mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.PrincipalIdealDomain

Goal state before:
pp: case inl
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
S : Ideal R
hpi : S.IsPrime
hS : S ≠ ⊥
T : Ideal R
x : R
hST : S ≤ T
hxS : x ∉ S
hxT : x ∈ T
z : R
hz : generator S = generator T * z
h : generator T ∈ S
⊢ 1 ∈ T

Tactic:
have hTS : T ≤ S := by
          rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]

Goal state after:
case inl
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
S : Ideal R
hpi : S.IsPrime
hS : S ≠ ⊥
T : Ideal R
x : R
hST : S ≤ T
hxS : x ∉ S
hxT : x ∈ T
z : R
hz : generator S = generator T * z
h : generator T ∈ S
hTS : T ≤ S
⊢ 1 ∈ T
Module: Mathlib.LinearAlgebra.Dimension.StrongRankCondition

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
x : M
h : Bijective ⇑(LinearMap.toSpanSingleton R M x)
⊢ finrank R R = 1

Tactic:
finrank_self

Goal state after:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
x : M
h : Bijective ⇑(LinearMap.toSpanSingleton R M x)
⊢ 1 = 1
Module: Mathlib.LinearAlgebra.Dimension.StrongRankCondition

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring S
inst✝¹ : Algebra R S
inst✝ : StrongRankCondition R
h : Bijective ⇑(algebraMap R S)
⊢ finrank R R = 1

Tactic:
finrank_self

Goal state after:
R : Type u_2
S : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring S
inst✝¹ : Algebra R S
inst✝ : StrongRankCondition R
h : Bijective ⇑(algebraMap R S)
⊢ 1 = 1
Module: Mathlib.Data.Matrix.Mul

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type v
inst✝² : NonUnitalSemiring α
inst✝¹ : Fintype m
inst✝ : Fintype n
L : Matrix l m α
M : Matrix m n α
N : Matrix n o α
⊢ L * M * N = L * (M * N)

Tactic:
by
  ext
  apply dotProduct_assoc

Goal state after:
Module: Mathlib.Data.Matrix.Basis

Goal state before:
pp: case intro.intro
m : Type u_2
n : Type u_3
α : Type u_7
β : Type u_8
inst✝⁵ : DecidableEq m
inst✝⁴ : DecidableEq n
inst✝³ : Finite m
inst✝² : Finite n
inst✝¹ : AddCommMonoid α
inst✝ : AddCommMonoid β
f g : Matrix m n α →+ β
h : ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (singleAddMonoidHom i j)
val✝¹ : Fintype m
val✝ : Fintype n
x : Matrix m n α
⊢ f (∑ i, ∑ j, single i j (x i j)) = g (∑ i, ∑ j, single i j (x i j))

Tactic:
simp_rw [map_sum]

Goal state after:
case intro.intro
m : Type u_2
n : Type u_3
α : Type u_7
β : Type u_8
inst✝⁵ : DecidableEq m
inst✝⁴ : DecidableEq n
inst✝³ : Finite m
inst✝² : Finite n
inst✝¹ : AddCommMonoid α
inst✝ : AddCommMonoid β
f g : Matrix m n α →+ β
h : ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (singleAddMonoidHom i j)
val✝¹ : Fintype m
val✝ : Fintype n
x : Matrix m n α
⊢ ∑ x_1, ∑ x_2, f (single x_1 x_2 (x x_1 x_2)) = ∑ x_1, ∑ x_2, g (single x_1 x_2 (x x_1 x_2))
Module: Mathlib.Data.Matrix.Mul

Goal state before:
pp: m : Type u_2
n : Type u_3
o : Type u_4
α : Type v
inst✝² : NonUnitalSemiring α
inst✝¹ : Fintype n
inst✝ : Fintype m
v : m → α
M : Matrix m n α
N : Matrix n o α
⊢ v ᵥ* M ᵥ* N = v ᵥ* (M * N)

Tactic:
by
  ext
  apply dotProduct_assoc

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Basic

Goal state before:
pp: case intro
R : Type u_1
M : Type u_2
ι : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
⊢ Submodule.span R ↑(Finset.image (⇑b) Finset.univ) = ⊤

Tactic:
simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]

Goal state after:
Module: Mathlib.RingTheory.AlgebraTower

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R S
inst✝² : Module S A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
ι : Type u_5
b : ι → S
ι' : Type u_6
c : ι' → A
hb : LinearIndependent R b
hc : LinearIndependent S c
⊢ Function.Injective ⇑(linearCombination R fun p ↦ b p.2 • c p.1)

Tactic:
linearCombination_smul

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R S
inst✝² : Module S A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
ι : Type u_5
b : ι → S
ι' : Type u_6
c : ι' → A
hb : LinearIndependent R b
hc : LinearIndependent S c
⊢ Function.Injective
    ⇑(↑R (linearCombination S c) ∘ₗ mapRange.linearMap (linearCombination R b) ∘ₗ ↑(curryLinearEquiv R))
Module: Mathlib.RingTheory.AlgebraTower

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R S
inst✝² : Module S A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
ι : Type u_5
ι' : Type u_6
b : Basis ι R S
c : Basis ι' S A
ij : ι × ι'
⊢ (b.smulTower c) ij = b ij.1 • c ij.2

Tactic:
obtain ⟨i, j⟩ := ij

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : Semiring S
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R S
inst✝² : Module S A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
ι : Type u_5
ι' : Type u_6
b : Basis ι R S
c : Basis ι' S A
i : ι
j : ι'
⊢ (b.smulTower c) (i, j) = b (i, j).1 • c (i, j).2
Module: Mathlib.Data.Finsupp.Multiset

Goal state before:
pp: ι : Type u_3
m n : ι →₀ ℕ
h : m < n
⊢ (m.sum fun x ↦ id) < n.sum fun x ↦ id

Tactic:
rw [← card_toMultiset, ← card_toMultiset]
  apply Multiset.card_lt_card
  exact toMultiset_strictMono h

Goal state after:
Module: Mathlib.RingTheory.TwoSidedIdeal.Basic

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
I : TwoSidedIdeal R
⊢ Function.Injective fun t ↦ {r | t.ringCon r 0}

Tactic:
by
    rintro ⟨t₁⟩ ⟨t₂⟩ (h : {x | _} = {x | _})
    congr 1
    refine RingCon.ext fun a b ↦ ⟨fun H ↦ ?_, fun H ↦ ?_⟩
    · have H' : a - b ∈ {x | t₁ x 0} := sub_self b ▸ t₁.sub H (t₁.refl b)
      rw [h] at H'
      convert! t₂.add H' (t₂.refl b) using 1 <;> abel
    · have H' : a - b ∈ {x | t₂ x 0} := sub_self b ▸ t₂.sub H (t₂.refl b)
      rw [← h] at H'
      convert! t₁.add H' (t₁.refl b) using 1 <;> abel

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Free

Goal state before:
pp: case pos
R : Type u
inst✝¹¹ : Semiring R
inst✝¹⁰ : StrongRankCondition R
S : Type u_2
T : Type u_3
inst✝⁹ : Semiring S
inst✝⁸ : Semiring T
inst✝⁷ : Module R T
inst✝⁶ : Module S T
inst✝⁵ : Module R S
inst✝⁴ : IsScalarTower R S T
inst✝³ : IsScalarTower S T T
inst✝² : FaithfulSMul S T
inst✝¹ : Module.Finite S T
inst✝ : Free R S
H : Module.Finite R S
this : Module.Finite R T
⊢ finrank R S ≤ finrank R T

Tactic:
exact finrank_bot_le_finrank_of_isScalarTower R S T

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

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Semiring B
inst✝ : Algebra R B
S T U : Subalgebra R A
h✝ : S ≤ T
this : Module ↥S ↥T := (inclusion h✝).toModule
x y : ↥S
h : ∀ (a : ↥T), x • a = y • a
⊢ ↑x = ↑y

Tactic:
by
    convert! Subtype.ext_iff.mp (h 1) using 1 <;> exact (mul_one _).symm

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.Lattice

Goal state before:
pp: case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
L : List (List A)
HL : ∀ t ∈ L, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ (List.map List.prod L).sum ∈ span R ↑(Submonoid.closure s)

Tactic:
induction L with
    | nil => exact zero_mem _
    | cons hd tl ih => ?_

Goal state after:
case a.cons
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : List A
tl : List (List A)
ih :
  (∀ t ∈ tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → (List.map List.prod tl).sum ∈ span R ↑(Submonoid.closure s)
HL : ∀ t ∈ hd :: tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ (List.map List.prod (hd :: tl)).sum ∈ span R ↑(Submonoid.closure s)
Module: Mathlib.Algebra.Algebra.Subalgebra.Lattice

Goal state before:
pp: case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ span R ↑(Submonoid.closure s) ≤ Subalgebra.toSubmodule (adjoin R s)

Tactic:
refine span_le.2 ?_

Goal state after:
case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ ↑(Submonoid.closure s) ⊆ ↑(Subalgebra.toSubmodule (adjoin R s))
Module: Mathlib.Algebra.Module.Projective

Goal state before:
pp: R : Type u_8
S : Type u_9
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
M : Type u_10
N : Type u_11
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R M
inst✝³ : Module S N
σ : R →+* S
σ' : S →+* R
inst✝² : RingHomInvPair σ σ'
inst✝¹ : RingHomInvPair σ' σ
e₂ : M ≃ₛₗ[σ] N
inst✝ : Projective R M
e₁ : R ≃+* S := RingHomInvPair.toRingEquiv σ σ'
f : M →ₗ[R] M →₀ R
hf : Function.LeftInverse ⇑(linearCombination R id) ⇑f
g : N →ₗ[S] N →₀ S :=
  { toFun := fun x ↦ mapRange ⇑e₁ ⋯ ((Finsupp.equivCongrLeft ↑e₂) (f (e₂.symm x))), map_add' := ⋯, map_smul' := ⋯ }
x : N
⊢ (linearCombination S id) (g x) = x

Tactic:
replace hf := congr(e₂ $(hf (e₂.symm x)))

Goal state after:
R : Type u_8
S : Type u_9
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
M : Type u_10
N : Type u_11
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R M
inst✝³ : Module S N
σ : R →+* S
σ' : S →+* R
inst✝² : RingHomInvPair σ σ'
inst✝¹ : RingHomInvPair σ' σ
e₂ : M ≃ₛₗ[σ] N
inst✝ : Projective R M
e₁ : R ≃+* S := RingHomInvPair.toRingEquiv σ σ'
f : M →ₗ[R] M →₀ R
g : N →ₗ[S] N →₀ S :=
  { toFun := fun x ↦ mapRange ⇑e₁ ⋯ ((Finsupp.equivCongrLeft ↑e₂) (f (e₂.symm x))), map_add' := ⋯, map_smul' := ⋯ }
x : N
hf : e₂ ((linearCombination R id) (f (e₂.symm x))) = e₂ (e₂.symm x)
⊢ (linearCombination S id) (g x) = x
Module: Mathlib.Data.Matrix.Composition

Goal state before:
pp: I : Type u_1
J : Type u_2
R : Type u_5
inst✝³ : DecidableEq I
inst✝² : DecidableEq J
inst✝¹ : Zero R
inst✝ : One R
i✝ j✝ : I × J
⊢ (comp I I J J R) 1 i✝ j✝ = 1 i✝ j✝

Tactic:
simp only [comp, Equiv.coe_fn_mk, one_apply, apply_ite]

Goal state after:
I : Type u_1
J : Type u_2
R : Type u_5
inst✝³ : DecidableEq I
inst✝² : DecidableEq J
inst✝¹ : Zero R
inst✝ : One R
i✝ j✝ : I × J
⊢ if i✝ = j✝ then (if i✝.1 = j✝.1 then 1 else 0) i✝.2 j✝.2 = 1 else (if i✝.1 = j✝.1 then 1 else 0) i✝.2 j✝.2 = 0
Module: Mathlib.Algebra.Star.Basic

Goal state before:
pp: R✝ : Type u
R : Type u_1
inst✝² : MulOneClass R
inst✝¹ : StarMul R
r : R
inst✝ : Invertible r
⊢ star (r * ⅟r) = 1

Tactic:
mul_invOf_self,

Goal state after:
R✝ : Type u
R : Type u_1
inst✝² : MulOneClass R
inst✝¹ : StarMul R
r : R
inst✝ : Invertible r
⊢ star 1 = 1
Module: Mathlib.Algebra.Star.SelfAdjoint

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝¹ : AddCommGroup R
inst✝ : StarAddMonoid R
x : R
hx : star x = -x
⊢ star (-x) = - -x

Tactic:
simp only [hx, star_neg]

Goal state after:
Module: Mathlib.Algebra.Star.Module

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : StarMul R
inst✝⁵ : TrivialStar R
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : StarAddMonoid A
inst✝¹ : StarModule R A
inst✝ : Invertible 2
x : A
hx : IsSelfAdjoint x
⊢ ⅟2 • 0 = ↑0

Tactic:
smul_zero,

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : StarMul R
inst✝⁵ : TrivialStar R
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : StarAddMonoid A
inst✝¹ : StarModule R A
inst✝ : Invertible 2
x : A
hx : IsSelfAdjoint x
⊢ 0 = ↑0
Module: Mathlib.Data.Matrix.Block

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
M : Matrix (n ⊕ o) (l ⊕ m) α
i : n ⊕ o
j : l ⊕ m
⊢ fromBlocks M.toBlocks₁₁ M.toBlocks₁₂ M.toBlocks₂₁ M.toBlocks₂₂ i j = M i j

Tactic:
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl

Goal state after:
Module: Mathlib.Data.Matrix.Block

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝ : Zero α
i : n ⊕ o
j : l ⊕ m
⊢ fromBlocks 0 0 0 0 i j = 0 i j

Tactic:
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl

Goal state after:
Module: Mathlib.Data.Matrix.Block

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝ : Add α
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
A' : Matrix n l α
B' : Matrix n m α
C' : Matrix o l α
D' : Matrix o m α
i : n ⊕ o
j : l ⊕ m
⊢ (fromBlocks A B C D + fromBlocks A' B' C' D') i j = fromBlocks (A + A') (B + B') (C + C') (D + D') i j

Tactic:
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_1
b : Basis ι R M
N : Submodule R M
ϕ : M →ₗ[R] R
hϕ : ∀ (ψ : M →ₗ[R] R), ¬Submodule.map ϕ N < Submodule.map ψ N
inst✝ : (Submodule.map ϕ N).IsPrincipal
hgen : generator (Submodule.map ϕ N) = 0
⊢ N = ⊥

Tactic:
rw [Submodule.eq_bot_iff]

Goal state after:
R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_1
b : Basis ι R M
N : Submodule R M
ϕ : M →ₗ[R] R
hϕ : ∀ (ψ : M →ₗ[R] R), ¬Submodule.map ϕ N < Submodule.map ψ N
inst✝ : (Submodule.map ϕ N).IsPrincipal
hgen : generator (Submodule.map ϕ N) = 0
⊢ ∀ x ∈ N, x = 0
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_1
N O : Submodule R M
b : Basis ι R ↥O
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst✝ : (ϕ.submoduleImage N).IsPrincipal
hgen : generator (ϕ.submoduleImage N) = 0
⊢ N = ⊥

Tactic:
rw [Submodule.eq_bot_iff]

Goal state after:
R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_1
N O : Submodule R M
b : Basis ι R ↥O
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst✝ : (ϕ.submoduleImage N).IsPrincipal
hgen : generator (ϕ.submoduleImage N) = 0
⊢ ∀ x ∈ N, x = 0
Module: Mathlib.LinearAlgebra.TensorProduct.Tower

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
x✝¹ : A ⊗[R] ↥p
x✝ : x✝¹ ∈ {t | ∃ m n, m ⊗ₜ[R] n = t}
a : A
m : ↥p
h : a ⊗ₜ[R] m = x✝¹
⊢ x✝¹ ∈ ↑(comap (LinearMap.baseChange A p.subtype) (span A ↑(map ((TensorProduct.mk R A M) 1) p)))

Tactic:
rw [← h, SetLike.mem_coe, mem_comap, LinearMap.baseChange_tmul, ← mul_one a, ← smul_eq_mul,
      ← smul_tmul']

Goal state after:
case refine_1
R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
x✝¹ : A ⊗[R] ↥p
x✝ : x✝¹ ∈ {t | ∃ m n, m ⊗ₜ[R] n = t}
a : A
m : ↥p
h : a ⊗ₜ[R] m = x✝¹
⊢ a • 1 ⊗ₜ[R] p.subtype m ∈ span A ↑(map ((TensorProduct.mk R A M) 1) p)
Module: Mathlib.LinearAlgebra.TensorProduct.Tower

Goal state before:
pp: R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p q : Submodule R M
h : p ≤ q
⊢ (LinearMap.baseChange A p.subtype).range ≤ baseChange A q

Tactic:
LinearMap.baseChange,

Goal state after:
R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p q : Submodule R M
h : p ≤ q
⊢ (AlgebraTensorModule.map LinearMap.id p.subtype).range ≤ baseChange A q
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
s : ↥S
⊢ 0 /ₒ s = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
s : ↥S
⊢ 0 /ₒ s = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: case c
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
r : R
s : ↥S
⊢ 0 * (r /ₒ s) = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
case c
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
r : R
s : ↥S
⊢ 0 /ₒ 1 * (r /ₒ s) = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: case c
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
r : R
s : ↥S
⊢ r /ₒ s * 0 = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
case c
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
r : R
s : ↥S
⊢ r /ₒ s * (0 /ₒ 1) = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
⊢ 0 = 1 /ₒ 1 ↔ 0 ∈ S

Tactic:
OreLocalization.zero_def,

Goal state after:
R : Type u_1
inst✝¹ : MonoidWithZero R
S : Submonoid R
inst✝ : OreSet S
⊢ 0 /ₒ 1 = 1 /ₒ 1 ↔ 0 ∈ S
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
k : Type u_2
l : Type u_3
m : Type u_4
n : Type u_5
inst✝¹ : DecidableEq n
inst✝ : Fintype n
f g : (n → R) →ₗ[R] m → R
⊢ (of fun i j ↦ (f + g) (Pi.single j 1) i) = (of fun i j ↦ f (Pi.single j 1) i) + of fun i j ↦ g (Pi.single j 1) i

Tactic:
by
    ext i j
    simp only [Pi.add_apply, LinearMap.add_apply, of_apply, Matrix.add_apply]

Goal state after:
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
s : ↥S
⊢ 0 /ₒ s = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
s : ↥S
⊢ 0 /ₒ s = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
x : OreLocalization S R
⊢ x • 0 = 0

Tactic:
induction x with | _ r s

Goal state after:
case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r : R
s : ↥S
⊢ (r /ₒ s) • 0 = 0
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r : R
s : ↥S
⊢ (r /ₒ s) • 0 = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r : R
s : ↥S
⊢ (r /ₒ s) • (0 /ₒ 1) = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r : R
s : ↥S
⊢ r • 0 /ₒ s = 0 /ₒ 1

Tactic:
smul_zero,

Goal state after:
case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r : R
s : ↥S
⊢ 0 /ₒ s = 0 /ₒ 1
Module: Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero

Goal state before:
pp: M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
P : Type u_3
inst✝ : CommMonoidWithZero P
f : S.LocalizationMap N
⊢ f 0 = 0

Tactic:
by
    have ⟨ms, eq⟩ := f.surj 0
    rw [← zero_mul, map_mul, ← eq, zero_mul, mul_zero]

Goal state after:
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : DistribMulAction R X
x y : OreLocalization S X
⊢ x + y = y + x

Tactic:
induction x with | _ r s

Goal state after:
case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : DistribMulAction R X
y : OreLocalization S X
r : X
s : ↥S
⊢ r /ₒ s + y = y + r /ₒ s
Module: Mathlib.RingTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddGroup X
inst✝ : DistribMulAction R X
x : OreLocalization S X
⊢ -x + x = 0

Tactic:
induction x with | _ r s

Goal state after:
case c
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddGroup X
inst✝ : DistribMulAction R X
r : X
s : ↥S
⊢ -(r /ₒ s) + r /ₒ s = 0
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
x : OreLocalization S X
⊢ 0 • x = 0

Tactic:
induction x with | _ r s

Goal state after:
case c
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r : X
s : ↥S
⊢ 0 • (r /ₒ s) = 0
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: case c
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r : X
s : ↥S
⊢ 0 • (r /ₒ s) = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
case c
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r : X
s : ↥S
⊢ (0 /ₒ 1) • (r /ₒ s) = 0
Module: Mathlib.RingTheory.OreLocalization.NonZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝³ : MonoidWithZero R
inst✝² : Nontrivial R
inst✝¹ : OreSet R⁰
inst✝ : NoZeroDivisors R
x : OreLocalization R⁰ R
h : x ≠ 0
⊢ x * x⁻¹ = 1

Tactic:
induction x with | _ r s

Goal state after:
case c
R : Type u_1
inst✝³ : MonoidWithZero R
inst✝² : Nontrivial R
inst✝¹ : OreSet R⁰
inst✝ : NoZeroDivisors R
r : R
s : ↥R⁰
h : r /ₒ s ≠ 0
⊢ r /ₒ s * (r /ₒ s)⁻¹ = 1
Module: Mathlib.RingTheory.OreLocalization.NonZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝³ : MonoidWithZero R
inst✝² : Nontrivial R
inst✝¹ : OreSet R⁰
inst✝ : NoZeroDivisors R
⊢ 0⁻¹ = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
R : Type u_1
inst✝³ : MonoidWithZero R
inst✝² : Nontrivial R
inst✝¹ : OreSet R⁰
inst✝ : NoZeroDivisors R
⊢ (0 /ₒ 1)⁻¹ = 0 /ₒ 1
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: case c
R : Type u_1
inst✝⁵ : Semiring R
S : Submonoid R
inst✝⁴ : OreSet S
X : Type u_2
inst✝³ : AddCommMonoid X
inst✝² : Module R X
R₀ : Type ?u.16
inst✝¹ : CommSemiring R₀
inst✝ : Algebra R₀ R
r : R₀
r₁ : R
s₁ : ↥S
⊢ (numeratorRingHom.comp (algebraMap R₀ R)) r * (r₁ /ₒ s₁) = r₁ /ₒ s₁ * (numeratorRingHom.comp (algebraMap R₀ R)) r

Tactic:
| _ r₁ s₁
     => 

Goal state after:
case c
R : Type u_1
inst✝⁵ : Semiring R
S : Submonoid R
inst✝⁴ : OreSet S
X : Type u_2
inst✝³ : AddCommMonoid X
inst✝² : Module R X
R₀ : Type ?u.16
inst✝¹ : CommSemiring R₀
inst✝ : Algebra R₀ R
r : R₀
r₁ : R
s₁ : ↥S
⊢ (numeratorRingHom.comp (algebraMap R₀ R)) r * (r₁ /ₒ s₁) = r₁ /ₒ s₁ * (numeratorRingHom.comp (algebraMap R₀ R)) r
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: R : Type u_1
inst✝⁴ : Semiring R
S : Submonoid R
inst✝³ : OreSet S
X : Type u_2
inst✝² : AddCommMonoid X
inst✝¹ : Module R X
T : Type u_3
inst✝ : Semiring T
f : R →+* T
fS : ↥S →* Tˣ
hf : ∀ (s : ↥S), f ↑s = ↑(fS s)
⊢ (universalMulHom (↑f) fS ⋯) 0 = 0

Tactic:
OreLocalization.zero_def,

Goal state after:
R : Type u_1
inst✝⁴ : Semiring R
S : Submonoid R
inst✝³ : OreSet S
X : Type u_2
inst✝² : AddCommMonoid X
inst✝¹ : Module R X
T : Type u_3
inst✝ : Semiring T
f : R →+* T
fS : ↥S →* Tˣ
hf : ∀ (s : ↥S), f ↑s = ↑(fS s)
⊢ (universalMulHom (↑f) fS ⋯) (0 /ₒ 1) = 0