mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.GeneralLinearGroup.AlgEquiv

Goal state before:
pp: case neg
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
⊢ ∃ T, ∀ (a : End K V), f a = (conjAlgEquiv K T) a

Tactic:
conjAlgEquiv_apply,

Goal state after:
case neg
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
⊢ ∃ T, ∀ (a : End K V), f a = ↑T ∘ₗ a ∘ₗ ↑T.symm
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: I : Type u_4
inst✝² : Fintype I
inst✝¹ : DecidableEq I
R : I → Type u_5
inst✝ : (i : I) → Semiring (R i)
⊢ CompleteOrthogonalIdempotents fun x ↦ Pi.single x 1

Tactic:
by
  refine ⟨⟨by simp [IsIdempotentElem, ← Pi.single_mul], ?_⟩, Finset.univ_sum_single 1⟩
  intro i j hij
  ext k
  by_cases hi : i = k
  · subst hi; simp [hij]
  · simp [hi]

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
x : M
⊢ x ∈ (f.genEigenspace μ) 0 ↔ x = 0

Tactic:
by
  rw [← Nat.cast_zero, mem_genEigenspace_nat, pow_zero, LinearMap.mem_ker, Module.End.one_apply]

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
n : ℕ∞
k : ℕ
ih : ∀ ⦃x : M⦄, ↑k ≤ n → ((f - μ • 1) ^ k) x = 0 → ((f - μ • 1) ^ k) (f x) = 0
x : M
hk : ↑(k + 1) ≤ n
hx : ((f - μ • 1) ^ (k + 1)) x = 0
⊢ ((f - μ • 1) ^ (k + 1)) (f x) = 0

Tactic:
rw [pow_succ, mul_apply] at hx ⊢
    simpa using ih (le_trans (by simp) hk) hx

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: R : Type u_1
e : R
inst✝ : Semigroup R
idem : IsIdempotentElem e
hc : IsMulCentral e
r : R
⊢ r ∈ corner e ↔ r ∈ Set.range fun x ↦ x * e

Tactic:
simp_rw [mem_corner_iff_mem_range_mul_left idem hc, (hc.comm _).eq]

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

Goal state before:
pp: K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
μ : K
n : ℕ
⊢ Submodule.map ((f - (algebraMap K (End K V)) μ) ^ n) (LinearMap.range f) ≤ f.genEigenrange μ ↑n

Tactic:
by rw [genEigenrange_nat]; apply LinearMap.map_le_range

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
n : ℕ
⊢ (of a ^ n).toWord = List.replicate n (a, true)

Tactic:
by
  rw [of, pow_mk, List.flatten_replicate_singleton, toWord]
  exact reduce_replicate _ _

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Criterion

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
q f g : R[X]
p : ℕ
hq_irr : Irreducible (map (algebraMap R K) q)
hq_monic : q.Monic
hf_lC : (algebraMap R K) f.leadingCoeff ≠ 0
hf_prim : f.IsPrimitive
hfmodP : map (algebraMap R K) f = C ((algebraMap R K) f.leadingCoeff) * map (algebraMap R K) q ^ p
hg_div : g ∣ f
P : Ideal R := ker (algebraMap R K)
hP : P.IsPrime
hgP : g.leadingCoeff ∉ P
map_dvd_pow_q : map (algebraMap R K) g ∣ map (algebraMap R K) q ^ p
m : ℕ
hm : m ≤ p
hf : Associated (map (algebraMap R K) g) (map (algebraMap R K) q ^ m)
r : R[X] := g - C g.leadingCoeff * q ^ m
hg : g = C g.leadingCoeff * q ^ m + r
u : K[X]ˣ
hu : map (algebraMap R K) q ^ m * ↑u = map (algebraMap R K) g
a : K
ha : IsUnit a
ha' : C a = ↑u
⊢ C ((map (algebraMap R K) q ^ m).leadingCoeff * (C a).leadingCoeff) = C a

Tactic:
leadingCoeff_C,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
q f g : R[X]
p : ℕ
hq_irr : Irreducible (map (algebraMap R K) q)
hq_monic : q.Monic
hf_lC : (algebraMap R K) f.leadingCoeff ≠ 0
hf_prim : f.IsPrimitive
hfmodP : map (algebraMap R K) f = C ((algebraMap R K) f.leadingCoeff) * map (algebraMap R K) q ^ p
hg_div : g ∣ f
P : Ideal R := ker (algebraMap R K)
hP : P.IsPrime
hgP : g.leadingCoeff ∉ P
map_dvd_pow_q : map (algebraMap R K) g ∣ map (algebraMap R K) q ^ p
m : ℕ
hm : m ≤ p
hf : Associated (map (algebraMap R K) g) (map (algebraMap R K) q ^ m)
r : R[X] := g - C g.leadingCoeff * q ^ m
hg : g = C g.leadingCoeff * q ^ m + r
u : K[X]ˣ
hu : map (algebraMap R K) q ^ m * ↑u = map (algebraMap R K) g
a : K
ha : IsUnit a
ha' : C a = ↑u
⊢ C ((map (algebraMap R K) q ^ m).leadingCoeff * a) = C a
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Basic

Goal state before:
pp: case refine_1
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 : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ R[x]
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ y * x ^ k ∈ R[x]

pp: case refine_2
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 : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ R[x]
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ (algebraMap R S) p * (y * x ^ k) = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k

Tactic:
· exact Subalgebra.mul_mem _ hy (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _)

Goal state after:
case refine_2
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 : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ R[x]
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ (algebraMap R S) p * (y * x ^ k) = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Basic

Goal state before:
pp: case refine_2
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 : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (R ∙ p)
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ R[x]
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ (algebraMap R S) p * (y * x ^ k) = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k

Tactic:
rw [← mul_assoc _ y, H]

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommRing R
A : Type u_2
B : Type u_3
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f : A →ₐ[R] B
hf : Function.Injective ⇑f
inj : Function.Injective ⇑(algebraMap R B)
cl : ∀ {x : B}, IsIntegral R x ↔ ∃ y, (algebraMap R B) y = x
⊢ Function.Injective (⇑f ∘ ⇑(algebraMap R A))

Tactic:
convert! inj
    aesop

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: A : Type u_2
inst✝³ : CommRing A
C : Type u_5
inst✝² : SetLike C A
inst✝¹ : SubringClass C A
S : C
T : Subring A
inst✝ : IsIntegrallyClosedIn (↥T) A
⊢ (∀ (a : A) (b : a ∈ S), (algebraMap (↥S) A) ⟨a, b⟩ ∈ T) ↔ ofClass S ≤ T

Tactic:
SetLike.le_def

Goal state after:
A : Type u_2
inst✝³ : CommRing A
C : Type u_5
inst✝² : SetLike C A
inst✝¹ : SubringClass C A
S : C
T : Subring A
inst✝ : IsIntegrallyClosedIn (↥T) A
⊢ (∀ (a : A) (b : a ∈ S), (algebraMap (↥S) A) ⟨a, b⟩ ∈ T) ↔ ∀ ⦃x : A⦄, x ∈ ofClass S → x ∈ T
Module: Mathlib.Algebra.Group.PNatPowAssoc

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝⁵ : Mul M
inst✝⁴ : Pow M ℕ+
inst✝³ : PNatPowAssoc M
inst✝² : Mul N
inst✝¹ : Pow N ℕ+
inst✝ : PNatPowAssoc N
x✝² x✝¹ : ℕ+
x✝ : M × N
⊢ x✝ ^ (x✝² + x✝¹) = x✝ ^ x✝² * x✝ ^ x✝¹

Tactic:
by ext <;> simp [ppow_add]

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
this : (Δ_[1] fun r ↦ r ^ j) = ∑ i ∈ range j, j.choose i • fun r ↦ r ^ i
i : ℕ
hi : i ∈ range j
⊢ j.choose i • 0 = 0

Tactic:
nsmul_zero

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
this : (Δ_[1] fun r ↦ r ^ j) = ∑ i ∈ range j, j.choose i • fun r ↦ r ^ i
i : ℕ
hi : i ∈ range j
⊢ 0 = 0
Module: Mathlib.Algebra.Group.Submonoid.Support

Goal state before:
pp: G : Type u_1
inst✝ : Group G
M : Submonoid G
hMp : M.IsMulPointed
hMs : M.IsMulSpanning
N : Submonoid G
hN : N.IsMulPointed
h : M ≤ N
⊢ N ≤ M

Tactic:
SetLike.le_def

Goal state after:
G : Type u_1
inst✝ : Group G
M : Submonoid G
hMp : M.IsMulPointed
hMs : M.IsMulSpanning
N : Submonoid G
hN : N.IsMulPointed
h : ∀ ⦃x : G⦄, x ∈ M → x ∈ N
⊢ ∀ ⦃x : G⦄, x ∈ N → x ∈ M
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: M : Type u_2
inst✝¹ : Monoid M
inst✝ : Subsingleton Mˣ
x : M
hx : x ≠ 1
h : Irreducible x
⊢ IsMulIndecomposable id univ x

Tactic:
by simpa using h.isUnit_or_isUnit

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: case h.left
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
X' : D
p : X' ⟶ X'
hp : p ≫ p = p
φ : X' ≅ (ε.inverse ⋙ ε.functor).obj X' := ε.counitIso.symm.app X'
Y : C
i : Y ⟶ ε.inverse.obj X'
e : ε.inverse.obj X' ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = ε.inverse.map p
⊢ ε.functor.map (𝟙 Y) = 𝟙 (ε.functor.obj Y)

Tactic:
ε.functor.map_id

Goal state after:
case h.left
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
X' : D
p : X' ⟶ X'
hp : p ≫ p = p
φ : X' ≅ (ε.inverse ⋙ ε.functor).obj X' := ε.counitIso.symm.app X'
Y : C
i : Y ⟶ ε.inverse.obj X'
e : ε.inverse.obj X' ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = ε.inverse.map p
⊢ 𝟙 (ε.functor.obj Y) = 𝟙 (ε.functor.obj Y)
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: case i_surj.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ j' ≠ 0

pp: case i_surj.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ (j'.pred ?i_surj.refine_1, i'.castSucc) ∈ S

pp: case i_surj.refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ φ (j'.pred ?i_surj.refine_1, i'.castSucc) ?i_surj.refine_2 = (i', j')

Tactic:
· rintro rfl
      simp only [Fin.val_zero, not_lt_zero] at hij'

Goal state after:
case i_surj.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ (j'.pred ⋯, i'.castSucc) ∈ S

case i_surj.refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ φ (j'.pred ⋯, i'.castSucc) ?i_surj.refine_2 = (i', j')
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f : X ⟶ Y
n : ℕ
⊢ ∑ j, f.app (op ⦋n + 1⦌) ≫ ((-1) ^ ↑j • Y.δ j) = ∑ j, ((-1) ^ ↑j • X.δ j) ≫ f.app (op ⦋n⦌)

Tactic:
refine Finset.sum_congr rfl fun _ _ => ?_

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f : X ⟶ Y
n : ℕ
x✝¹ : Fin (n + 2)
x✝ : x✝¹ ∈ Finset.univ
⊢ f.app (op ⦋n + 1⦌) ≫ ((-1) ^ ↑x✝¹ • Y.δ x✝¹) = ((-1) ^ ↑x✝¹ • X.δ x✝¹) ≫ f.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case «0»
n : ℕ
i : Fin n
⊢ ↑((Hom.toOrderHom (mkOfSucc i ≫ δ i.succ.castSucc)) ((fun i ↦ i) ⟨0, ⋯⟩)) =
    ↑((Hom.toOrderHom (intervalEdge (↑i) 2 ⋯)) ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
simp only [δ, len_mk, Nat.reduceAdd, mkHom, comp_toOrderHom, Hom.toOrderHom_mk,
      Fin.zero_eta, OrderHom.comp_coe, OrderEmbedding.toOrderHom_coe, Function.comp_apply,
      mkOfSucc_homToOrderHom_zero, Fin.succAboveOrderEmb_apply,
      Fin.castSucc_succAbove_castSucc, Fin.succAbove_succ_self]

Goal state after:
case «0»
n : ℕ
i : Fin n
⊢ ↑i.castSucc.castSucc = ↑((Hom.toOrderHom (intervalEdge (↑i) 2 ⋯)) 0)
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: case pos
n : ℕ
i : Fin (n + 2)
j m : ℕ
h : m + (j + 1) = n + 1
hi' : ↑i ≤ j + 1
k : Fin (⦋n⦌.len + 1)
hk : ↑k < j
hk' : i ≤ k.castSucc
⊢ ↑((ConcreteCategory.hom (σ₀Iter (j + 1) ⋯)) k.succ) = 0

Tactic:
σ₀Iter_coe_eq_of_lt ..

