mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.TrivSqZeroExt.Basic

Goal state before:
pp: case h2
R : Type u
M : Type v
inst✝⁴ : DivisionSemiring R
inst✝³ : AddCommGroup M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : Module R M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
a b : tsze R M
⊢ (a * b)⁻¹.snd = (b⁻¹ * a⁻¹).snd

Tactic:
simp only [snd_inv, snd_mul, fst_mul, fst_inv]

Goal state after:
case h2
R : Type u
M : Type v
inst✝⁴ : DivisionSemiring R
inst✝³ : AddCommGroup M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : Module R M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
a b : tsze R M
⊢ -((a.fst * b.fst)⁻¹ •> (a.fst •> b.snd + a.snd <• b.fst) <• (a.fst * b.fst)⁻¹) =
    b.fst⁻¹ •> -(a.fst⁻¹ •> a.snd <• a.fst⁻¹) + -(b.fst⁻¹ •> b.snd <• b.fst⁻¹) <• a.fst⁻¹
Module: Mathlib.RingTheory.Coalgebra.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
⊢ ((AlgebraTensorModule.curry (LinearMap.rTensor (A ⊗[R] B) counit ∘ₗ comul)) x) y =
    ((AlgebraTensorModule.curry ((mk S S (A ⊗[R] B)) 1)) x) y

Tactic:
convert!
      congr((TensorProduct.lid S _).symm
        (TensorProduct.lid _ _ $(rTensor_counit_comul (R := S) x) ⊗ₜ[R]
          TensorProduct.lid _ _ $(rTensor_counit_comul (R := R) y)))

Goal state after:
case e'_2
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
⊢ ((AlgebraTensorModule.curry (LinearMap.rTensor (A ⊗[R] B) counit ∘ₗ comul)) x) y =
    (TensorProduct.lid S (A ⊗[R] B)).symm
      ((TensorProduct.lid S A) ((LinearMap.rTensor A counit) (comul x)) ⊗ₜ[R]
        (TensorProduct.lid R B) ((LinearMap.rTensor B counit) (comul y)))

case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
⊢ ((AlgebraTensorModule.curry ((mk S S (A ⊗[R] B)) 1)) x) y =
    (TensorProduct.lid S (A ⊗[R] B)).symm
      ((TensorProduct.lid S A) (1 ⊗ₜ[S] x) ⊗ₜ[R] (TensorProduct.lid R B) (1 ⊗ₜ[R] y))
Module: Mathlib.RingTheory.Coalgebra.TensorProduct

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
A : Type u_3
B : Type u_4
inst✝²⁷ : CommSemiring R✝
inst✝²⁶ : CommSemiring S✝
inst✝²⁵ : AddCommMonoid A
inst✝²⁴ : AddCommMonoid B
inst✝²³ : Algebra R✝ S✝
inst✝²² : Module R✝ A
inst✝²¹ : Module S✝ A
inst✝²⁰ : Module R✝ B
inst✝¹⁹ : IsScalarTower R✝ S✝ A
R : Type u_5
S : Type u_6
M : Type u_7
N : Type u_8
P : Type u_9
Q : Type u_10
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : CommSemiring S
inst✝¹⁶ : Algebra R S
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : AddCommMonoid N
inst✝¹³ : AddCommMonoid P
inst✝¹² : AddCommMonoid Q
inst✝¹¹ : Module R M
inst✝¹⁰ : Module R N
inst✝⁹ : Module R P
inst✝⁸ : Module R Q
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower R S M
inst✝⁵ : Coalgebra S M
inst✝⁴ : Module S N
inst✝³ : IsScalarTower R S N
inst✝² : Coalgebra S N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
f : M →ₗc[S] N
g : P →ₗc[R] Q
x : M
y : P
⊢ (_root_.TensorProduct.map (AlgebraTensorModule.map ↑f ↑g) (AlgebraTensorModule.map ↑f ↑g))
      ((AlgebraTensorModule.tensorTensorTensorComm R S R S M M P P) (comul x ⊗ₜ[R] comul y)) =
    (AlgebraTensorModule.tensorTensorTensorComm R S R S N N Q Q) (comul (f x) ⊗ₜ[R] comul (g y))

Tactic:
simp only [← CoalgHomClass.map_comp_comul_apply]

Goal state after:
R✝ : Type u_1
S✝ : Type u_2
A : Type u_3
B : Type u_4
inst✝²⁷ : CommSemiring R✝
inst✝²⁶ : CommSemiring S✝
inst✝²⁵ : AddCommMonoid A
inst✝²⁴ : AddCommMonoid B
inst✝²³ : Algebra R✝ S✝
inst✝²² : Module R✝ A
inst✝²¹ : Module S✝ A
inst✝²⁰ : Module R✝ B
inst✝¹⁹ : IsScalarTower R✝ S✝ A
R : Type u_5
S : Type u_6
M : Type u_7
N : Type u_8
P : Type u_9
Q : Type u_10
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : CommSemiring S
inst✝¹⁶ : Algebra R S
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : AddCommMonoid N
inst✝¹³ : AddCommMonoid P
inst✝¹² : AddCommMonoid Q
inst✝¹¹ : Module R M
inst✝¹⁰ : Module R N
inst✝⁹ : Module R P
inst✝⁸ : Module R Q
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower R S M
inst✝⁵ : Coalgebra S M
inst✝⁴ : Module S N
inst✝³ : IsScalarTower R S N
inst✝² : Coalgebra S N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
f : M →ₗc[S] N
g : P →ₗc[R] Q
x : M
y : P
⊢ (_root_.TensorProduct.map (AlgebraTensorModule.map ↑f ↑g) (AlgebraTensorModule.map ↑f ↑g))
      ((AlgebraTensorModule.tensorTensorTensorComm R S R S M M P P) (comul x ⊗ₜ[R] comul y)) =
    (AlgebraTensorModule.tensorTensorTensorComm R S R S N N Q Q)
      ((_root_.TensorProduct.map ↑f ↑f) (comul x) ⊗ₜ[R] (_root_.TensorProduct.map ↑g ↑g) (comul y))
Module: Mathlib.Order.Heyting.Hom

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝⁴ : FunLike F α β
inst✝³ : HeytingAlgebra α
inst✝² : HeytingAlgebra β
inst✝¹ : HeytingAlgebra γ
inst✝ : HeytingAlgebra δ
f g : HeytingHom α β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr

Goal state after:
Module: Mathlib.Order.Heyting.Hom

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝⁴ : FunLike F α β
inst✝³ : CoheytingAlgebra α
inst✝² : CoheytingAlgebra β
inst✝¹ : CoheytingAlgebra γ
inst✝ : CoheytingAlgebra δ
f g : CoheytingHom α β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr

Goal state after:
Module: Mathlib.Order.Heyting.Hom

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝⁴ : FunLike F α β
inst✝³ : BiheytingAlgebra α
inst✝² : BiheytingAlgebra β
inst✝¹ : BiheytingAlgebra γ
inst✝ : BiheytingAlgebra δ
f g : BiheytingHom α β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
x : R
inst✝ : Away x S
s : S
⊢ s * (algebraMap R S) (x ^ (sec x s).2) = (algebraMap R S) (sec x s).1

Tactic:
simp only [IsLocalization.Away.sec, ← IsLocalization.sec_spec]
  congr
  exact (IsLocalization.sec (Submonoid.powers x) s).2.property.choose_spec

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : S
⊢ ∃ a, (algebraMap R S) a = x

Tactic:
obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.exists_mk'_eq (.powers e) x

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ ∃ a, (algebraMap R S) a = mk' S x ⟨(fun x ↦ e ^ x) n, ⋯⟩
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ e ^ (n + 1) * x = e ^ 1 * x

Tactic:
· rw [he.pow_succ_eq]; ring

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ e * x = e * y
H' : Function.Surjective ⇑(algebraMap R S)
r : ↥(Submonoid.powers e)
⊢ IsUnit ((algebraMap R S) ↑r)

Tactic:
by
    obtain ⟨r, n, rfl⟩ := r
    simp [show algebraMap R S e = 1 by rw [← (algebraMap R S).map_one, H, mul_one, he]]

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}
H' : Function.Surjective ⇑(algebraMap R S)
x y : R
⊢ x - y ∈ (LinearMap.toSpanSingleton R R e).ker ↔ e * x = e * y

Tactic:
LinearMap.mem_ker,

Goal state after:
R : Type u_4
S : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}
H' : Function.Surjective ⇑(algebraMap R S)
x y : R
⊢ (LinearMap.toSpanSingleton R R e) (x - y) = 0 ↔ e * x = e * y
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
S : Type u_2
inst✝ : CommSemiring S
f : R →+* S
a b : R
H : ∀ (a_1 : S), ∃ b_1 m, f b_1 = f (a * b) ^ m * a_1
⊢ ∀ (s : S) (n : ℕ), ∃ c l m k, f (a ^ (k + n) * c) = f (a ^ (k + l) * b ^ m) * s

Tactic:
intro s n

Goal state after:
R : Type u_1
inst✝¹ : CommSemiring R
S : Type u_2
inst✝ : CommSemiring S
f : R →+* S
a b : R
H : ∀ (a_1 : S), ∃ b_1 m, f b_1 = f (a * b) ^ m * a_1
s : S
n : ℕ
⊢ ∃ c l m k, f (a ^ (k + n) * c) = f (a ^ (k + l) * b ^ m) * s
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
I✝ I : Ideal R
inst✝ : I.IsTwoSided
x r : R
xJ : x ∈ I.jacobson
⊢ x * r ∈ I.jacobson

Tactic:
apply mem_sInf.mpr
    intro 𝔪 𝔪_mem
    by_cases r𝔪 : r ∈ 𝔪
    · apply 𝔪.smul_mem _ r𝔪
    -- 𝔪₀ := { a : R | a*r ∈ 𝔪 }
    let 𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R (S := Rᵐᵒᵖ) R (.op r)) 𝔪
    suffices x ∈ 𝔪₀ by simpa [𝔪₀] using this
    have I𝔪₀ : I ≤ 𝔪₀ := fun i iI =>
      𝔪_mem.left (I.mul_mem_right _ iI)
    have 𝔪₀_maximal : IsMaximal 𝔪₀ := by
      refine isMaximal_iff.mpr ⟨
        fun h => r𝔪 (by simpa [𝔪₀] using h),
        fun J b 𝔪₀J b𝔪₀ bJ => ?_⟩
      let K : Ideal R := Ideal.span {b*r} ⊔ 𝔪
      have ⟨s, y, y𝔪, sbyr⟩ :=
        mem_span_singleton_sup.mp <|
          mul_mem_left _ r <|
            (isMaximal_iff.mp 𝔪_mem.right).right K (b * r)
            le_sup_right b𝔪₀
            (mem_sup_left <| mem_span_singleton_self _)
      have : 1 - s * b ∈ 𝔪₀ := by
        rw [mul_one, add_comm, ← eq_sub_iff_add_eq] at sbyr
        rw [sbyr, ← mul_assoc] at y𝔪
        simp [𝔪₀, sub_mul, y𝔪]
      have : 1 - s * b + s * b ∈ J := by
        apply add_mem (𝔪₀J this) (J.mul_mem_left _ bJ)
      simpa using this
    exact mem_sInf.mp xJ ⟨I𝔪₀, 𝔪₀_maximal⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.StrictInitial

Goal state before:
pp: case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasStrictTerminalObjects C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : HasLimit F
i : J
H : (j : J) → j ≠ i → IsTerminal (F.obj j)
inst✝ : Subsingleton (i ⟶ i)
j k : J
⊢ ∀ (f : j ⟶ k),
    (((Functor.const J).obj (F.1 i)).map f ≫
        if h : k = i then eqToHom ⋯ else (H k h).from (((Functor.const J).obj (F.1 i)).obj k)) =
      (if h : j = i then eqToHom ⋯ else (H j h).from (((Functor.const J).obj (F.1 i)).obj j)) ≫ F.map f

Tactic:
f

Goal state after:
case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasStrictTerminalObjects C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : HasLimit F
i : J
H : (j : J) → j ≠ i → IsTerminal (F.obj j)
inst✝ : Subsingleton (i ⟶ i)
j k : J
f : j ⟶ k
⊢ (((Functor.const J).obj (F.1 i)).map f ≫
      if h : k = i then eqToHom ⋯ else (H k h).from (((Functor.const J).obj (F.1 i)).obj k)) =
    (if h : j = i then eqToHom ⋯ else (H j h).from (((Functor.const J).obj (F.1 i)).obj j)) ≫ F.map f
Module: Mathlib.CategoryTheory.Limits.Shapes.StrictInitial

Goal state before:
pp: case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasStrictTerminalObjects C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : HasLimit F
i : J
H : (j : J) → j ≠ i → IsTerminal (F.obj j)
inst✝ : Subsingleton (i ⟶ i)
j k : J
f : j ⟶ k
h : ¬k = i
h_1 : ¬j = i
⊢ ((Functor.const J).obj (F.1 i)).map f ≫ (H k h).from (((Functor.const J).obj (F.1 i)).obj k) =
    (H j h_1).from (((Functor.const J).obj (F.1 i)).obj j) ≫ F.map f

Tactic:
· apply (H _ h).hom_ext

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.StrictInitial

Goal state before:
pp: case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasStrictTerminalObjects C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : HasLimit F
i : J
H : (j : J) → j ≠ i → IsTerminal (F.obj j)
inst✝ : Subsingleton (i ⟶ i)
j✝ : J
h : ¬j✝ = i
⊢ limit.π F i ≫ (H j✝ h).from (((Functor.const J).obj (F.1 i)).obj j✝) = 𝟙 (limit F) ≫ limit.π F j✝

Tactic:
· apply (H _ h).hom_ext

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝³ : CommSemiring R
x : R
B : Type u_2
inst✝² : CommSemiring B
inst✝¹ : Algebra R B
inst✝ : IsLocalization.Away x B
a : R
b : B
m d : ℤ
⊢ selfZPow x B m * mk' B a 1 * selfZPow x B (-d) = b → selfZPow x B m * mk' B a 1 = selfZPow x B d * b

Tactic:
intro h
    have := congr_arg (fun s : B => s * selfZPow x B d) h
    rwa [mul_assoc, mul_assoc, selfZPow_neg_mul, mul_one, mul_comm b _] at this

Goal state after:
Module: Mathlib.RingTheory.Localization.Integer

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
a : S
⊢ ∃ b, IsInteger R (↑b • a)

Tactic:
simp_rw [Algebra.smul_def, mul_comm _ a]

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
a : S
⊢ ∃ b, IsInteger R (a * (algebraMap R S) ↑b)
Module: Mathlib.RingTheory.Localization.Module

Goal state before:
pp: R : Type u_1
Rₛ : Type u_2
inst✝¹⁰ : CommSemiring R
S : Submonoid R
inst✝⁹ : CommSemiring Rₛ
inst✝⁸ : Algebra R Rₛ
inst✝⁷ : IsLocalization S Rₛ
M : Type u_3
Mₛ : Type u_4
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid Mₛ
inst✝³ : Module R Mₛ
inst✝² : Module Rₛ Mₛ
inst✝¹ : IsScalarTower R Rₛ Mₛ
f : M →ₗ[R] Mₛ
inst✝ : IsLocalizedModule S f
ι : Type u_5
v : ι → M
hv : ∀ (s : Finset ι) (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i
t : Finset ι
g₁ g₂ : ι → Rₛ
i : ι
hi : i ∈ t
a : ↥S
fg : ι ⊕ ι → R
hfg :
  (∀ (x : ↥t), (algebraMap R Rₛ) (fg (Sum.inl ↑x)) = ↑a • g₁ ↑x) ∧
    ∀ (x : ↥t), (algebraMap R Rₛ) (fg (Sum.inr ↑x)) = ↑a • g₂ ↑x
eq : ∑ x, ↑a • g₁ ↑x • (⇑f ∘ v) ↑x = ∑ x, ↑a • g₂ ↑x • (⇑f ∘ v) ↑x
⊢ g₁ i = g₂ i

Tactic:
← smul_assoc,

Goal state after:
R : Type u_1
Rₛ : Type u_2
inst✝¹⁰ : CommSemiring R
S : Submonoid R
inst✝⁹ : CommSemiring Rₛ
inst✝⁸ : Algebra R Rₛ
inst✝⁷ : IsLocalization S Rₛ
M : Type u_3
Mₛ : Type u_4
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid Mₛ
inst✝³ : Module R Mₛ
inst✝² : Module Rₛ Mₛ
inst✝¹ : IsScalarTower R Rₛ Mₛ
f : M →ₗ[R] Mₛ
inst✝ : IsLocalizedModule S f
ι : Type u_5
v : ι → M
hv : ∀ (s : Finset ι) (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i
t : Finset ι
g₁ g₂ : ι → Rₛ
i : ι
hi : i ∈ t
a : ↥S
fg : ι ⊕ ι → R
hfg :
  (∀ (x : ↥t), (algebraMap R Rₛ) (fg (Sum.inl ↑x)) = ↑a • g₁ ↑x) ∧
    ∀ (x : ↥t), (algebraMap R Rₛ) (fg (Sum.inr ↑x)) = ↑a • g₂ ↑x
eq : ∑ x, (↑a • g₁ ↑x) • (⇑f ∘ v) ↑x = ∑ x, (↑a • g₂ ↑x) • (⇑f ∘ v) ↑x
⊢ g₁ i = g₂ i
Module: Mathlib.CategoryTheory.Monoidal.Mon

Goal state before:
pp: C : Type u₁
inst✝¹¹ : Category.{v₁, u₁} C
inst✝¹⁰ : MonoidalCategory C
M✝ N✝ O X : C
inst✝⁹ : MonObj M✝
inst✝⁸ : MonObj N✝
inst✝⁷ : MonObj O
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : MonoidalCategory D
E : Type u₃
inst✝⁴ : Category.{v₃, u₃} E
inst✝³ : MonoidalCategory E
F F' : C ⥤ D
G : D ⥤ E
inst✝² : SymmetricCategory C
inst✝¹ : SymmetricCategory D
inst✝ : F.LaxBraided
M N : Mon C
⊢ «μ» F.mapMon M N ≫ F.mapMon.map (β_ M N).hom = (β_ (F.mapMon.obj M) (F.mapMon.obj N)).hom ≫ «μ» F.mapMon N M

Tactic:
ext; exact Functor.LaxBraided.braided ..

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝⁹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
inst✝⁶ : IsLocalization S A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
M' : Type u_4
inst✝³ : AddCommMonoid M'
inst✝² : Module R M'
inst✝¹ : Module A M'
inst✝ : IsScalarTower R A M'
f : M →ₗ[R] M'
h : IsBaseChange A f
⊢ IsLocalizedModule S f

Tactic:
let : Module A (LocalizedModule S M) := LocalizedModule.moduleOfIsLocalization ..

Goal state after:
R : Type u_1
inst✝⁹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
inst✝⁶ : IsLocalization S A
M : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
M' : Type u_4
inst✝³ : AddCommMonoid M'
inst✝² : Module R M'
inst✝¹ : Module A M'
inst✝ : IsScalarTower R A M'
f : M →ₗ[R] M'
h : IsBaseChange A f
this : Module A (LocalizedModule S M) := LocalizedModule.moduleOfIsLocalization
⊢ IsLocalizedModule S f
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝¹⁰ : CommSemiring A
inst✝⁹ : Algebra R A
inst✝⁸ : IsLocalization S A
M₁ : Type u_5
M₂ : Type u_6
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module A M₁
inst✝² : Module A M₂
inst✝¹ : IsScalarTower R A M₁
inst✝ : IsScalarTower R A M₂
x✝¹ : M₁
x✝ : M₂
r : R
s : ↥S
⊢ (↑s • mk' A r s • x✝¹) ⊗ₜ[R] x✝ = (↑s • x✝¹) ⊗ₜ[R] (mk' A r s • x✝)

Tactic:
← smul_assoc,

Goal state after:
R : Type u_1
inst✝¹¹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝¹⁰ : CommSemiring A
inst✝⁹ : Algebra R A
inst✝⁸ : IsLocalization S A
M₁ : Type u_5
M₂ : Type u_6
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module A M₁
inst✝² : Module A M₂
inst✝¹ : IsScalarTower R A M₁
inst✝ : IsScalarTower R A M₂
x✝¹ : M₁
x✝ : M₂
r : R
s : ↥S
⊢ ((↑s • mk' A r s) • x✝¹) ⊗ₜ[R] x✝ = (↑s • x✝¹) ⊗ₜ[R] (mk' A r s • x✝)
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝¹⁰ : CommSemiring A
inst✝⁹ : Algebra R A
inst✝⁸ : IsLocalization S A
M₁ : Type u_5
M₂ : Type u_6
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module A M₁
inst✝² : Module A M₂
inst✝¹ : IsScalarTower R A M₁
inst✝ : IsScalarTower R A M₂
x✝¹ : M₁
x✝ : M₂
r : R
s : ↥S
⊢ ((↑s • mk' A r s) • x✝¹) ⊗ₜ[R] x✝ = x✝¹ ⊗ₜ[R] (↑s • mk' A r s • x✝)

Tactic:
← smul_assoc,

Goal state after:
R : Type u_1
inst✝¹¹ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝¹⁰ : CommSemiring A
inst✝⁹ : Algebra R A
inst✝⁸ : IsLocalization S A
M₁ : Type u_5
M₂ : Type u_6
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module A M₁
inst✝² : Module A M₂
inst✝¹ : IsScalarTower R A M₁
inst✝ : IsScalarTower R A M₂
x✝¹ : M₁
x✝ : M₂
r : R
s : ↥S
⊢ ((↑s • mk' A r s) • x✝¹) ⊗ₜ[R] x✝ = x✝¹ ⊗ₜ[R] ((↑s • mk' A r s) • x✝)
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
a✝ : LocalizedModule S A
a : A
s : ↥S
⊢ 1 • 1 • ((a, s).1 * (0, 1).1) = 1 • ((0, 1).2 * (a, s).2) • 0

Tactic:
by simp only [mul_zero, smul_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
inst✝ : PreservesLimit (cospan h i) F
s : IsPullback f g h i
⊢ IsPullback (F.map f) (F.map g) (F.map h) (F.map i)

Tactic:
by
  refine
    IsPullback.of_isLimit' (F.map_commSq s.toCommSq)
      (IsLimit.equivOfNatIsoOfIso (cospanCompIso F h i) _ _ (WalkingCospan.ext ?_ ?_ ?_)
        (isLimitOfPreserves F s.isLimit))
  · rfl
  · simp
  · simp

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: case h
R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
a✝ : LocalizedModule S A
a₁ : A
s₁ : ↥S
b✝ : LocalizedModule S A
a₂ : A
s₂ : ↥S
c✝ : LocalizedModule S A
a₃ : A
s₃ : ↥S
⊢ 1 • (s₁ * s₂ * (s₁ * s₃)) • (a₁ * (s₃ • a₂ + s₂ • a₃)) =
    1 • (s₁ * (s₂ * s₃)) • ((s₁ * s₃) • (a₁ * a₂) + (s₁ * s₂) • (a₁ * a₃))

Tactic:
simp only [← mul_assoc, mul_right_comm, mul_add, mul_smul_comm, smul_add, smul_smul, one_mul]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
H :
  ∀ ⦃X Y Z : C⦄ (f : X ⟶ Z) (g : Y ⟶ Z) [HasPullback f g],
    ∃ P fst snd, IsPullback fst snd f g ∧ IsPullback (F.map fst) (F.map snd) (F.map f) (F.map g)
X Y Z : C
⊢ ∀ (f : X ⟶ Z) (g : Y ⟶ Z), PreservesLimit (cospan f g) F

Tactic:
f

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
H :
  ∀ ⦃X Y Z : C⦄ (f : X ⟶ Z) (g : Y ⟶ Z) [HasPullback f g],
    ∃ P fst snd, IsPullback fst snd f g ∧ IsPullback (F.map fst) (F.map snd) (F.map f) (F.map g)
X Y Z : C
f : X ⟶ Z
⊢ ∀ (g : Y ⟶ Z), PreservesLimit (cospan f g) F
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
H :
  ∀ ⦃X Y Z : C⦄ (f : Z ⟶ X) (g : Z ⟶ Y) [HasPushout f g],
    ∃ P inl inr, IsPushout f g inl inr ∧ IsPushout (F.map f) (F.map g) (F.map inl) (F.map inr)
X Y Z : C
⊢ ∀ (f : Z ⟶ X) (g : Z ⟶ Y), PreservesColimit (span f g) F

Tactic:
f

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
H :
  ∀ ⦃X Y Z : C⦄ (f : Z ⟶ X) (g : Z ⟶ Y) [HasPushout f g],
    ∃ P inl inr, IsPushout f g inl inr ∧ IsPushout (F.map f) (F.map g) (F.map inl) (F.map inr)
X Y Z : C
f : Z ⟶ X
⊢ ∀ (g : Z ⟶ Y), PreservesColimit (span f g) F
Module: Mathlib.RingTheory.FreeCommRing

Goal state before:
pp: case refine_3
α : Type u
x : FreeCommRing α
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hxs : x.IsSupported s
p : α
hps : p ∈ s
n : FreeCommRing α
ih : (map Subtype.val) ((restriction s) n) = n
⊢ of ↑⟨p, hps⟩ * (map Subtype.val) ((restriction s) n) = of p * n

Tactic:
ih

Goal state after:
case refine_3
α : Type u
x : FreeCommRing α
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hxs : x.IsSupported s
p : α
hps : p ∈ s
n : FreeCommRing α
ih : (map Subtype.val) ((restriction s) n) = n
⊢ of ↑⟨p, hps⟩ * n = of p * n
Module: Mathlib.RingTheory.FreeCommRing

Goal state before:
pp: α : Type u
x : FreeRing α
L : FreeMonoid α
hd : α
tl : FreeMonoid α
ih : (FreeMonoid.lift FreeCommRing.of) tl = FreeAbelianGroup.of ↑tl
⊢ FreeCommRing.of hd * (FreeMonoid.lift FreeCommRing.of) tl = FreeAbelianGroup.of ↑(FreeMonoid.of hd * tl)

Tactic:
ih

Goal state after:
α : Type u
x : FreeRing α
L : FreeMonoid α
hd : α
tl : FreeMonoid α
ih : (FreeMonoid.lift FreeCommRing.of) tl = FreeAbelianGroup.of ↑tl
⊢ FreeCommRing.of hd * FreeAbelianGroup.of ↑tl = FreeAbelianGroup.of ↑(FreeMonoid.of hd * tl)