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

Goal state before:
pp: case c.c.e_a.e_a
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)
r₁ : R
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
h₃ : ↑(s₃ * s₁) = r₃ * ↑s₂
h₃' : s₃ * s₁ = ⟨r₃ * ↑s₂, ⋯⟩
⊢ f r₃ * f ↑s₂ = f ↑⟨r₃ * ↑s₂, ⋯⟩

Tactic:
]

Goal state after:
case c.c.e_a.e_a
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)
r₁ : R
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
h₃ : ↑(s₃ * s₁) = r₃ * ↑s₂
h₃' : s₃ * s₁ = ⟨r₃ * ↑s₂, ⋯⟩
⊢ f r₃ * f ↑s₂ = f ↑⟨r₃ * ↑s₂, ⋯⟩
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
b : Basis m R S
r : R
i✝ j✝ : m
⊢ Matrix.diagonal (fun x ↦ r) i✝ j✝ = Matrix.diagonal (fun i ↦ r) i✝ j✝

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
b : Basis m R S
x : S
i j : m
⊢ (b.repr (x * b j)) i = (b.repr (x * b j)) i

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
b : Basis m R S
x x' : S
h : (leftMulMatrix b) x = (leftMulMatrix b) x'
⊢ ((lmul R S) x') 1 = ((lmul R S) x') 1

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
S : Submonoid R
inst✝ : OreSet S
hS : S ≤ nonZeroDivisorsLeft R
r₁ r₂ : R
h : ∃ u v, u • r₂ = v • r₁ ∧ ↑u * ↑1 = v * ↑1
⊢ r₁ = r₂

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
S : Submonoid R
inst✝ : OreSet S
hS : S ≤ nonZeroDivisorsLeft R
r₁ r₂ : R
h : ∃ u v, u • r₂ = v • r₁ ∧ ↑u * ↑1 = v * ↑1
⊢ r₁ = r₂
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
S : Submonoid R
inst✝ : OreSet S
hS : S ≤ nonZeroDivisorsLeft R
r₁ r₂ : R
u : ↥S
v : R
h₁ : ↑u * (r₂ - r₁) = 0
h₂ : ↑u = v
⊢ r₁ = r₂

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
S : Submonoid R
inst✝ : OreSet S
hS : S ≤ nonZeroDivisorsLeft R
r₁ r₂ : R
u : ↥S
v : R
h₁ : ↑u * (r₂ - r₁) = 0
h₂ : ↑u = v
⊢ r₁ = r₂
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case right
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
⊢ s i ∈ N

Tactic:
]

Goal state after:
case right
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
⊢ s i ∈ N
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring T
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra S T
inst✝⁵ : Algebra R T
inst✝⁴ : IsScalarTower R S T
m : Type u_4
n : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b : Basis m R S
c : Basis n S T
x : S
i : m
k : n
j : m
k' : n
⊢ (leftMulMatrix b) (if (i, k).2 = (j, k').2 then (algebraMap S S) x else 0) (i, k).1 (j, k').1 =
    if (i, k).2 = (j, k').2 then (leftMulMatrix b) x (i, k).1 (j, k').1 else 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring T
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra S T
inst✝⁵ : Algebra R T
inst✝⁴ : IsScalarTower R S T
m : Type u_4
n : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b : Basis m R S
c : Basis n S T
x : S
i : m
k : n
j : m
k' : n
⊢ (leftMulMatrix b) (if (i, k).2 = (j, k').2 then (algebraMap S S) x else 0) (i, k).1 (j, k').1 =
    if (i, k).2 = (j, k').2 then (leftMulMatrix b) x (i, k).1 (j, k').1 else 0
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 : ∃ 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
a : ι → R
ha : ∀ (i : ι), a i ≠ 0
ha' : ∀ (i : ι), a i • s i ∈ N
A : R := ∏ i, a i
⊢ ∀ a_1 ∈ Finset.univ, a a_1 ≠ 0

Tactic:
]

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 : ∃ 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
a : ι → R
ha : ∀ (i : ι), a i ≠ 0
ha' : ∀ (i : ι), a i • s i ∈ N
A : R := ∏ i, a i
⊢ ∀ a_1 ∈ Finset.univ, a a_1 ≠ 0
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring T
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra S T
inst✝⁵ : Algebra R T
inst✝⁴ : IsScalarTower R S T
m : Type u_4
n : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b : Basis m R S
c : Basis n S T
x : S
i j : m
k : n
⊢ (leftMulMatrix b) x i j = (leftMulMatrix b) x i j

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring T
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra S T
inst✝⁵ : Algebra R T
inst✝⁴ : IsScalarTower R S T
m : Type u_4
n : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b : Basis m R S
c : Basis n S T
x : S
i j : m
k k' : n
h : k ≠ k'
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ ∑ j, Matrix.single ij.1 ij.2 1 j k • b₂ j = if ij.2 = k then b₂ ij.1 else 0

Tactic:
]

Goal state after:
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ ∑ j, Matrix.single ij.1 ij.2 1 j k • b₂ j = if ij.2 = k then b₂ ij.1 else 0
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:
]

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
⊢ ∀ m ∈ range s, φ m ∈ 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✝ : ∃ 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
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 ⋯
i : ι
⊢ (∏ j ∈ {i}ᶜ, a j) • a i • s i = (∏ i ∈ {i}ᶜ, a i) • a i • s i

Tactic:
]

Goal state after:
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
⊢ (n : ℕ) × Basis (Fin n) R M

Tactic:
classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    have := exists_maximal_linearIndepOn R s
    let I : Set ι := this.choose
    obtain
      ⟨indepI : LinearIndependent R (s ∘ (fun x => x) : I → M), hI :
        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
      this.choose_spec
    let N := span R (range <| (s ∘ (fun x => x) : I → M))
    -- same as `span R (s '' I)` but more convenient
    let _sI : I → N := fun i ↦ ⟨s i.1, subset_span (mem_range_self i)⟩
    -- `s` restricted to `I` is a basis of `N`
    let sI_basis : Basis I R N := Basis.span indepI
    -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
    have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N := by
      intro i
      by_cases hi : i ∈ I
      · use 1, zero_ne_one.symm
        rw [one_smul]
        exact subset_span (mem_range_self (⟨i, hi⟩ : I))
      · simpa [image_eq_range s I] using! hI i hi
    choose a ha ha' using exists_a
    let A := ∏ i, a i
    have hA : A ≠ 0 := by
      rw [Finset.prod_ne_zero_iff]
      simpa using! ha
    -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
    let φ : M →ₗ[R] M := LinearMap.lsmul R M A
    have : LinearMap.ker φ = ⊥ := LinearMap.ker_lsmul hA
    let ψ := LinearEquiv.ofInjective φ (LinearMap.ker_eq_bot.mp this)
    have : LinearMap.range φ ≤ N := by
      -- as announced, `A • M ⊆ N`
      suffices ∀ i, φ (s i) ∈ N by
        rw [LinearMap.range_eq_map, ← hs, map_span_le]
        rintro _ ⟨i, rfl⟩
        apply this
      intro i
      calc
        (∏ j ∈ {i}ᶜ, a j) • a i • s i ∈ N := N.smul_mem _ (ha' i)
        _ = (∏ j, a j) • s i := by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
    -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
    obtain ⟨n, b : Basis (Fin n) R (LinearMap.range φ)⟩ := Submodule.basisOfPidOfLE this sI_basis
    -- hence `M` is free.
    exact ⟨n, b.map ψ.symm⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: ι : Type u_1
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : CommSemiring R
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
f : End A (ι → M)
i : ι
x : M
j : ι
⊢ f (∑ x_1, Pi.single x_1 (Pi.single i x x_1)) j = f (Pi.single i x) j

Tactic:
]

Goal state after:
ι : Type u_1
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : CommSemiring R
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
f : End A (ι → M)
i : ι
x : M
j : ι
⊢ f (∑ x_1, Pi.single x_1 (Pi.single i x x_1)) j = f (Pi.single i x) j
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: ι : Type u_1
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : CommSemiring R
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
f g : End A (ι → M)
i✝ j✝ : ι
x✝ : M
⊢ f (g (Pi.single j✝ x✝)) i✝ = f (∑ x, Pi.single x (g (Pi.single j✝ x✝) x)) i✝

Tactic:
]

Goal state after:
ι : Type u_1
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
R : Type u_2
inst✝⁶ : CommSemiring R
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
f g : End A (ι → M)
i✝ j✝ : ι
x✝ : M
⊢ f (g (Pi.single j✝ x✝)) i✝ = f (∑ x, Pi.single x (g (Pi.single j✝ x✝) x)) i✝
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝² : CommRing R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
i : Fin n
c : Fin n →₀ R
hm✝ : (c.sum fun i x ↦ x • ↑(snf.bN i)) ∈ N
hm : ⟨c.sum fun i t ↦ t • ↑(snf.bN i), hm✝⟩ = c.sum fun i t ↦ t • ⟨↑(snf.bN i), ⋯⟩
⊢ (snf.bM.repr (c.sum fun i x ↦ x • ↑(snf.bN i))) (snf.f i) =
    (snf.bN.repr (snf.a i • ⟨c.sum fun i x ↦ x • ↑(snf.bN i), hm✝⟩)) i

Tactic:
hm,

Goal state after:
ι : Type u_1
R : Type u_2
inst✝² : CommRing R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
i : Fin n
c : Fin n →₀ R
hm✝ : (c.sum fun i x ↦ x • ↑(snf.bN i)) ∈ N
hm : ⟨c.sum fun i t ↦ t • ↑(snf.bN i), hm✝⟩ = c.sum fun i t ↦ t • ⟨↑(snf.bN i), ⋯⟩
⊢ (snf.bM.repr (c.sum fun i x ↦ x • ↑(snf.bN i))) (snf.f i) =
    (snf.bN.repr (snf.a i • c.sum fun i t ↦ t • ⟨↑(snf.bN i), ⋯⟩)) i
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: ι✝ : Type u_1
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : DecidableEq ι✝
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
ι : Type u_5
inst✝¹ : Finite ι
inst✝ : IsStablyFiniteRing A
this : Fintype ι
⊢ IsStablyFiniteRing (Matrix ι ι A)

Tactic:
]

Goal state after:
ι✝ : Type u_1
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : DecidableEq ι✝
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
M : Type u_4
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
ι : Type u_5
inst✝¹ : Finite ι
inst✝ : IsStablyFiniteRing A
this : Fintype ι
⊢ IsStablyFiniteRing (Matrix ι ι A)
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
surj : Surjective ⇑f
this : ↑(LinearEquiv.ofBijective f ⋯).symm ∘ₗ ↑(LinearEquiv.ofBijective f ⋯) = LinearMap.id
⊢ ↑(LinearEquiv.ofBijective f ⋯).symm * 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
surj : Surjective ⇑f
this : ↑(LinearEquiv.ofBijective f ⋯).symm ∘ₗ ↑(LinearEquiv.ofBijective f ⋯) = LinearMap.id
⊢ ↑(LinearEquiv.ofBijective f ⋯).symm * f = 1
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : M →ₗ[R] M
hf : ∀ (x : M), f x ∈ N
hf' : ∀ x ∈ N, f x ∈ N
i : Fin n
⊢ (snf.bN.repr ((f.restrict hf') (snf.bN i))) i = (snf.bN.repr (snf.a i • ⟨f (snf.bM (snf.f i)), ⋯⟩)) i

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : M →ₗ[R] M
hf : ∀ (x : M), f x ∈ N
hf' : ∀ x ∈ N, f x ∈ N
i : Fin n
⊢ (snf.bN.repr ((f.restrict hf') (snf.bN i))) i = (snf.bN.repr (snf.a i • ⟨f (snf.bM (snf.f i)), ⋯⟩)) i
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))

Tactic:
subst N_bot

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))
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_le_O : ⊥ ≤ M0
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case intro
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
N : Submodule R M
inst✝ : Finite ι
b : Basis ι R M
h : finrank R ↥N = finrank R M
val✝ : Fintype ι
bM : Basis ι R M
bN : Basis (Fin (Fintype.card ι)) R ↥N
f : Fin (Fintype.card ι) ↪ ι
a : Fin (Fintype.card ι) → R
snf : ∀ (i : Fin (Fintype.card ι)), ↑(bN i) = a i • bM (f i)
e : Fin (Fintype.card ι) ≃ ι := Equiv.ofBijective ⇑f ⋯
⊢ ∃ b' a ab', ∀ (i : ι), ↑(ab' i) = a i • b' i

Tactic:
have fe : ∀ i, f (e.symm i) = i := e.apply_symm_apply

Goal state after:
case intro
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
N : Submodule R M
inst✝ : Finite ι
b : Basis ι R M
h : finrank R ↥N = finrank R M
val✝ : Fintype ι
bM : Basis ι R M
bN : Basis (Fin (Fintype.card ι)) R ↥N
f : Fin (Fintype.card ι) ↪ ι
a : Fin (Fintype.card ι) → R
snf : ∀ (i : Fin (Fintype.card ι)), ↑(bN i) = a i • bM (f i)
e : Fin (Fintype.card ι) ≃ ι := Equiv.ofBijective ⇑f ⋯
fe : ∀ (i : ι), f (e.symm i) = i
⊢ ∃ b' a ab', ∀ (i : ι), ↑(ab' i) = a i • b' i
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case intro
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
S : Type u_4
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra R S
inst✝ : Finite ι
b : Basis ι R S
I : Ideal S
hI : I ≠ ⊥
fst✝ : ℕ
bS : Basis ι R S
bI : Basis (Fin fst✝) R ↥(restrictScalars R I)
f✝ : Fin fst✝ ↪ ι
a✝ : Fin fst✝ → R
snf✝ : ∀ (i : Fin fst✝), ↑(bI i) = a✝ i • bS (f✝ i)
val✝ : Fintype ι
⊢ Fintype.card (Fin fst✝) = Fintype.card ι

Tactic:
]

Goal state after:
case intro
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
S : Type u_4
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra R S
inst✝ : Finite ι
b : Basis ι R S
I : Ideal S
hI : I ≠ ⊥
fst✝ : ℕ
bS : Basis ι R S
bI : Basis (Fin fst✝) R ↥(restrictScalars R I)
f✝ : Fin fst✝ ↪ ι
a✝ : Fin fst✝ → R
snf✝ : ∀ (i : Fin fst✝), ↑(bI i) = a✝ i • bS (f✝ i)
val✝ : Fintype ι
⊢ Fintype.card (Fin fst✝) = Fintype.card ι
Module: Mathlib.Algebra.Algebra.RestrictScalars

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : CommSemiring R
inst✝² : Semiring S
inst✝¹ : Algebra R S
inst✝ : Module S M
r : R
s : S
x : M
⊢ (addEquiv R S M).symm ((algebraMap R S) r • s • x) = r • (addEquiv R S M).symm (s • x)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
M : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : CommSemiring R
inst✝² : Semiring S
inst✝¹ : Algebra R S
inst✝ : Module S M
r : R
s : S
x : M
⊢ (addEquiv R S M).symm ((algebraMap R S) r • s • x) = r • (addEquiv R S M).symm (s • x)
Module: Mathlib.Algebra.Star.TensorProduct

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : AddCommMonoid A
inst✝⁶ : StarAddMonoid A
inst✝⁵ : Module R A
inst✝⁴ : StarModule R A
inst✝³ : AddCommMonoid B
inst✝² : StarAddMonoid B
inst✝¹ : Module R B
inst✝ : StarModule R B
x : A ⊗[R] B
⊢ (congr ((starLinearEquiv R).trans (starLinearEquiv R)) ((starLinearEquiv R).trans (starLinearEquiv R))) x = x

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : AddCommMonoid A
inst✝⁶ : StarAddMonoid A
inst✝⁵ : Module R A
inst✝⁴ : StarModule R A
inst✝³ : AddCommMonoid B
inst✝² : StarAddMonoid B
inst✝¹ : Module R B
inst✝ : StarModule R B
x : A ⊗[R] B
⊢ (congr ((starLinearEquiv R).trans (starLinearEquiv R)) ((starLinearEquiv R).trans (starLinearEquiv R))) x = x