Goal state after:
case pos
n : ℕ
i : Fin (n + 2)
j m : ℕ
h : m + (j + 1) = n + 1
hi' : ↑i ≤ j + 1
k : Fin (⦋n⦌.len + 1)
hk : ↑k < j
hk' : i ≤ k.castSucc
⊢ 0 = 0
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: case neg
n : ℕ
i : Fin (n + 2)
j m : ℕ
h : m + (j + 1) = n + 1
hi' : ↑i ≤ j + 1
k : Fin (⦋n⦌.len + 1)
hk : ↑k < j
hk' : k.castSucc < i
⊢ ↑((ConcreteCategory.hom (σ₀Iter (j + 1) ⋯)) k.castSucc) = 0

Tactic:
σ₀Iter_coe_eq_of_lt ..

Goal state after:
case neg
n : ℕ
i : Fin (n + 2)
j m : ℕ
h : m + (j + 1) = n + 1
hi' : ↑i ≤ j + 1
k : Fin (⦋n⦌.len + 1)
hk : ↑k < j
hk' : k.castSucc < i
⊢ 0 = 0
Module: Mathlib.Order.Fin.Finset

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableEq α
a x : α
hab : a < x
x✝ : ↥{a, x}
hx : x ∈ {a, x}
⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩

Tactic:
· exact ⟨1, rfl⟩

Goal state after:
Module: Mathlib.Order.Fin.Finset

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableEq α
a c x : α
hab : a < x
hbc : x < c
x✝ : ↥{a, x, c}
hx : x ∈ {a, x, c}
⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨x, ⋯⟩, ⟨c, ⋯⟩] a_1 = ⟨x, hx⟩

pp: case inr.inr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableEq α
a b : α
hab : a < b
x : α
hbc : b < x
x✝ : ↥{a, b, x}
hx : x ∈ {a, b, x}
⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨b, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩

Tactic:
· exact ⟨1, rfl⟩

Goal state after:
case inr.inr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableEq α
a b : α
hab : a < b
x : α
hbc : b < x
x✝ : ↥{a, b, x}
hx : x ∈ {a, b, x}
⊢ ∃ a_1, ![⟨a, ⋯⟩, ⟨b, ⋯⟩, ⟨x, ⋯⟩] a_1 = ⟨x, hx⟩
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : ChainComplex V ℕ
X : V
f : C.X 0 ⟶ X
w : C.d 1 0 ≫ f = 0
i j : ℕ
⊢ (C.augment f w).d (i + 1) (j + 1) = C.d i j

Tactic:
by
  cases i <;> rfl

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : SimplexCategoryᵒᵖ
⊢ (ed.s' ≫ X.left.map (SimplexCategory.isTerminalZero.from (unop n)).op) ≫ X.hom.app n = 𝟙 X.right

Tactic:
rw [assoc, dsimp% X.hom.naturality, comp_id]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : SimplexCategoryᵒᵖ
⊢ ed.s' ≫ X.hom.app (op ⦋0⦌) = 𝟙 X.right
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case zero
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
hm : #S = 0
⊢ (face S).toSSet.HasDimensionLT d

pp: case succ
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m + 1
⊢ (face S).toSSet.HasDimensionLT d

Tactic:
· obtain rfl : S = ∅ := by rwa [← Finset.card_eq_zero]
    rw [face_empty]
    infer_instance

Goal state after:
case succ
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m + 1
⊢ (face S).toSSet.HasDimensionLT d
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
⊢ objEquiv.symm (𝟙 (unop (op ⦋n⦌))) ∈ Δ[n].nonDegenerate n

Tactic:
by
  rw [mem_nonDegenerate_iff_strictMono]
  exact fun _ _ h ↦ h

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : j ≤ i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑j + l = ↑i
⊢ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ ≫ X.left.σ i.succ =
    X.left.σ i ≫ X.left.δ₀Iter ↑j.castSucc ⋯ ≫ ed.s (↑k + 1) ≫ X.left.σ₀Iter ↑j.castSucc ⋯

Tactic:
X.left.σ_δ₀Iter'_assoc _ _ ⟨l, by grind⟩
        (by grind),

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : j ≤ i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑j + l = ↑i
⊢ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ ≫ X.left.σ i.succ =
    X.left.δ₀Iter (↑j.castSucc)
        (CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_32 n j k hk) ≫
      X.left.σ ⟨l, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_41 n i j k hk l hl⟩ ≫
        ed.s (↑k + 1) ≫ X.left.σ₀Iter ↑j.castSucc ⋯
Module: Mathlib.Algebra.Homology.HomologicalBicomplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
x✝¹ x✝ : HomologicalComplex₂ C c₁ c₂
φ₁ φ₂ : x✝¹ ⟶ x✝
h : (toGradedObjectFunctor C c₁ c₂).map φ₁ = (toGradedObjectFunctor C c₁ c₂).map φ₂
⊢ φ₁ = φ₂

Tactic:
ext i₁ i₂
    exact congr_fun h ⟨i₁, i₂⟩

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} C₄
inst✝¹ : Category.{v_5, u_5} C₁₂
inst✝ : Category.{v_6, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
e : F ≅ F'
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
⊢ mapTrifunctorMapNatTrans e.hom I₁ I₂ I₃ ≫ mapTrifunctorMapNatTrans e.inv I₁ I₂ I₃ = 𝟙 (mapTrifunctor F I₁ I₂ I₃)

Tactic:
by
    ext X₁ X₂ X₃ ⟨i₁, i₂, i₃⟩
    dsimp
    simp only [← NatTrans.comp_app, e.hom_inv_id, NatTrans.id_app]

Goal state after:
Module: Mathlib.Algebra.Homology.Bifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D
inst✝⁴ : HasZeroMorphisms C₁
inst✝³ : HasZeroMorphisms C₂
inst✝² : HasZeroMorphisms D
F : C₁ ⥤ C₂ ⥤ D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
K₁ : HomologicalComplex C₁ c₁
K₂ K₂' : HomologicalComplex C₂ c₂
φ : K₂ ⟶ K₂'
⊢ ∀ (i₁ i₁' : I₁) (i₂ : I₂),
    c₁.Rel i₁ i₁' →
      ((GradedObject.mapBifunctor F I₁ I₂).obj K₁.X).map φ.f (i₁, i₂) ≫
          ((HomologicalComplex₂.ofGradedObject c₁ c₂ (((GradedObject.mapBifunctor F I₁ I₂).obj K₁.X).obj K₂'.X)
                    (fun i₁ i₁' i₂ ↦ (F.map (K₁.d i₁ i₁')).app (K₂'.X i₂))
                    (fun i₁ i₂ i₂' ↦ (F.obj (K₁.X i₁)).map (K₂'.d i₂ i₂')) ⋯ ⋯ ⋯ ⋯ ⋯).d
                i₁ i₁').f
            i₂ =
        ((HomologicalComplex₂.ofGradedObject c₁ c₂ (((GradedObject.mapBifunctor F I₁ I₂).obj K₁.X).obj K₂.X)
                    (fun i₁ i₁' i₂ ↦ (F.map (K₁.d i₁ i₁')).app (K₂.X i₂))
                    (fun i₁ i₂ i₂' ↦ (F.obj (K₁.X i₁)).map (K₂.d i₂ i₂')) ⋯ ⋯ ⋯ ⋯ ⋯).d
                i₁ i₁').f
            i₂ ≫
          ((GradedObject.mapBifunctor F I₁ I₂).obj K₁.X).map φ.f (i₁', i₂)

Tactic:
by dsimp; intros; rw [NatTrans.naturality]

Goal state after:
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j j' : J
h : ¬c.Rel j' j
i₁ : I₁
i₂ : I₂
h' : c₁.π c₂ c (i₁, i₂) = j
⊢ (K₁.ιMapBifunctor K₂ F c i₁ i₂ j h' ≫
      (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).totalDesc fun i₁ i₂ x ↦
        c₁.ε₁ c₂ c (c₁.prev i₁, i₂) •
          (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
            (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c (c₁.prev i₁) i₂ j') =
    K₁.ιMapBifunctor K₂ F c i₁ i₂ j h' ≫ 0

Tactic:
rw [comp_zero, HomologicalComplex₂.ι_totalDesc]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j j' : J
h : ¬c.Rel j' j
i₁ : I₁
i₂ : I₂
h' : c₁.π c₂ c (i₁, i₂) = j
⊢ c₁.ε₁ c₂ c (c₁.prev i₁, i₂) •
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
        (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c (c₁.prev i₁) i₂ j' =
    0
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j j' : J
h : ¬c.Rel j' j
i₁ : I₁
i₂ : I₂
h' : c₁.π c₂ c (i₁, i₂) = j
h₃ : ¬c₁.Rel (c₁.prev i₁) i₁
⊢ c₁.ε₁ c₂ c (c₁.prev i₁, i₂) •
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
        (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c (c₁.prev i₁) i₂ j' =
    0

Tactic:
· rw [h₁.zero _ _ h₃, Functor.map_zero, zero_app, zero_comp, smul_zero]

Goal state after: