(f' i j h)) i₂) p₂ = q₂
i : ι
h1 : i₁ ≤ i
h2 : i₂ ≤ i
x : G i₂
⊢ ((of G (fun i j h ↦ ⇑(f' i j h)) i).comp (f' i₂ i h2)) x = (of G (fun i j h ↦ ⇑(f' i j h)) i₂) x
Module: Mathlib.Algebra.ContinuedFractions.Computation.Translations

Goal state before:
pp: case inr.inl
K : Type u_1
inst✝³ : DivisionRing K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
v : K
inst✝ : IsStrictOrderedRing K
n : ℕ
h : fract v ≠ 0
h₁ : (of (fract v)⁻¹).s.get? n = none
⊢ (of (fract v)⁻¹).TerminatedAt n

Tactic:
terminatedAt_iff_s_none

Goal state after:
case inr.inl
K : Type u_1
inst✝³ : DivisionRing K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
v : K
inst✝ : IsStrictOrderedRing K
n : ℕ
h : fract v ≠ 0
h₁ : (of (fract v)⁻¹).s.get? n = none
⊢ (of (fract v)⁻¹).s.get? n = none
Module: Mathlib.Algebra.ContinuedFractions.Computation.Translations

Goal state before:
pp: case inr
K : Type u_1
inst✝³ : DivisionRing K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
v : K
n : ℕ
inst✝ : IsStrictOrderedRing K
h : fract v ≠ 0
⊢ (of v).convs' (n + 1) = ↑⌊v⌋ + 1 / (of (fract v)⁻¹).convs' n

Tactic:
rw [convs', of_h_eq_floor, add_right_inj, convs'Aux_succ_some (of_s_head h)]

Goal state after:
case inr
K : Type u_1
inst✝³ : DivisionRing K
inst✝² : LinearOrder K
inst✝¹ : FloorRing K
v : K
n : ℕ
inst✝ : IsStrictOrderedRing K
h : fract v ≠ 0
⊢ { a := 1, b := ↑⌊(fract v)⁻¹⌋ }.a / ({ a := 1, b := ↑⌊(fract v)⁻¹⌋ }.b + convs'Aux (of v).s.tail n) =
    1 / (of (fract v)⁻¹).convs' n
Module: Mathlib.Data.Nat.Fib.Basic

Goal state before:
pp: n : ℕ
⊢ fib (n + 1) = ∑ k ∈ Finset.range n, fib k + 1

Tactic:
by
  induction n with
  | zero => simp
  | succ n ih =>
    calc
      fib (n + 2) = fib n + fib (n + 1) := fib_add_two
      _ = (fib n + ∑ k ∈ Finset.range n, fib k) + 1 := by rw [ih, add_assoc]
      _ = (∑ k ∈ Finset.range (n + 1), fib k) + 1 := by simp [Finset.range_add_one]

Goal state after:
Module: Mathlib.Data.Seq.Basic

Goal state before:
pp: case bisim
α : Type u
SS : Seq (Seq1 (Seq1 α))
⊢ Seq.IsBisimulation fun s1 s2 ↦ ∃ s SS, s1 = s.append SS.join.join ∧ s2 = s.append (Seq.map join SS).join

pp: case r
α : Type u
SS : Seq (Seq1 (Seq1 α))
⊢ ∃ s SS_1, SS.join.join = s.append SS_1.join.join ∧ (Seq.map join SS).join = s.append (Seq.map join SS_1).join

Tactic:
· rintro s1 s2 ⟨s, SS, rfl, rfl⟩
    cases s with
    | nil =>
      cases SS with
      | nil => simp
      | cons S SS =>
        obtain ⟨⟨x, s⟩, S⟩ := S
        cases s with
        | nil => simpa using ⟨_, _, rfl, rfl⟩
        | cons x s => simpa using ⟨Seq.cons x (append s (Seq.join S)), SS, by simp, by simp⟩
    | cons _ s => simpa using ⟨s, SS, rfl, rfl⟩

Goal state after:
case r
α : Type u
SS : Seq (Seq1 (Seq1 α))
⊢ ∃ s SS_1, SS.join.join = s.append SS_1.join.join ∧ (Seq.map join SS).join = s.append (Seq.map join SS_1).join
Module: Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv

Goal state before:
pp: case inr
K : Type u_1
n : ℕ
g : GenContFract K
inst✝ : Field K
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? n = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt n
a b : K
s_nth_eq : g.s.get? n = some { a := a, b := b }
⊢ g.convs (n + 1) = (g.squashGCF n).convs n

Tactic:
have b_ne_zero : b ≠ 0 := nth_partDen_ne_zero (partDen_eq_s_b s_nth_eq)

Goal state after:
case inr
K : Type u_1
n : ℕ
g : GenContFract K
inst✝ : Field K
nth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? n = some b → b ≠ 0
not_terminatedAt_n : ¬g.TerminatedAt n
a b : K
s_nth_eq : g.s.get? n = some { a := a, b := b }
b_ne_zero : b ≠ 0
⊢ g.convs (n + 1) = (g.squashGCF n).convs n
Module: Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv

Goal state before:
pp: K : Type u_1
n : ℕ
g : GenContFract K
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
s_pos : ∀ {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

Tactic:
induction n generalizing g with

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : Fintype I
⊢ (Pairwise fun x1 x2 ↦ e x1 * e x2 = 0) ∧ ∑ i, e i = 1 → ∀ (i : I), IsIdempotentElem (e i)

Tactic:
intro ⟨ortho, complete⟩ i

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : Fintype I
ortho : Pairwise fun x1 x2 ↦ e x1 * e x2 = 0
complete : ∑ i, e i = 1
i : I
⊢ IsIdempotentElem (e i)
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
⊢ ⇑f ∘ e' ∘ ⇑(Fintype.equivFin I).toEmbedding = e

Tactic:
ext x

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.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
x y : R
⊢ CompleteOrthogonalIdempotents ![x, y] ↔ IsIdempotentElem x ∧ y = 1 - x

Tactic:
rw [pair_iff'ₛ, ← eq_sub_iff_add_eq', ← and_assoc, and_congr_left_iff]

Goal state after:
R : Type u_1
inst✝ : Ring R
x y : R
⊢ y = 1 - x → (x * y = 0 ∧ y * x = 0 ↔ IsIdempotentElem x)
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case inl
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
n : ℕ
e : Fin n → S
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : Fin n), e i ∈ f.range
h✝ : Subsingleton R
⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e

Tactic:
choose e' he' using he'

Goal state after:
case inl
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
n : ℕ
e : Fin n → S
he : CompleteOrthogonalIdempotents e
h✝ : Subsingleton R
e' : Fin n → R
he' : ∀ (i : Fin n), f (e' i) = e i
⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
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
⊢ ⇑f ∘ e' ∘ ⇑(Fintype.equivFin I) = e

Tactic:
ext x

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.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matrix n I = matrix n J
⊢ I = J

Tactic:
ext x

Goal state after:
R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matrix n I = matrix n J
x : R
⊢ x ∈ I ↔ x ∈ J
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: case pos
R : Type u_1
inst✝² : Ring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : Ideal R
x : R
xIJ : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I
p q : n
M : Matrix n n R
z : R
zMx : z * M q p * x + z - 1 ∈ I
N : Matrix n n R := 1 - ∑ i, single i q (if i = q then 1 - z else M i p * x * z)
i : n
iq : i = q
⊢ (N * M * single p q x + N - 1) i q ∈ ↑I.toAddSubmonoid

Tactic:
simp [iq, N, zMx, single, mul_apply, sum_apply, ite_and, sub_mul]

Goal state after:
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
⊢ f.genEigenrange μ ↑k = ((f - μ • 1) ^ k).range

Tactic:
ext x

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.genEigenrange μ ↑k ↔ x ∈ ((f - μ • 1) ^ k).range
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case refine_1.none.none
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
I : Type u_3
inst✝ : Fintype I
f : R →+* S
e₀ : R
he₀ : IsIdempotentElem e₀
hfe₀ : RingHom.ker f = Ideal.span {e₀}
e : I → S
he : CompleteOrthogonalIdempotents e
e' : I → R
he' : ∀ (i : I), f (e' i) = e i
k : I → R
hk : ∀ (i : I), e' i * e' i - e' i = e₀ * k i
h : none ≠ none
⊢ Option.rec e₀ ((1 - e₀) • e') none * Option.rec e₀ ((1 - e₀) • e') none = 0

Tactic:
simp at h

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case refine_1.none.some
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
I : Type u_3
inst✝ : Fintype I
f : R →+* S
e₀ : R
he₀ : IsIdempotentElem e₀
hfe₀ : RingHom.ker f = Ideal.span {e₀}
e : I → S
he : CompleteOrthogonalIdempotents e
e' : I → R
he' : ∀ (i : I), f (e' i) = e i
k : I → R
hk : ∀ (i : I), e' i * e' i - e' i = e₀ * k i
j : I
h : none ≠ some j
⊢ e₀ * ((1 - e₀) * e' j) = 0

Tactic:
linear_combination - he₀.eq * e' j

Goal state after:
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
⊢ LinearMap.restrict f ⋯ = μ • LinearMap.id

Tactic:
ext x

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 : ↥(f.eigenspace μ)
⊢ ↑((LinearMap.restrict f ⋯) x) = ↑((μ • LinearMap.id) x)
Module: Mathlib.LinearAlgebra.Trace

Goal state before:
pp: case neg
R : Type u
inst✝² : CommSemiring R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f g : M →ₗ[R] M
H : ¬∃ s, Nonempty (Basis (↥s) R M)
⊢ (trace R M) (f * g) = (trace R M) (g * f)

Tactic:
rw [trace, dif_neg H, LinearMap.zero_apply, LinearMap.zero_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: case insert
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
x : M
hx : x ∈ (f.genEigenspace μ₁) k
hx' : x ∈ (f.genEigenspace μ₂) k ⊔ s.sup fun μ ↦ (f.genEigenspace μ) k
⊢ x ∈ (f.genEigenspace μ₂) k

Tactic:
obtain ⟨y, hy, z, hz, rfl⟩ := Submodule.mem_sup.mp hx'

Goal state after:
case insert
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
y : M
hy : y ∈ (f.genEigenspace μ₂) k
z : M
hz : z ∈ s.sup fun μ ↦ (f.genEigenspace μ) k
hx : y + z ∈ (f.genEigenspace μ₁) k
hx' : y + z ∈ (f.genEigenspace μ₂) k ⊔ s.sup fun μ ↦ (f.genEigenspace μ) k
⊢ y + z ∈ (f.genEigenspace μ₂) k
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: case insert
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
y z : M
hz : z ∈ s.sup fun μ ↦ (f.genEigenspace μ) k
hx : y + z ∈ (f.genEigenspace μ₁) k
g : End R M := f - μ₂ • 1
hy : ∃ l, ∃ (_ : ↑l ≤ k), y ∈ ((f - μ₂ • 1) ^ l).ker
⊢ ∃ l, ∃ (_ : ↑l ≤ k), y + z ∈ ((f - μ₂ • 1) ^ l).ker

Tactic:
peel hy with l hlk hl

Goal state after:
case insert.h.h
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
y z : M
hz : z ∈ s.sup fun μ ↦ (f.genEigenspace μ) k
hx : y + z ∈ (f.genEigenspace μ₁) k
g : End R M := f - μ₂ • 1
hy : ∃ l, ∃ (_ : ↑l ≤ k), y ∈ ((f - μ₂ • 1) ^ l).ker
l : ℕ
hlk : ↑l ≤ k
hl : y ∈ ((f - μ₂ • 1) ^ l).ker
⊢ y + z ∈ ((f - μ₂ • 1) ^ l).ker
Module: Mathlib.LinearAlgebra.Trace

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : M →ₗ[R] M
g : N →ₗ[R] N
⊢ (trace R (M ⊗[R] N)) (map f g) = (trace R M) f * (trace R N) g

Tactic:
have h := LinearMap.ext_iff.1 (LinearMap.ext_iff.1 (trace_tensorProduct R M N) f) g
  simp only [compr₂_apply, mapBilinear_apply, compl₁₂_apply, lsmul_apply,
    smul_eq_mul] at h
  exact h

Goal state after:
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
p : Submodule R M
k : ℕ∞
μ : R
hfp : ∀ x ∈ p, f x ∈ p
⊢ (genEigenspace (LinearMap.restrict f hfp) μ) k = Submodule.comap p.subtype ((f.genEigenspace μ) k)

Tactic:
ext x

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
p : Submodule R M
k : ℕ∞
μ : R
hfp : ∀ x ∈ p, f x ∈ p
x : ↥p
⊢ x ∈ (genEigenspace (LinearMap.restrict f hfp) μ) k ↔ x ∈ Submodule.comap p.subtype ((f.genEigenspace μ) k)
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: K : Type v
V : Type w
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
μ : K
⊢ Disjoint (f.genEigenrange μ ↑(finrank K V)) ((f.genEigenspace μ) ↑(finrank K V))

Tactic:
have h :=
    calc
      Submodule.comap ((f - μ • 1) ^ finrank K V)
        (f.genEigenspace μ (finrank K V)) =
          LinearMap.ker ((f - algebraMap _ _ μ) ^ finrank K V *
            (f - algebraMap K (End K V) μ) ^ finrank K V) := by
              rw [genEigenspace_nat, ← LinearMap.ker_comp]; rfl
      _ = f.genEigenspace μ (finrank K V + finrank K V : ℕ) := by
              simp_rw [← pow_add, genEigenspace_nat]; rfl
      _ = f.genEigenspace μ (finrank K V) := by
              rw [genEigenspace_eq_genEigenspace_finrank_of_le]; lia
  rw [disjoint_iff_inf_le, genEigenrange_nat, LinearMap.range_eq_map,
    Submodule.map_inf_eq_map_inf_comap, top_inf_eq, h, genEigenspace_nat]
  apply Submodule.map_comap_le

Goal state after:
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 ρ : R
⊢ f.HasEigenvalue (μ - -ρ) ↔ f.HasEigenvalue (μ + ρ)

Tactic:
sub_neg_eq_add

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 ρ : R
⊢ f.HasEigenvalue (μ + ρ) ↔ f.HasEigenvalue (μ + ρ)
Module: Mathlib.Algebra.Group.MinimalAxioms

Goal state before:
pp: G : Type u
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : One G
assoc : ∀ (a b c : G), a * b * c = a * (b * c)
one_mul : ∀ (a : G), 1 * a = a
inv_mul_cancel : ∀ (a : G), a⁻¹ * a = 1
a✝ a : G
⊢ (a * a⁻¹)⁻¹ * (a * (a⁻¹ * a * a⁻¹)) = 1

Tactic:
rw [inv_mul_cancel, one_mul, inv_mul_cancel]

Goal state after:
Module: Mathlib.Algebra.Field.Periodic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
f : α → β
c : α
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Archimedean α
h : Periodic f c
hc✝ : c ≠ 0
a : α
hc : c < 0
⊢ f '' uIcc a (a + c) = range f

Tactic:
rw [uIcc_of_ge (add_le_of_nonpos_right hc.le), ← h.neg.image_Icc (neg_pos.2 hc) (a + c),
      add_neg_cancel_right]

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: case neg.nil
α : Type u_1
inst✝ : DecidableEq α
p : Prop
L1 : List (α × Bool)
b' : Bool
tail : List (α × Bool)
y : α
r : reduce L1 = (y, !b') :: tail
h : ¬(y = y ∧ b' = !!b')
⊢ p

Tactic:
simp at h

Goal state after:
Module: Mathlib.Algebra.FreeMonoid.FreeSemigroup

Goal state before:
pp: α : Type u_1
⊢ Set.range ⇑toFreeMonoid = {1}ᶜ

Tactic:
ext x

Goal state after:
α : Type u_1
x : FreeMonoid α
⊢ x ∈ Set.range ⇑toFreeMonoid ↔ x ∈ {1}ᶜ
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Basic

Goal state before:
pp: R : Type u
inst✝² : CommRing R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : eval x (Polynomial.map (algebraMap R S) f) = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
⊢ ∃ y ∈ R[x], (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree

Tactic:
eval_eq_sum_range,

Goal state after:
R : Type u
inst✝² : CommRing R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx :
  ∑ i ∈ range ((Polynomial.map (algebraMap R S) f).natDegree + 1), (Polynomial.map (algebraMap R S) f).coeff i * x ^ i =
    0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
⊢ ∃ y ∈ R[x], (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
x : R
hroot : eval x f = 0
hmo : f.Monic
i : ℕ
hi : f.natDegree ≤ i
k : ℕ
hk : i = f.natDegree + k
⊢ x ^ f.natDegree ∈ 𝓟

Tactic:
eval_eq_sum_range,

Goal state after:
R : Type u
inst✝ : CommRing R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
x : R
hroot : ∑ i ∈ range (f.natDegree + 1), f.coeff i * x ^ i = 0
hmo : f.Monic
i : ℕ
hi : f.natDegree ≤ i
k : ℕ
hk : i = f.natDegree + k
⊢ x ^ f.natDegree ∈ 𝓟
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: case refine_1
R : Type u_4
S : Type u_5
T : Type u_6
U : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : CommRing U
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : Algebra R U
inst✝⁵ : Algebra S U
inst✝⁴ : Algebra T U
inst✝³ : IsScalarTower S T U
inst✝² : IsScalarTower R T U
inst✝¹ : IsIntegralClosure S R T
inst✝ : IsIntegrallyClosedIn T U
⊢ Function.Injective ⇑(algebraMap S U)

Tactic:
rw [IsScalarTower.algebraMap_eq S T U]

Goal state after:
case refine_1
R : Type u_4
S : Type u_5
T : Type u_6
U : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : CommRing U
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : Algebra R U
inst✝⁵ : Algebra S U
inst✝⁴ : Algebra T U
inst✝³ : IsScalarTower S T U
inst✝² : IsScalarTower R T U
inst✝¹ : IsIntegralClosure S R T
inst✝ : IsIntegrallyClosedIn T U
⊢ Function.Injective ⇑((algebraMap T U).comp (algebraMap S T))
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_3
c₁ c₂ c₃ : R
inst✝ : CommRing R
a : ℍ[R,c₁,c₂,c₃]
x : R
h : a = ↑x
⊢ a = ↑a.re

Tactic:
rw [h, re_coe]

Goal state after:
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
ih : ∀ {j : ℕ}, j < n → (Δ_[1]^[n] fun r ↦ r ^ j) = 0
j : ℕ
h : j < n + 1
⊢ (Δ_[1] fun r ↦ r ^ j) = ∑ i ∈ range j, j.choose i • fun r ↦ r ^ i

Tactic:
ext x

Goal state after:
R : Type u_3
inst✝ : CommRing R
n : ℕ
ih : ∀ {j : ℕ}, j < n → (Δ_[1]^[n] fun r ↦ r ^ j) = 0
j : ℕ
h : j < n + 1
x : R
⊢ Δ_[1] (fun r ↦ r ^ j) x = (∑ i ∈ range j, j.choose i • fun r ↦ r ^ i) x
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
IH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !
⊢ (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i

Tactic:
ext x

Goal state after:
R : Type u_3
inst✝ : CommRing R
n : ℕ
IH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !
x : R
⊢ Δ_[1] (fun r ↦ r ^ (n + 1)) x = (∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i) x
Module: Mathlib.Algebra.Group.Pointwise.Set.Card

Goal state before:
pp: case inr
M : Type u_2
inst✝¹ : Mul M
s t : Set M
inst✝ : IsCancelMul M
h : (s * t).Finite
⊢ #↑s * #↑t < ℵ₀

Tactic:
aesop (add simp [Cardinal.mul_lt_aleph0_iff, finite_mul])

Goal state after:
Module: Mathlib.Algebra.Group.Submonoid.Saturation

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
s : Submonoid M
x : M
⊢ x ∈ s.saturation ↔ ∃ c, x * c ∈ s ∧ True

Tactic:
and_true,

Goal state after:
M : Type u_1
inst✝ : CommMonoid M
s : Submonoid M
x : M
⊢ x ∈ s.saturation ↔ ∃ c, x * c ∈ s
Module: Mathlib.Algebra.Group.Submonoid.Saturation

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
x : M
⊢ (∃ y, x * y = 1) ↔ IsUnit x

Tactic:
isUnit_iff_exists_inv

Goal state after: