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

Goal state before:
pp: A : Type u_4
inst✝⁵ : CommRing A
K : Type u_5
inst✝⁴ : Field K
L : Type u_7
inst✝³ : Field L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : Algebra A L
f : L →ₐ[A] K
⊢ IsFractionRing A L

Tactic:
refine IsFractionRing.of_algEquiv <| .symm <| .ofBijective f ⟨f.injective, fun x ↦ ?_⟩

Goal state after:
A : Type u_4
inst✝⁵ : CommRing A
K : Type u_5
inst✝⁴ : Field K
L : Type u_7
inst✝³ : Field L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : Algebra A L
f : L →ₐ[A] K
x : K
⊢ ∃ a, f a = x
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: A : Type u_8
B : Type u_9
C : Type u_10
inst✝²¹ : CommRing A
inst✝²⁰ : CommRing B
inst✝¹⁹ : CommRing C
inst✝¹⁸ : Algebra A B
inst✝¹⁷ : Algebra A C
FA : Type u_12
FB : Type u_13
FC : Type u_14
inst✝¹⁶ : Field FA
inst✝¹⁵ : Field FB
inst✝¹⁴ : Field FC
inst✝¹³ : Algebra A FA
inst✝¹² : Algebra B FB
inst✝¹¹ : Algebra C FC
inst✝¹⁰ : IsFractionRing A FA
inst✝⁹ : IsFractionRing B FB
inst✝⁸ : IsFractionRing C FC
inst✝⁷ : Algebra A FB
inst✝⁶ : IsScalarTower A B FB
inst✝⁵ : Algebra A FC
inst✝⁴ : IsScalarTower A C FC
inst✝³ : Algebra FA FB
inst✝² : IsScalarTower A FA FB
inst✝¹ : Algebra FA FC
inst✝ : IsScalarTower A FA FC
f : B ≃ₐ[A] C
a✝ : FB
⊢ (AlgEquiv.restrictScalars A (fieldEquivOfAlgEquiv FA FB FC f)) a✝ = (algEquivOfAlgEquiv f) a✝

Tactic:
rfl

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
M : Submonoid R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
m : A
⊢ (algebraMap R (Localization M)) ((algebraMap A R) m) = (algebraMap A (Localization M)) m

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: case add
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
J₁ : Submodule R N
fg₁ : J₁.FG
y₁ : ↥J₁ ⊗[R] M
J₂ : Submodule R N
fg₂ : J₂.FG
y₂ : ↥J₂ ⊗[R] M
⊢ (rTensor M J₁.subtype) y₁ + (rTensor M J₂.subtype) y₂ =
    (rTensor M ((J₁ ⊔ J₂).subtype ∘ₗ inclusion ⋯)) y₁ + (rTensor M ((J₁ ⊔ J₂).subtype ∘ₗ inclusion ⋯)) y₂

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
s : Set (N ⊗[R] M)
hs : Finite ↑s
J : N ⊗[R] M → Submodule R N
fg : ∀ (x : N ⊗[R] M), (J x).FG
y : (x : N ⊗[R] M) → ↥(J x) ⊗[R] M
eq : ∀ (x : N ⊗[R] M), x = (rTensor M (J x).subtype) (y x)
x : N ⊗[R] M
hx : x ∈ s
⊢ (rTensor M ((⨆ x, J ↑x).subtype ∘ₗ inclusion ⋯)) (y x) = (rTensor M (J x).subtype) (y x)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
I : Submodule R N
J : Submodule R ↥I
fg : J.FG
y : ↥J ⊗[R] M
⊢ (rTensor M J.subtype) y = (rTensor M (inclusion ⋯ ∘ₗ I.subtype.submoduleMap J)) y

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
I : Submodule R N
s : Set (↥I ⊗[R] M)
hs : Finite ↑s
J : ↥I ⊗[R] M → Submodule R N
fg : ∀ (x : ↥I ⊗[R] M), (J x).FG
hle : ∀ (x : ↥I ⊗[R] M), J x ≤ I
y : (x : ↥I ⊗[R] M) → ↥(J x) ⊗[R] M
eq : ∀ (x : ↥I ⊗[R] M), x = (rTensor M (inclusion ⋯)) (y x)
x : ↥I ⊗[R] M
hx : x ∈ s
⊢ (rTensor M (inclusion ⋯ ∘ₗ inclusion ⋯)) (y x) = (rTensor M (inclusion ⋯)) (y x)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : Module.Finite R M
s : Finset M
hs : Submodule.span R ↑s = ⊤
⊢ Module.Finite A (TensorProduct R A M)

Tactic:
refine ⟨⟨s.image (TensorProduct.mk R A M 1), eq_top_iff.mpr ?_⟩⟩

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : Module.Finite R M
s : Finset M
hs : Submodule.span R ↑s = ⊤
⊢ ⊤ ≤ Submodule.span A ↑(Finset.image (⇑((TensorProduct.mk R A M) 1)) s)
Module: Mathlib.Algebra.Algebra.Epi

Goal state before:
pp: case zero
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : Module.Finite R A
h : Algebra.IsEpi R A
R' : Submodule R A := (Algebra.linearMap R A).range
h✝ : Nontrivial (A ⧸ R')
⊢ 0 = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : AddGroupWithOne β
n : ℕ
⊢ e.invFun ↑n = ↑n

Tactic:
rfl

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

Goal state before:
pp: R : Type uR
R' : Type u_1
S : Type uS
T : Type u_2
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝⁴ : CommSemiring R
inst✝³ : Ring A
inst✝² : Algebra R A
inst✝¹ : Semiring B
inst✝ : Algebra R B
⊢ instRing.toAddCommGroup = addCommGroup

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Semiring B
inst✝ : Algebra R B
a✝ : B ⊗[R] A
⊢ (TensorProduct.comm R A B).symm a✝ = (TensorProduct.comm R B A) a✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Algebra.Algebra.Spectrum.Quasispectrum

Goal state before:
pp: R : Type u_3
S : Type u_4
A : Type u_5
inst✝⁵ : Semifield R
inst✝⁴ : Semifield S
inst✝³ : Ring A
inst✝² : Algebra R S
inst✝¹ : Algebra R A
inst✝ : Algebra S A
a : A
f : S → R
⊢ QuasispectrumRestricts a f ↔ SpectrumRestricts a f

Tactic:
rfl

Goal state after:
Module: Mathlib.Algebra.Group.Pi.Units

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
x : (i : ι) → M i
hx : IsUnit x
i : ι
⊢ ↑(MulEquiv.piUnits hx.unit i) = ↑⋯.unit

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case ha
R : Type uR
A : Type uA
B : Type uB
C : Type uC
D : Type uD
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : Semiring D
inst✝ : Algebra R D
f : A →ₐ[R] C
g : B →ₐ[R] D
x✝ : A
⊢ (((↑(TensorProduct.comm R C D)).comp (map f g)).comp includeLeft) x✝ =
    (((map g f).comp ↑(TensorProduct.comm R A B)).comp includeLeft) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case hb
R : Type uR
A : Type uA
B : Type uB
C : Type uC
D : Type uD
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : Semiring D
inst✝ : Algebra R D
f : A →ₐ[R] C
g : B →ₐ[R] D
x✝ : B
⊢ ((AlgHom.restrictScalars R ((↑(TensorProduct.comm R C D)).comp (map f g))).comp includeRight) x✝ =
    ((AlgHom.restrictScalars R ((map g f).comp ↑(TensorProduct.comm R A B))).comp includeRight) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
C : Type uC
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : Semiring C
inst✝ : Algebra R C
x✝² : A
x✝¹ : B
x✝ : C
⊢ ((AlgebraTensorModule.curry ((AlgebraTensorModule.curry ↑↑(leftComm R A B C)) x✝²)) x✝¹) x✝ =
    ((AlgebraTensorModule.curry ((AlgebraTensorModule.curry ↑(_root_.TensorProduct.leftComm R A B C)) x✝²)) x✝¹) x✝

