mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
i : ι
hi : i ∈ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ N

pp: case neg
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
i : ι
hi : i ∉ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ N

Tactic:
·

Goal state after:
case pos
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
i : ι
hi : i ∈ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ N
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
i : ι
hi : i ∉ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ N

Tactic:
·

Goal state after:
case neg
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
i : ι
hi : i ∉ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ N
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this✝¹ : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this✝¹.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
a : ι → R
ha : ∀ (i : ι), a i ≠ 0
ha' : ∀ (i : ι), a i • s i ∈ N
A : R := ∏ i, a i
hA : A ≠ 0
φ : M →ₗ[R] M := (LinearMap.lsmul R M) A
this✝ : φ.ker = ⊥
ψ : M ≃ₗ[R] ↥φ.range := LinearEquiv.ofInjective φ ⋯
this : ∀ (i : ι), φ (s i) ∈ N
⊢ ∀ m ∈ range s, φ m ∈ N

Tactic:
rintro _ ⟨i, rfl⟩

Goal state after:
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
b : ι → M
inst✝³ : IsPrincipalIdealRing R
inst✝² : IsDomain R
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : IsTorsionFree R M
this✝¹ : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' s_1)
I : Set ι := this✝¹.choose
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x ↦ ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
N : Submodule R M := span R (range (s ∘ fun x ↦ ↑x))
_sI : ↑I → ↥N := fun i ↦ ⟨s ↑i, ⋯⟩
sI_basis : Basis (↑I) R ↥N := Basis.span indepI
a : ι → R
ha : ∀ (i : ι), a i ≠ 0
ha' : ∀ (i : ι), a i • s i ∈ N
A : R := ∏ i, a i
hA : A ≠ 0
φ : M →ₗ[R] M := (LinearMap.lsmul R M) A
this✝ : φ.ker = ⊥
ψ : M ≃ₗ[R] ↥φ.range := LinearEquiv.ofInjective φ ⋯
this : ∀ (i : ι), φ (s i) ∈ N
i : ι
⊢ φ (s i) ∈ N
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: case inl
ι : Type u_5
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
R : Type u_8
M : Type u_9
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : Basis ι R M
a : ι → R
s : End R M
hs : ∀ (k : ι), s (b k) = a k • b k
i j : ι
⊢ s (if j = j then b i else 0) - (b.end (i, j)) (s (b j)) =
    (if j = j then a i • b i else 0) - if j = j then a j • b i else 0

pp: case inr
ι : Type u_5
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
R : Type u_8
M : Type u_9
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : Basis ι R M
a : ι → R
s : End R M
hs : ∀ (k : ι), s (b k) = a k • b k
i j k : ι
hjk : j ≠ k
⊢ s (if j = k then b i else 0) - (b.end (i, j)) (s (b k)) =
    (if j = k then a i • b i else 0) - if j = k then a j • b i else 0

Tactic:
·

Goal state after:
case inl
ι : Type u_5
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
R : Type u_8
M : Type u_9
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : Basis ι R M
a : ι → R
s : End R M
hs : ∀ (k : ι), s (b k) = a k • b k
i j : ι
⊢ s (if j = j then b i else 0) - (b.end (i, j)) (s (b j)) =
    (if j = j then a i • b i else 0) - if j = j then a j • b i else 0
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: case inr
ι : Type u_5
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
R : Type u_8
M : Type u_9
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : Basis ι R M
a : ι → R
s : End R M
hs : ∀ (k : ι), s (b k) = a k • b k
i j k : ι
hjk : j ≠ k
⊢ s (if j = k then b i else 0) - (b.end (i, j)) (s (b k)) =
    (if j = k then a i • b i else 0) - if j = k then a j • b i else 0

Tactic:
·

Goal state after:
case inr
ι : Type u_5
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
R : Type u_8
M : Type u_9
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : Basis ι R M
a : ι → R
s : End R M
hs : ∀ (k : ι), s (b k) = a k • b k
i j k : ι
hjk : j ≠ k
⊢ s (if j = k then b i else 0) - (b.end (i, j)) (s (b k)) =
    (if j = k then a i • b i else 0) - if j = k then a j • b i else 0
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: case refine_1
A : Type u_3
inst✝ : Semiring A
h : ∀ (n : ℕ) {a b : End A (Fin n → A)}, a * b = 1 → b * a = 1
n : ℕ
f : End A (Fin n → A)
surj : Surjective ⇑f
⊢ Injective ⇑f

pp: case refine_2
A : Type u_3
inst✝ : Semiring A
h : ∀ (n : ℕ) (f : End A (Fin n → A)), Surjective ⇑f → Injective ⇑f
n : ℕ
f g : End A (Fin n → A)
eq : f * g = 1
⊢ g * f = 1

Tactic:
·

Goal state after:
case refine_1
A : Type u_3
inst✝ : Semiring A
h : ∀ (n : ℕ) {a b : End A (Fin n → A)}, a * b = 1 → b * a = 1
n : ℕ
f : End A (Fin n → A)
surj : Surjective ⇑f
⊢ Injective ⇑f
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: case refine_2
A : Type u_3
inst✝ : Semiring A
h : ∀ (n : ℕ) (f : End A (Fin n → A)), Surjective ⇑f → Injective ⇑f
n : ℕ
f g : End A (Fin n → A)
eq : f * g = 1
⊢ g * f = 1

Tactic:
·

Goal state after:
case refine_2
A : Type u_3
inst✝ : Semiring A
h : ∀ (n : ℕ) (f : End A (Fin n → A)), Surjective ⇑f → Injective ⇑f
n : ℕ
f g : End A (Fin n → A)
eq : f * g = 1
⊢ g * f = 1
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
M0 : Submodule R M
ih :
  ∀ N' ≤ M0,
    ∀ x ∈ M0,
      (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) →
        ∀ N ≤ N', ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
N : Submodule R M
N_le_O : N ≤ M0
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_bot : N = ⊥
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))

pp: case neg
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
M0 : Submodule R M
ih :
  ∀ N' ≤ M0,
    ∀ x ∈ M0,
      (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) →
        ∀ N ≤ N', ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
N : Submodule R M
N_le_O : N ≤ M0
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_bot : ¬N = ⊥
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))

Tactic:
·

Goal state after:
case pos
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
M0 : Submodule R M
ih :
  ∀ N' ≤ M0,
    ∀ x ∈ M0,
      (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) →
        ∀ N ≤ N', ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
N : Submodule R M
N_le_O : N ≤ M0
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_bot : N = ⊥
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_1
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x₁ x₂ : Localization (Submonoid.comap (Pi.evalRingHom R i) S)
eq : (mapPiEvalRingHom S) x₁ = (mapPiEvalRingHom S) x₂
⊢ x₁ = x₂

pp: case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x : Localization S
⊢ ∃ a, (mapPiEvalRingHom S) a = x

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x₁ x₂ : Localization (Submonoid.comap (Pi.evalRingHom R i) S)
eq : (mapPiEvalRingHom S) x₁ = (mapPiEvalRingHom S) x₂
⊢ x₁ = x₂
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_1.inl
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
r₁ r₂ : (i : ι) → R i
j : ι
S : Submonoid (R j)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R j) S
s₁ s₂ : ↥T
s : ↥S
hs :
  ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₂, ⋯⟩ * (Pi.evalRingHom R j) r₁) =
    ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₁, ⋯⟩ * (Pi.evalRingHom R j) r₂)
⊢ (↑⟨update 0 j ↑s, ⋯⟩ * (↑s₂ * r₁)) j = (↑⟨update 0 j ↑s, ⋯⟩ * (↑s₁ * r₂)) j

pp: case refine_1.inr
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
r₁ : (i : ι) → R i
s₁ : ↥T
r₂ : (i : ι) → R i
s₂ : ↥T
s : ↥S
hs :
  ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₂, ⋯⟩ * (Pi.evalRingHom R i) r₁) =
    ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₁, ⋯⟩ * (Pi.evalRingHom R i) r₂)
j : ι
ne : j ≠ i
⊢ (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₂ * r₁)) j = (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₁ * r₂)) j

Tactic:
·

Goal state after:
case refine_1.inl
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
r₁ r₂ : (i : ι) → R i
j : ι
S : Submonoid (R j)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R j) S
s₁ s₂ : ↥T
s : ↥S
hs :
  ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₂, ⋯⟩ * (Pi.evalRingHom R j) r₁) =
    ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₁, ⋯⟩ * (Pi.evalRingHom R j) r₂)
⊢ (↑⟨update 0 j ↑s, ⋯⟩ * (↑s₂ * r₁)) j = (↑⟨update 0 j ↑s, ⋯⟩ * (↑s₁ * r₂)) j
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_1.inr
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
r₁ : (i : ι) → R i
s₁ : ↥T
r₂ : (i : ι) → R i
s₂ : ↥T
s : ↥S
hs :
  ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₂, ⋯⟩ * (Pi.evalRingHom R i) r₁) =
    ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₁, ⋯⟩ * (Pi.evalRingHom R i) r₂)
j : ι
ne : j ≠ i
⊢ (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₂ * r₁)) j = (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₁ * r₂)) j

Tactic:
·

Goal state after:
case refine_1.inr
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
r₁ : (i : ι) → R i
s₁ : ↥T
r₂ : (i : ι) → R i
s₂ : ↥T
s : ↥S
hs :
  ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₂, ⋯⟩ * (Pi.evalRingHom R i) r₁) =
    ↑s * (↑⟨(Pi.evalRingHom R i) ↑s₁, ⋯⟩ * (Pi.evalRingHom R i) r₂)
j : ι
ne : j ≠ i
⊢ (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₂ * r₁)) j = (↑⟨update 0 i ↑s, ⋯⟩ * (↑s₁ * r₂)) j
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x : Localization S
⊢ ∃ a, (mapPiEvalRingHom S) a = x

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x : Localization S
⊢ ∃ a, (mapPiEvalRingHom S) a = x
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
⊢ Injective ⇑(Algebra.ofId R S)

pp: case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
⊢ Surjective ⇑(Algebra.ofId R S)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
⊢ Injective ⇑(Algebra.ofId R S)
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
⊢ Surjective ⇑(Algebra.ofId R S)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M N : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
H : M ≤ IsUnit.submonoid R
⊢ Surjective ⇑(Algebra.ofId R S)
Module: Mathlib.RingTheory.Adjoin.Basic

Goal state before:
pp: case a
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
t : Set B
⊢ adjoin R (⇑(LinearMap.inl R A B) '' (s ∪ {1}) ∪ ⇑(LinearMap.inr R A B) '' (t ∪ {1})) ≤ (adjoin R s).prod (adjoin R t)

pp: case a
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
t : Set B
⊢ (adjoin R s).prod (adjoin R t) ≤ adjoin R (⇑(LinearMap.inl R A B) '' (s ∪ {1}) ∪ ⇑(LinearMap.inr R A B) '' (t ∪ {1}))

Tactic:
·

Goal state after:
case a
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
t : Set B
⊢ adjoin R (⇑(LinearMap.inl R A B) '' (s ∪ {1}) ∪ ⇑(LinearMap.inr R A B) '' (t ∪ {1})) ≤ (adjoin R s).prod (adjoin R t)
Module: Mathlib.RingTheory.Adjoin.Basic

Goal state before:
pp: case a
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
t : Set B
⊢ (adjoin R s).prod (adjoin R t) ≤ adjoin R (⇑(LinearMap.inl R A B) '' (s ∪ {1}) ∪ ⇑(LinearMap.inr R A B) '' (t ∪ {1}))

Tactic:
·

Goal state after:
case a
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
t : Set B
⊢ (adjoin R s).prod (adjoin R t) ≤ adjoin R (⇑(LinearMap.inl R A B) '' (s ∪ {1}) ∪ ⇑(LinearMap.inr R A B) '' (t ∪ {1}))
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: case inl
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommRing P
A : Type u_4
inst✝ : CommRing A
K : Type u_5
z : ℚ
x1 : ℤ
x2 : ↥ℤ⁰
hx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1
hx2 : ↑x2 < 0
⊢ ∃ x, z * (algebraMap ℤ ℚ) ↑x.2 = (algebraMap ℤ ℚ) x.1

pp: case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommRing P
A : Type u_4
inst✝ : CommRing A
K : Type u_5
z : ℚ
x1 : ℤ
x2 : ↥ℤ⁰
hx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1
hx2 : 0 < ↑x2
⊢ ∃ x, z * (algebraMap ℤ ℚ) ↑x.2 = (algebraMap ℤ ℚ) x.1

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommRing P
A : Type u_4
inst✝ : CommRing A
K : Type u_5
z : ℚ
x1 : ℤ
x2 : ↥ℤ⁰
hx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1
hx2 : ↑x2 < 0
⊢ ∃ x, z * (algebraMap ℤ ℚ) ↑x.2 = (algebraMap ℤ ℚ) x.1
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case map_units
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ (y : ↥M), IsUnit ((algebraMap R P) ↑y)

pp: case surj
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ (z : P), ∃ x, z * (algebraMap R P) ↑x.2 = (algebraMap R P) x.1

pp: case exists_of_eq
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ {x y : R}, (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * x = ↑c * y

Tactic:
·

Goal state after:
case map_units
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ (y : ↥M), IsUnit ((algebraMap R P) ↑y)
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case surj
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ (z : P), ∃ x, z * (algebraMap R P) ↑x.2 = (algebraMap R P) x.1

pp: case exists_of_eq
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ {x y : R}, (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * x = ↑c * y

Tactic:
·

Goal state after:
case surj
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ (z : P), ∃ x, z * (algebraMap R P) ↑x.2 = (algebraMap R P) x.1
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommRing P
A : Type u_4
inst✝ : CommRing A
K : Type u_5
z : ℚ
x1 : ℤ
x2 : ↥ℤ⁰
hx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1
hx2 : 0 < ↑x2
⊢ ∃ x, z * (algebraMap ℤ ℚ) ↑x.2 = (algebraMap ℤ ℚ) x.1

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommRing P
A : Type u_4
inst✝ : CommRing A
K : Type u_5
z : ℚ
x1 : ℤ
x2 : ↥ℤ⁰
hx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1
hx2 : 0 < ↑x2
⊢ ∃ x, z * (algebraMap ℤ ℚ) ↑x.2 = (algebraMap ℤ ℚ) x.1
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case exists_of_eq
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ {x y : R}, (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * x = ↑c * y

Tactic:
·

Goal state after:
case exists_of_eq
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ {x y : R}, (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * x = ↑c * y
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : CommRing R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : FaithfulSMul R K
surj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y
inj : Function.Injective ⇑(algebraMap R K)
this✝ : NoZeroDivisors R
this : Nontrivial R
z : K
x : R
eq : z = (algebraMap R K) x / (algebraMap R K) 0
⊢ ∃ x, z * (algebraMap R K) ↑x.2 = (algebraMap R K) x.1

pp: case inr
R : Type u_1
inst✝³ : CommRing R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : FaithfulSMul R K
surj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y
inj : Function.Injective ⇑(algebraMap R K)
this✝ : NoZeroDivisors R
this : Nontrivial R
z : K
x y : R
eq : z = (algebraMap R K) x / (algebraMap R K) y
hy : y ≠ 0
⊢ ∃ x, z * (algebraMap R K) ↑x.2 = (algebraMap R K) x.1

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝³ : CommRing R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : FaithfulSMul R K
surj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y
inj : Function.Injective ⇑(algebraMap R K)
this✝ : NoZeroDivisors R
this : Nontrivial R
z : K
x : R
eq : z = (algebraMap R K) x / (algebraMap R K) 0
⊢ ∃ x, z * (algebraMap R K) ↑x.2 = (algebraMap R K) x.1