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

Goal state before:
pp: case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
k : ℕ
hk : k ≠ 0
z : A
hz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
J : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx : x ∈ Submodule.span A {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hy : y ∈ Submodule.span A {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hx_pow : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hy_pow : ∀ (n : ℕ), n ≠ 0 → hI.dpow n y ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
n : ℕ
hn : n ≠ 0
m : ℕ
a✝ : m ∈ Finset.range (n + 1)
hm0 : m = 0
⊢ hI.dpow 0 x * hI.dpow (n - 0) y ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}

Tactic:
;

Goal state after:
case pos
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ ↑I
k : ℕ
hk : k ≠ 0
z : A
hz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
J : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx : x ∈ Submodule.span A {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hy : y ∈ Submodule.span A {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hx_pow : ∀ (n : ℕ), n ≠ 0 → hI.dpow n x ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
hy_pow : ∀ (n : ℕ), n ≠ 0 → hI.dpow n y ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
n : ℕ
hn : n ≠ 0
m : ℕ
a✝ : m ∈ Finset.range (n + 1)
hm0 : m = 0
⊢ hI.dpow 0 x * hI.dpow (n - 0) y ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}
Module: Mathlib.RingTheory.Extension.ExtendScalars

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.Ring
x_in : x ∈ P.ker
property✝ : Cotangent.mk ⟨x, x_in⟩ ∈ P.extendScalars.cotangentComplex.ker
⊢ (P.cotangentComplex ∘ₗ ↑P.cotangentExtendScalarsEquiv ∘ₗ h1Cotangentι) ⟨Cotangent.mk ⟨x, x_in⟩, property✝⟩ =
    1 ⊗ₜ[P.Ring] (D R P.Ring) x

pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.Ring
x_in : x ∈ P.ker
property✝ : Cotangent.mk ⟨x, x_in⟩ ∈ P.extendScalars.cotangentComplex.ker
⊢ 1 ⊗ₜ[P.Ring] (D R P.Ring) x =
    (H1Cotangent.δ R P.Ring S ∘ₗ H1Cotangent.map (Hom.ofAlgHom (ofId P.Ring (Generators.self P.Ring S).Ring) ⋯))
      ⟨Cotangent.mk ⟨x, x_in⟩, property✝⟩

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.Ring
x_in : x ∈ P.ker
property✝ : Cotangent.mk ⟨x, x_in⟩ ∈ P.extendScalars.cotangentComplex.ker
⊢ (P.cotangentComplex ∘ₗ ↑P.cotangentExtendScalarsEquiv ∘ₗ h1Cotangentι) ⟨Cotangent.mk ⟨x, x_in⟩, property✝⟩ =
    1 ⊗ₜ[P.Ring] (D R P.Ring) x

R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.Ring
x_in : x ∈ P.ker
property✝ : Cotangent.mk ⟨x, x_in⟩ ∈ P.extendScalars.cotangentComplex.ker
⊢ 1 ⊗ₜ[P.Ring] (D R P.Ring) x =
    (H1Cotangent.δ R P.Ring S ∘ₗ H1Cotangent.map (Hom.ofAlgHom (ofId P.Ring (Generators.self P.Ring S).Ring) ⋯))
      ⟨Cotangent.mk ⟨x, x_in⟩, property✝⟩
Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal

Goal state before:
pp: A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
⊢ hI.IsSubDPIdeal (RingHom.ker f ⊓ I)

Tactic:
rw [isSubDPIdeal_inf_iff]

Goal state after:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
J : Ideal B
hJ : DividedPowers J
f : A →+* B
hf : hI.IsDPMorphism hJ f
⊢ ∀ {n : ℕ} {a b : A}, a ∈ I → b ∈ I → a - b ∈ RingHom.ker f → hI.dpow n a - hI.dpow n b ∈ RingHom.ker f
Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal

Goal state before:
pp: A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
J : Ideal B
hf : Function.Surjective ⇑f
hIJ : J = Ideal.map f I
hIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a b : A
hb : b ∈ I
hy : f b ∈ J
⊢ dpow hI f n (f a * f b) = f a ^ n * dpow hI f n (f b)

Tactic:
rw [dpow_apply' hI hIf hb, ← _root_.map_mul, ← map_pow,
      dpow_apply' hI hIf (mul_mem_left I a hb), hI.dpow_mul hb, _root_.map_mul]

Goal state after:
Module: Mathlib.RingTheory.Frobenius

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : IsDomain S
ζ : S
q : ℕ := Nat.card (R ⧸ Ideal.under R Q)
hζ : ζ ^ 0 = 1
hk' : ↑0 ∉ Q
⊢ False

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : IsDomain S
ζ : S
q : ℕ := Nat.card (R ⧸ Ideal.under R Q)
hζ : ζ ^ 0 = 1
hk' : ↑0 ∉ Q
⊢ False
Module: Mathlib.RingTheory.GradedAlgebra.AlgHom

Goal state before:
pp: R : Type u_1
A : Type u_6
ι : Type u_10
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : 𝒜 →ₐᵍ[R] 𝒜
n : ℕ
x✝ : A
⊢ (f ^ Nat.zero) x✝ = (⇑f)^[Nat.zero] x✝

Tactic:
;

Goal state after:
R : Type u_1
A : Type u_6
ι : Type u_10
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : 𝒜 →ₐᵍ[R] 𝒜
n : ℕ
x✝ : A
⊢ (f ^ Nat.zero) x✝ = (⇑f)^[Nat.zero] x✝
Module: Mathlib.RingTheory.GradedAlgebra.AlgHom

Goal state before:
pp: R : Type u_1
A : Type u_6
ι : Type u_10
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : 𝒜 →ₐᵍ[R] 𝒜
n x✝¹ : ℕ
ih : ⇑(f ^ x✝¹) = (⇑f)^[x✝¹]
x✝ : A
⊢ (f ^ x✝¹.succ) x✝ = (⇑f)^[x✝¹.succ] x✝

Tactic:
;

Goal state after:
R : Type u_1
A : Type u_6
ι : Type u_10
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : 𝒜 →ₐᵍ[R] 𝒜
n x✝¹ : ℕ
ih : ⇑(f ^ x✝¹) = (⇑f)^[x✝¹]
x✝ : A
⊢ (f ^ x✝¹.succ) x✝ = (⇑f)^[x✝¹.succ] x✝
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : AddMonoid ι
inst✝⁴ : Semiring A
inst✝³ : SetLike σ A
inst✝² : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝¹ : DecidableEq ι
inst✝ : GradedRing 𝒜
R : Subsemiring A
x : Subtype (IsHomogeneousElem 𝒜)
left✝ : x ∈ Subtype.val ⁻¹' ↑R
⊢ IsHomogeneousElem 𝒜 ↑x

Tactic:
;

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : AddMonoid ι
inst✝⁴ : Semiring A
inst✝³ : SetLike σ A
inst✝² : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝¹ : DecidableEq ι
inst✝ : GradedRing 𝒜
R : Subsemiring A
x : Subtype (IsHomogeneousElem 𝒜)
left✝ : x ∈ Subtype.val ⁻¹' ↑R
⊢ IsHomogeneousElem 𝒜 ↑x
Module: Mathlib.RingTheory.Frobenius

Goal state before:
pp: case e'_3
R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
G : Type u_3
inst✝⁸ : Group G
inst✝⁷ : MulSemiringAction G S
inst✝⁶ : SMulCommClass G R S
Q : Ideal S
inst✝⁵ : Finite G
inst✝⁴ : Algebra.IsInvariant R S G
inst✝³ : Q.IsPrime
inst✝² : Finite (S ⧸ Q)
Q' : Ideal S
inst✝¹ : Q'.IsPrime
inst✝ : Finite (S ⧸ Q')
H : Ideal.under R Q = Ideal.under R Q'
hP : (Ideal.under R Q).IsPrime
h₂ : Ideal.under R Q = Ideal.under R Q'
⊢ arithFrobAt R G Q = ⋯.choose ⟨Q, ⋯⟩

Tactic:
;

Goal state after:
case e'_3
R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
G : Type u_3
inst✝⁸ : Group G
inst✝⁷ : MulSemiringAction G S
inst✝⁶ : SMulCommClass G R S
Q : Ideal S
inst✝⁵ : Finite G
inst✝⁴ : Algebra.IsInvariant R S G
inst✝³ : Q.IsPrime
inst✝² : Finite (S ⧸ Q)
Q' : Ideal S
inst✝¹ : Q'.IsPrime
inst✝ : Finite (S ⧸ Q')
H : Ideal.under R Q = Ideal.under R Q'
hP : (Ideal.under R Q).IsPrime
h₂ : Ideal.under R Q = Ideal.under R Q'
⊢ arithFrobAt R G Q = ⋯.choose ⟨Q, ⋯⟩
Module: Mathlib.RingTheory.Frobenius

Goal state before:
pp: case e'_4
R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
G : Type u_3
inst✝⁸ : Group G
inst✝⁷ : MulSemiringAction G S
inst✝⁶ : SMulCommClass G R S
Q : Ideal S
inst✝⁵ : Finite G
inst✝⁴ : Algebra.IsInvariant R S G
inst✝³ : Q.IsPrime
inst✝² : Finite (S ⧸ Q)
Q' : Ideal S
inst✝¹ : Q'.IsPrime
inst✝ : Finite (S ⧸ Q')
H : Ideal.under R Q = Ideal.under R Q'
hP : (Ideal.under R Q').IsPrime
h₁ : Ideal.under R Q' = Ideal.under R Q
⊢ arithFrobAt R G Q' = ⋯.choose ⟨Q', ⋯⟩

Tactic:
;

Goal state after:
case e'_4
R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
G : Type u_3
inst✝⁸ : Group G
inst✝⁷ : MulSemiringAction G S
inst✝⁶ : SMulCommClass G R S
Q : Ideal S
inst✝⁵ : Finite G
inst✝⁴ : Algebra.IsInvariant R S G
inst✝³ : Q.IsPrime
inst✝² : Finite (S ⧸ Q)
Q' : Ideal S
inst✝¹ : Q'.IsPrime
inst✝ : Finite (S ⧸ Q')
H : Ideal.under R Q = Ideal.under R Q'
hP : (Ideal.under R Q').IsPrime
h₁ : Ideal.under R Q' = Ideal.under R Q
⊢ arithFrobAt R G Q' = ⋯.choose ⟨Q', ⋯⟩
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: case mk
R : Type u
inst✝² : CommRing R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
k : ℕ
N₂ : G(k, M; R)
toSubmodule✝ : Submodule R M
finite_quotient✝ : Module.Finite R (M ⧸ toSubmodule✝)
projective_quotient✝ : Projective R (M ⧸ toSubmodule✝)
rankAtStalk_eq✝ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝) p = k
h :
  { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
        rankAtStalk_eq := rankAtStalk_eq✝ }.toSubmodule =
    N₂.toSubmodule
⊢ { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
      rankAtStalk_eq := rankAtStalk_eq✝ } =
    N₂

Tactic:
;

Goal state after:
case mk
R : Type u
inst✝² : CommRing R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
k : ℕ
N₂ : G(k, M; R)
toSubmodule✝ : Submodule R M
finite_quotient✝ : Module.Finite R (M ⧸ toSubmodule✝)
projective_quotient✝ : Projective R (M ⧸ toSubmodule✝)
rankAtStalk_eq✝ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝) p = k
h :
  { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
        rankAtStalk_eq := rankAtStalk_eq✝ }.toSubmodule =
    N₂.toSubmodule
⊢ { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
      rankAtStalk_eq := rankAtStalk_eq✝ } =
    N₂
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: case mk.mk
R : Type u
inst✝² : CommRing R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
k : ℕ
toSubmodule✝¹ : Submodule R M
finite_quotient✝¹ : Module.Finite R (M ⧸ toSubmodule✝¹)
projective_quotient✝¹ : Projective R (M ⧸ toSubmodule✝¹)
rankAtStalk_eq✝¹ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝¹) p = k
toSubmodule✝ : Submodule R M
finite_quotient✝ : Module.Finite R (M ⧸ toSubmodule✝)
projective_quotient✝ : Projective R (M ⧸ toSubmodule✝)
rankAtStalk_eq✝ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝) p = k
h :
  { toSubmodule := toSubmodule✝¹, finite_quotient := finite_quotient✝¹, projective_quotient := projective_quotient✝¹,
        rankAtStalk_eq := rankAtStalk_eq✝¹ }.toSubmodule =
    { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
        rankAtStalk_eq := rankAtStalk_eq✝ }.toSubmodule
⊢ { toSubmodule := toSubmodule✝¹, finite_quotient := finite_quotient✝¹, projective_quotient := projective_quotient✝¹,
      rankAtStalk_eq := rankAtStalk_eq✝¹ } =
    { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
      rankAtStalk_eq := rankAtStalk_eq✝ }

Tactic:
;

Goal state after:
case mk.mk
R : Type u
inst✝² : CommRing R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
k : ℕ
toSubmodule✝¹ : Submodule R M
finite_quotient✝¹ : Module.Finite R (M ⧸ toSubmodule✝¹)
projective_quotient✝¹ : Projective R (M ⧸ toSubmodule✝¹)
rankAtStalk_eq✝¹ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝¹) p = k
toSubmodule✝ : Submodule R M
finite_quotient✝ : Module.Finite R (M ⧸ toSubmodule✝)
projective_quotient✝ : Projective R (M ⧸ toSubmodule✝)
rankAtStalk_eq✝ : ∀ (p : PrimeSpectrum R), rankAtStalk (M ⧸ toSubmodule✝) p = k
h :
  { toSubmodule := toSubmodule✝¹, finite_quotient := finite_quotient✝¹, projective_quotient := projective_quotient✝¹,
        rankAtStalk_eq := rankAtStalk_eq✝¹ }.toSubmodule =
    { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
        rankAtStalk_eq := rankAtStalk_eq✝ }.toSubmodule
⊢ { toSubmodule := toSubmodule✝¹, finite_quotient := finite_quotient✝¹, projective_quotient := projective_quotient✝¹,
      rankAtStalk_eq := rankAtStalk_eq✝¹ } =
    { toSubmodule := toSubmodule✝, finite_quotient := finite_quotient✝, projective_quotient := projective_quotient✝,
      rankAtStalk_eq := rankAtStalk_eq✝ }
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A : Type w
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f : A →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g : B →ₐ[R] C
N : G(k, A ⊗[R] M; A)
algInst✝² : Algebra A B := ⋯
algInst✝¹ : Algebra B C := ⋯
algInst✝ : Algebra A C := ⋯
⊢ algebraMap A C = (algebraMap B C).comp (algebraMap A B)

Tactic:
;

Goal state after:
R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A : Type w
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f : A →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g : B →ₐ[R] C
N : G(k, A ⊗[R] M; A)
algInst✝² : Algebra A B := ⋯
algInst✝¹ : Algebra B C := ⋯
algInst✝ : Algebra A C := ⋯
⊢ algebraMap A C = (algebraMap B C).comp (algebraMap A B)
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A✝ : Type w
inst✝⁵ : CommRing A✝
inst✝⁴ : Algebra R A✝
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f : A✝ →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g : B →ₐ[R] C
A : CommAlgCat R
N : G(k, ↑A ⊗[R] M; ↑A)
⊢ (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom (𝟙 A)))) N = (ConcreteCategory.hom (𝟙 G(k, ↑A ⊗[R] M; ↑A))) N

Tactic:
;

Goal state after:
R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A✝ : Type w
inst✝⁵ : CommRing A✝
inst✝⁴ : Algebra R A✝
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f : A✝ →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g : B →ₐ[R] C
A : CommAlgCat R
N : G(k, ↑A ⊗[R] M; ↑A)
⊢ (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom (𝟙 A)))) N = (ConcreteCategory.hom (𝟙 G(k, ↑A ⊗[R] M; ↑A))) N
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A : Type w
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f✝ : A →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g✝ : B →ₐ[R] C
X✝ Y✝ Z✝ : CommAlgCat R
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
N : G(k, ↑X✝ ⊗[R] M; ↑X✝)
⊢ (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom (f ≫ g)))) N =
    (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom f) ≫ ↾map (CommAlgCat.Hom.hom g))) N

Tactic:
;

Goal state after:
R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A : Type w
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f✝ : A →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g✝ : B →ₐ[R] C
X✝ Y✝ Z✝ : CommAlgCat R
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
N : G(k, ↑X✝ ⊗[R] M; ↑X✝)
⊢ (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom (f ≫ g)))) N =
    (ConcreteCategory.hom (↾map (CommAlgCat.Hom.hom f) ≫ ↾map (CommAlgCat.Hom.hom g))) N
Module: Mathlib.RingTheory.PowerSeries.WellKnown

Goal state before:
pp: case zero
S : Type u_1
inst✝ : CommRing S
d n✝ : ℕ
⊢ (coeff n✝) (mk 1 ^ (0 + 1)) = (coeff n✝) (mk fun n ↦ ↑((0 + n).choose 0))

Tactic:
;

Goal state after:
case zero
S : Type u_1
inst✝ : CommRing S
d n✝ : ℕ
⊢ (coeff n✝) (mk 1 ^ (0 + 1)) = (coeff n✝) (mk fun n ↦ ↑((0 + n).choose 0))
Module: Mathlib.RingTheory.HahnSeries.Binomial

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : LinearOrder Γ
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
inst✝³ : CommRing R
inst✝² : BinomialRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A⟦Γ⟧
hx : 0 < (x - 1).orderTop
r : R
n : ℕ
g : Γ
hg : g ∈ ((binomialFamily x r) n).support
hn : n = 0
⊢ 0 ≤ g

pp: case neg
Γ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : LinearOrder Γ
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
inst✝³ : CommRing R
inst✝² : BinomialRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A⟦Γ⟧
hx : 0 < (x - 1).orderTop
r : R
n : ℕ
g : Γ
hg : g ∈ ((binomialFamily x r) n).support
hn : ¬n = 0
⊢ 0 ≤ g

Tactic:
;

Goal state after:
case pos
Γ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : LinearOrder Γ
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
inst✝³ : CommRing R
inst✝² : BinomialRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A⟦Γ⟧
hx : 0 < (x - 1).orderTop
r : R
n : ℕ
g : Γ
hg : g ∈ ((binomialFamily x r) n).support
hn : n = 0
⊢ 0 ≤ g

case neg
Γ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : LinearOrder Γ
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
inst✝³ : CommRing R
inst✝² : BinomialRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : A⟦Γ⟧
hx : 0 < (x - 1).orderTop
r : R
n : ℕ
g : Γ
hg : g ∈ ((binomialFamily x r) n).support
hn : ¬n = 0
⊢ 0 ≤ g
Module: Mathlib.RingTheory.HahnSeries.HEval

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
h : 0 < x.orderTop
⊢ (powerSeriesFamily x (a * b)).hsum = ((powerSeriesFamily x a).mul (powerSeriesFamily x b)).hsum

pp: case neg
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
h : ¬0 < x.orderTop
⊢ (powerSeriesFamily x (a * b)).hsum = ((powerSeriesFamily x a).mul (powerSeriesFamily x b)).hsum

Tactic:
;

Goal state after:
case pos
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
h : 0 < x.orderTop
⊢ (powerSeriesFamily x (a * b)).hsum = ((powerSeriesFamily x a).mul (powerSeriesFamily x b)).hsum

case neg
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
h : ¬0 < x.orderTop
⊢ (powerSeriesFamily x (a * b)).hsum = ((powerSeriesFamily x a).mul (powerSeriesFamily x b)).hsum
Module: Mathlib.RingTheory.FormalGroup.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
σ✝ : Type u_3
τ : Type u_4
σ : Type u_5
F : FormalGroup R
f : R →+* S
g₁ g₂ : MvPowerSeries (Fin 3) R
i : Fin (Nat.succ 0).succ
⊢ ![(MvPowerSeries.map f) g₁, (MvPowerSeries.map f) g₂] i = (MvPowerSeries.map f) (![g₁, g₂] i)

Tactic:
;

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
σ✝ : Type u_3
τ : Type u_4
σ : Type u_5
F : FormalGroup R
f : R →+* S
g₁ g₂ : MvPowerSeries (Fin 3) R
i : Fin (Nat.succ 0).succ
⊢ ![(MvPowerSeries.map f) g₁, (MvPowerSeries.map f) g₂] i = (MvPowerSeries.map f) (![g₁, g₂] i)
Module: Mathlib.RingTheory.FormalGroup.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
F : FormalGroup R
x : Fin (Nat.succ 0).succ
⊢ ![0, 0] x = 0 x

Tactic:
;

Goal state after:
R : Type u_1
inst✝ : CommRing R
F : FormalGroup R
x : Fin (Nat.succ 0).succ
⊢ ![0, 0] x = 0 x
Module: Mathlib.RingTheory.HahnSeries.Valuation

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : x = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

pp: case neg
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

Tactic:
;

Goal state after:
case pos
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : x = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

case neg
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop
Module: Mathlib.RingTheory.HahnSeries.Valuation

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
hy : y = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

pp: case neg
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
hy : ¬y = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

Tactic:
;

Goal state after:
case pos
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
hy : y = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop

case neg
Γ : Type u_1
R : Type u_2
inst✝⁴ : AddCancelCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : Ring R
inst✝ : IsDomain R
x y : R⟦Γ⟧
hx : ¬x = 0
hy : ¬y = 0
⊢ (x * y).orderTop = x.orderTop + y.orderTop
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
hx : 0 < x.orderTop
g : Γ
hpwo : (⋃ n, (x ^ n).support).IsPWO
h0 : x = 0
⊢ {a | ((fun n ↦ x ^ n) a).coeff g ≠ 0}.Finite

pp: case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
hx : 0 < x.orderTop
g : Γ
hpwo : (⋃ n, (x ^ n).support).IsPWO
h0 : ¬x = 0
⊢ {a | ((fun n ↦ x ^ n) a).coeff g ≠ 0}.Finite

Tactic:
;

Goal state after:
case pos
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
hx : 0 < x.orderTop
g : Γ
hpwo : (⋃ n, (x ^ n).support).IsPWO
h0 : x = 0
⊢ {a | ((fun n ↦ x ^ n) a).coeff g ≠ 0}.Finite

case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
hx : 0 < x.orderTop
g : Γ
hpwo : (⋃ n, (x ^ n).support).IsPWO
h0 : ¬x = 0
⊢ {a | ((fun n ↦ x ^ n) a).coeff g ≠ 0}.Finite