Tactic:
rfl

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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : SMulCommClass R B B
inst✝¹ : IsScalarTower R B B
inst✝ : StarModule R B
φ : A →⋆ₙₐ[R] B
hφ : Function.Surjective ⇑φ
r : R
a : A
⊢ (starMap φ) (mk (r, a)) = inl r + ↑(φ a)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
x✝ : S
⊢ ((AlgHom.restrictScalars R (lmul'' R)).comp includeRight) x✝ =
    ((AlgHom.restrictScalars R ((↑(TensorProduct.lid S S)).comp (mapOfCompatibleSMul S R S S S))).comp includeRight) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case ha
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Semiring B
inst✝³ : CommSemiring S
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : Algebra R S
f : A →ₐ[R] S
g : B →ₐ[R] S
x✝ : A
⊢ (((lmul' R).comp (map f g)).comp includeLeft) x✝ = ((lift f g ⋯).comp includeLeft) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case hb
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Semiring B
inst✝³ : CommSemiring S
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : Algebra R S
f : A →ₐ[R] S
g : B →ₐ[R] S
x✝ : B
⊢ ((AlgHom.restrictScalars R ((lmul' R).comp (map f g))).comp includeRight) x✝ =
    ((AlgHom.restrictScalars R (lift f g ⋯)).comp includeRight) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: R : Type uR
S : Type uS
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : CompatibleSMul R S S S
x✝ : S
⊢ ((AlgHom.restrictScalars R ↑(lmulEquiv R S)).comp includeRight) x✝ =
    ((AlgHom.restrictScalars R ↑(lidOfCompatibleSMul R S S)).comp includeRight) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case ha
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Semiring B
inst✝³ : CommSemiring S
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : Algebra R S
f : A →ₐ[R] S
g : B →ₐ[R] S
x✝ : A
⊢ ((productMap f g).comp includeLeft) x✝ = (((lmul' R).comp (map f g)).comp includeLeft) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Maps

Goal state before:
pp: case hb
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Semiring B
inst✝³ : CommSemiring S
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : Algebra R S
f : A →ₐ[R] S
g : B →ₐ[R] S
x✝ : B
⊢ ((AlgHom.restrictScalars R (productMap f g)).comp includeRight) x✝ =
    ((AlgHom.restrictScalars R ((lmul' R).comp (map f g))).comp includeRight) x✝

Tactic:
rfl

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

Goal state before:
pp: A : Type u_1
𝕜 : Type u_2
inst✝⁵ : Ring A
inst✝⁴ : PartialOrder A
inst✝³ : CommSemiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Algebra 𝕜 A
inst✝ : NonnegSpectrumClass 𝕜 A
a : A
ha : IsStrictlyPositive a
x : 𝕜
hx : x ∈ spectrum 𝕜 a
⊢ 0 ≤ x

Tactic:
grind

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

Goal state before:
pp: case refine_2.refine_1
R : Type u_1
inst✝² : NonUnitalSemiring R
inst✝¹ : PartialOrder R
inst✝ : StarRing R
h_le_iff : ∀ (x y : R), x ≤ y ↔ ∃ s, y = x + star s * s
p : R
hp : p ∈ AddSubmonoid.closure (range fun s ↦ star s * s)
s x : R
⊢ x ≤ x + (fun s ↦ star s * s) s

Tactic:
exact (h_le_iff _ _).mpr ⟨s, rfl⟩

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Algebra R A
ι : Type u_4
inst✝ : Nonempty ι
S : ι → Subalgebra R A
hS : ∀ (i : ι), IsMulCommutative ↥(S i)
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) S
⊢ IsMulCommutative ↥(⨆ i, S i)

Tactic:
simpa [isMulCommutative_iff, ← SetLike.mem_coe, coe_iSup_of_directed dir,
    Subsemiring.coe_iSup_of_directed dir] using Subsemiring.isMulCommutative_iSup dir

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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Subalgebra R A
ι : Type u_4
inst✝ : Nonempty ι
K : ι → Subalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →ₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : Subalgebra R A
hT : T ≤ iSup K
compat : ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)), (f i) ⟨x, hxi⟩ = (f j) ⟨x, hxj⟩ :=
  iSupLift._proof_2 K dir f hf
⊢ ∀ (x y : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) compat ↑(iSup K) ⋯ (x + y) =
      Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) compat ↑(iSup K) ⋯ x +
        Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) compat ↑(iSup K) ⋯ y

Tactic:
apply Set.iUnionLift_binary (coe_iSup_of_directed (K := K) dir) dir _ (fun _ => (· + ·))

Goal state after:
case hopi
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Subalgebra R A
ι : Type u_4
inst✝ : Nonempty ι
K : ι → Subalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →ₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : Subalgebra R A
hT : T ≤ iSup K
compat : ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)), (f i) ⟨x, hxi⟩ = (f j) ⟨x, hxj⟩ := ⋯
⊢ ∀ (i : ι) (x y : ↑↑(K i)), Set.inclusion ⋯ (x + y) = Set.inclusion ⋯ x + Set.inclusion ⋯ y

case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Subalgebra R A
ι : Type u_4
inst✝ : Nonempty ι
K : ι → Subalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →ₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : Subalgebra R A
hT : T ≤ iSup K
compat : ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)), (f i) ⟨x, hxi⟩ = (f j) ⟨x, hxj⟩ := ⋯
⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: case refine_2
R : Type u
M : Type v
ι : Type w
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
s t : Set ι
f : ι → M
hst : Disjoint s t
h : LinearIndepOn R f (s ∪ t)
⊢ LinearIndepOn R (⇑(span R (f '' s)).mkQ ∘ f) t

Tactic:
apply (h.mono subset_union_right).map

Goal state after:
case refine_2
R : Type u
M : Type v
ι : Type w
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
s t : Set ι
f : ι → M
hst : Disjoint s t
h : LinearIndepOn R f (s ∪ t)
⊢ Disjoint (span R (range fun x ↦ f ↑x)) (span R (f '' s)).mkQ.ker
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: R✝ : Type u_1
A : Type u_2
inst✝⁷ : NonUnitalRing R✝
inst✝⁶ : PartialOrder R✝
inst✝⁵ : StarRing R✝
inst✝⁴ : StarOrderedRing R✝
p q : R✝
R : Type u_3
inst✝³ : NonUnitalRing R
inst✝² : LinearOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
r : R
⊢ 0 ≤ r ∨ 0 ≤ -r

Tactic:
grind

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

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
inst✝¹ : Free R M
ι : Type v
inst✝ : Fintype ι
⊢ toNat (lift.{v, v} #ι * lift.{v, v} (Module.rank R M)) = toNat #ι * toNat (Module.rank R M)

Tactic:
toNat_mul,

Goal state after:
R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
inst✝¹ : Free R M
ι : Type v
inst✝ : Fintype ι
⊢ toNat (lift.{v, v} #ι) * toNat (lift.{v, v} (Module.rank R M)) = toNat #ι * toNat (Module.rank R M)
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: F : Type u_2
E : Type u_3
inst✝² : CommSemiring F
inst✝¹ : Semiring E
inst✝ : Algebra F E
⊢ Module.rank F ↥⊤ = Module.rank F ↥(Subalgebra.toSubmodule ⊤)

Tactic:
rfl

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

Goal state before:
pp: F : Type u_2
E : Type u_3
inst✝² : CommSemiring F
inst✝¹ : Semiring E
inst✝ : Algebra F E
⊢ finrank F ↥⊤ = finrank F ↥(Subalgebra.toSubmodule ⊤)

Tactic:
rfl

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

Goal state before:
pp: F : Type u_2
E : Type u_3
inst✝⁶ : CommRing F
inst✝⁵ : IsDomain F
inst✝⁴ : Ring E
inst✝³ : Algebra F E
inst✝² : StrongRankCondition F
inst✝¹ : IsTorsionFree F E
inst✝ : Nontrivial E
⊢ Module.rank F ↥(toSubmodule ⊥) = 1

Tactic:
Algebra.toSubmodule_bot,

Goal state after:
F : Type u_2
E : Type u_3
inst✝⁶ : CommRing F
inst✝⁵ : IsDomain F
inst✝⁴ : Ring E
inst✝³ : Algebra F E
inst✝² : StrongRankCondition F
inst✝¹ : IsTorsionFree F E
inst✝ : Nontrivial E
⊢ Module.rank F ↥1 = 1
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : Star A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : Star B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : StarHomClass F A B
φ : F
⊢ ↑(NonUnitalStarAlgHom.range φ) = Set.range ⇑φ

Tactic:
rfl

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁸ : CommSemiring R
inst✝⁷ : NonUnitalSemiring A
inst✝⁶ : StarRing A
inst✝⁵ : Module R A
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
S : ι → NonUnitalStarSubalgebra R A
hS : ∀ (i : ι), IsMulCommutative ↥(S i)
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) S
⊢ IsMulCommutative ↥(⨆ i, S i)

Tactic:
simpa [isMulCommutative_iff, ← SetLike.mem_coe, NonUnitalSubsemiring.coe_iSup_of_directed dir,
    coe_iSup_of_directed dir] using NonUnitalSubsemiring.isMulCommutative_iSup dir

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

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
n : ℕ
hn : finrank R M = n.succ
⊢ 0 < finrank R M

Tactic:
by rw [hn]; exact n.succ_pos

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

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : NonUnitalSemiring A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module R A
inst✝¹³ : NonUnitalSemiring B
inst✝¹² : StarRing B
inst✝¹¹ : Module R B
inst✝¹⁰ : FunLike F A B
inst✝⁹ : NonUnitalAlgHomClass F R A B
inst✝⁸ : StarHomClass F A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝⁷ : StarRing R
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : StarModule R A
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
i j : ι
x : A
hxi : x ∈ ↑(K i)
hxj : x ∈ ↑(K j)
k : ι
hik : (fun x1 x2 ↦ x1 ≤ x2) (K i) (K k)
hjk : (fun x1 x2 ↦ x1 ≤ x2) (K j) (K k)
⊢ ((f k).comp (inclusion hik)) ⟨x, hxi⟩ = ((f k).comp (inclusion hjk)) ⟨x, hxj⟩

Tactic:
rfl

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
inst✝ : Module.Finite R M
N : Submodule R M
⊢ finrank R (M ⧸ N) + finrank R ↥N ≤ finrank R M

Tactic:
have := nontrivial_of_invariantBasisNumber R

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
inst✝ : Module.Finite R M
N : Submodule R M
this : Nontrivial R
⊢ finrank R (M ⧸ N) + finrank R ↥N ≤ finrank R M
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : StrongRankCondition R
v : M
n : v ≠ 0
h : ∀ (w : M), ∃ c, c • v = w
⊢ Module.rank R M = 1

Tactic:
have := nontrivial_of_invariantBasisNumber R

Goal state after:
R : Type u
M : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : StrongRankCondition R
v : M
n : v ≠ 0
h : ∀ (w : M), ∃ c, c • v = w
this : Nontrivial R
⊢ Module.rank R M = 1
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : StarRing A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
a : A
inst✝ : IsStarNormal a
⊢ ∀ a_1 ∈ {a}, ∀ b ∈ {a}, a_1 * star b = star b * a_1

Tactic:
grind

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

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
inst✝ : FiniteDimensional K ↥p
f : V →ₗ[K] V
h : ∀ x ∈ p, f x ∈ p
h' : Disjoint p f.ker
x : V
hx : x ∈ comap f p
y : V
hy : y ∈ p
hxy : f y = f x
⊢ x - y ∈ f.ker

Tactic:
simp [hxy]

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝ : CommSemiring R
⊢ Quot.mk (Rel R X) (Pre.ofScalar 0) = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f : E →ₛₗ.[σ] F
x : E
⊢ x ∈ (- -f).domain ↔ x ∈ f.domain

Tactic:
rfl

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
s : S
φ : R[X]
⊢ eval₂ f s φ = eval₂ f s φ

Tactic:
rfl

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f g : E →ₛₗ.[σ] F
x : E
⊢ x ∈ (f - g).domain ↔ x ∈ (f + -g).domain

Tactic:
rfl

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
x : S
p : R[ℕ]
⊢ ∑ n ∈ p.coeff.support, f (p.coeff n) * x ^ n = (liftNC ↑f ⇑((powersHom S) x)) p

Tactic:
rfl

Goal state after: