mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case refine_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
hue : u + e = 1
he : ∀ (j : ι), j ≠ i → e ∈ I j
⊢ (Quotient.mk (I i)) e = 1

pp: case refine_2
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
hue : u + e = 1
he : ∀ (j : ι), j ≠ i → e ∈ I j
⊢ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) e = 0

Tactic:
·

Goal state after:
case refine_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
hue : u + e = 1
he : ∀ (j : ι), j ≠ i → e ∈ I j
⊢ (Quotient.mk (I i)) e = 1
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝³ : Field R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
s : S
h1 : 1 ∉ s
algHom : Unitization R ↥s →ₐ[R] ↥(Algebra.adjoin R ↑s) := (unitization s).codRestrict (Algebra.adjoin R ↑s) ⋯
⊢ Function.Injective ⇑algHom

pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝³ : Field R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
s : S
h1 : 1 ∉ s
algHom : Unitization R ↥s →ₐ[R] ↥(Algebra.adjoin R ↑s) := (unitization s).codRestrict (Algebra.adjoin R ↑s) ⋯
x : ↥(Algebra.adjoin R ↑s)
⊢ ∃ a, algHom a = x

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝³ : Field R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
s : S
h1 : 1 ∉ s
algHom : Unitization R ↥s →ₐ[R] ↥(Algebra.adjoin R ↑s) := (unitization s).codRestrict (Algebra.adjoin R ↑s) ⋯
⊢ Function.Injective ⇑algHom
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case refine_2
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
hue : u + e = 1
he : ∀ (j : ι), j ≠ i → e ∈ I j
⊢ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) e = 0

Tactic:
·

Goal state after:
case refine_2
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
hue : u + e = 1
he : ∀ (j : ι), j ≠ i → e ∈ I j
⊢ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) e = 0
Module: Mathlib.Algebra.GroupWithZero.Units.Fintype

Goal state before:
pp: case inl
α : Type u_1
inst✝ : GroupWithZero α
h✝ : Finite { a // a ≠ 0 }
⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }

pp: case inr
α : Type u_1
inst✝ : GroupWithZero α
h✝ : Infinite { a // a ≠ 0 }
⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : GroupWithZero α
h✝ : Finite { a // a ≠ 0 }
⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝³ : Field R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
s : S
h1 : 1 ∉ s
algHom : Unitization R ↥s →ₐ[R] ↥(Algebra.adjoin R ↑s) := (unitization s).codRestrict (Algebra.adjoin R ↑s) ⋯
x : ↥(Algebra.adjoin R ↑s)
⊢ ∃ a, algHom a = x

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝³ : Field R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
s : S
h1 : 1 ∉ s
algHom : Unitization R ↥s →ₐ[R] ↥(Algebra.adjoin R ↑s) := (unitization s).codRestrict (Algebra.adjoin R ↑s) ⋯
x : ↥(Algebra.adjoin R ↑s)
⊢ ∃ a, algHom a = x
Module: Mathlib.Algebra.GroupWithZero.Units.Fintype

Goal state before:
pp: case inr
α : Type u_1
inst✝ : GroupWithZero α
h✝ : Infinite { a // a ≠ 0 }
⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : GroupWithZero α
h✝ : Infinite { a // a ≠ 0 }
⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
i : ι
⊢ (Quotient.mk (I i)) (f i * e i) = g i

pp: case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
i : ι
⊢ ∀ (x : ι), x ≠ i → (Quotient.mk (I i)) (f x * e x) = 0

Tactic:
·

Goal state after:
case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
i : ι
⊢ (Quotient.mk (I i)) (f i * e i) = g i
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case inl
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Even n
⊢ ↑n = 0 ↔ 2 ∣ n

pp: case inr
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Odd n
⊢ ↑n = 0 ↔ 2 ∣ n

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Even n
⊢ ↑n = 0 ↔ 2 ∣ n
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case inr
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Odd n
⊢ ↑n = 0 ↔ 2 ∣ n

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Odd n
⊢ ↑n = 0 ↔ 2 ∣ n
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case hp
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharP R 2
⊢ ∃ a, Even a

pp: case hn
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a

Tactic:
·

Goal state after:
case hp
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharP R 2
⊢ ∃ a, Even a
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case hn
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a

Tactic:
·

Goal state after:
case hn
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
i : ι
⊢ ∀ (x : ι), x ≠ i → (Quotient.mk (I i)) (f x * e x) = 0

Tactic:
·

Goal state after:
case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
i : ι
⊢ ∀ (x : ι), x ≠ i → (Quotient.mk (I i)) (f x * e x) = 0
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_1
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing R
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
f : A →⋆ₙₐ[R] C
S : StarSubalgebra R C
h : (starLift f).range ≤ S
⊢ NonUnitalStarAlgHom.range f ≤ S.toNonUnitalStarSubalgebra

pp: case refine_2
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing R
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
f : A →⋆ₙₐ[R] C
S : StarSubalgebra R C
h : NonUnitalStarAlgHom.range f ≤ S.toNonUnitalStarSubalgebra
⊢ (starLift f).range ≤ S

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing R
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
f : A →⋆ₙₐ[R] C
S : StarSubalgebra R C
h : (starLift f).range ≤ S
⊢ NonUnitalStarAlgHom.range f ≤ S.toNonUnitalStarSubalgebra
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_2
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing R
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
f : A →⋆ₙₐ[R] C
S : StarSubalgebra R C
h : NonUnitalStarAlgHom.range f ≤ S.toNonUnitalStarSubalgebra
⊢ (starLift f).range ≤ S

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing R
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
f : A →⋆ₙₐ[R] C
S : StarSubalgebra R C
h : NonUnitalStarAlgHom.range f ≤ S.toNonUnitalStarSubalgebra
⊢ (starLift f).range ≤ S
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case hp
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 2
⊢ ∃ a, Even a

pp: case hn
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a

Tactic:
·

Goal state after:
case hp
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 2
⊢ ∃ a, Even a
Module: Mathlib.Algebra.CharP.Two

Goal state before:
pp: case hn
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a

Tactic:
·

Goal state after:
case hn
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 2
⊢ ∃ a, ¬Even a
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁷ : Field R
inst✝⁶ : StarRing R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
inst✝ : StarMemClass S A
s : S
h1 : 1 ∉ s
starAlgHom : Unitization R ↥s →⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s) :=
  (unitization s).codRestrict (StarAlgebra.adjoin R ↑s) ⋯
⊢ Function.Injective ⇑starAlgHom

pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁷ : Field R
inst✝⁶ : StarRing R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
inst✝ : StarMemClass S A
s : S
h1 : 1 ∉ s
starAlgHom : Unitization R ↥s →⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s) :=
  (unitization s).codRestrict (StarAlgebra.adjoin R ↑s) ⋯
x : ↥(StarAlgebra.adjoin R ↑s)
⊢ ∃ a, starAlgHom a = x

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁷ : Field R
inst✝⁶ : StarRing R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
inst✝ : StarMemClass S A
s : S
h1 : 1 ∉ s
starAlgHom : Unitization R ↥s →⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s) :=
  (unitization s).codRestrict (StarAlgebra.adjoin R ↑s) ⋯
⊢ Function.Injective ⇑starAlgHom
Module: Mathlib.Algebra.Algebra.Subalgebra.Unitization

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁷ : Field R
inst✝⁶ : StarRing R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
inst✝ : StarMemClass S A
s : S
h1 : 1 ∉ s
starAlgHom : Unitization R ↥s →⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s) :=
  (unitization s).codRestrict (StarAlgebra.adjoin R ↑s) ⋯
x : ↥(StarAlgebra.adjoin R ↑s)
⊢ ∃ a, starAlgHom a = x

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁷ : Field R
inst✝⁶ : StarRing R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : SetLike S A
hSA : NonUnitalSubringClass S A
hSRA : SMulMemClass S R A
inst✝ : StarMemClass S A
s : S
h1 : 1 ∉ s
starAlgHom : Unitization R ↥s →⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s) :=
  (unitization s).codRestrict (StarAlgebra.adjoin R ↑s) ⋯
x : ↥(StarAlgebra.adjoin R ↑s)
⊢ ∃ a, starAlgHom a = x
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case «0».«1»
R✝ : Type u
S : Type v
F : Type w
inst✝³ : Ring R✝
inst✝² : Semiring S
ι✝ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I J : Ideal R
coprime : IsCoprime I J
f : Fin 2 → Ideal R := ![I, J]
h : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ (IsCoprime on f) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

pp: case «1».«0»
R✝ : Type u
S : Type v
F : Type w
inst✝³ : Ring R✝
inst✝² : Semiring S
ι✝ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I J : Ideal R
coprime : IsCoprime I J
f : Fin 2 → Ideal R := ![I, J]
h : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ (IsCoprime on f) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
·

Goal state after:
case «0».«1»
R✝ : Type u
S : Type v
F : Type w
inst✝³ : Ring R✝
inst✝² : Semiring S
ι✝ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I J : Ideal R
coprime : IsCoprime I J
f : Fin 2 → Ideal R := ![I, J]
h : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ (IsCoprime on f) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case «1».«0»
R✝ : Type u
S : Type v
F : Type w
inst✝³ : Ring R✝
inst✝² : Semiring S
ι✝ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I J : Ideal R
coprime : IsCoprime I J
f : Fin 2 → Ideal R := ![I, J]
h : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ (IsCoprime on f) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
·

Goal state after:
case «1».«0»
R✝ : Type u
S : Type v
F : Type w
inst✝³ : Ring R✝
inst✝² : Semiring S
ι✝ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I J : Ideal R
coprime : IsCoprime I J
f : Fin 2 → Ideal R := ![I, J]
h : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ (IsCoprime on f) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inl
a : ℕ
a1 : a ≠ 1
left✝ : Prime a
hp : Prime (a * 1)
⊢ a * 1 = a

pp: case inr
j : ℕ
left✝ : Prime j
a1 : 1 ≠ 1
hp : Prime (1 * j)
⊢ 1 * j = 1

Tactic:
·

Goal state after:
case inl
a : ℕ
a1 : a ≠ 1
left✝ : Prime a
hp : Prime (a * 1)
⊢ a * 1 = a
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inr
j : ℕ
left✝ : Prime j
a1 : 1 ≠ 1
hp : Prime (1 * j)
⊢ 1 * j = 1

Tactic:
·

Goal state after:
case inr
j : ℕ
left✝ : Prime j
a1 : 1 ≠ 1
hp : Prime (1 * j)
⊢ 1 * j = 1
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inl
b : ℕ
hdvd : ∀ (p : ℕ), Prime p → p ∣ 1 → p * 1 ∣ b
⊢ 1 ∣ b

pp: case inr
a b : ℕ
hdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b
ha : a ≠ 1
⊢ a ∣ b

Tactic:
·

Goal state after:
case inl
b : ℕ
hdvd : ∀ (p : ℕ), Prime p → p ∣ 1 → p * 1 ∣ b
⊢ 1 ∣ b
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case mp
m n : ℕ
⊢ ¬m.Coprime n → ∃ p, Prime p ∧ p ∣ m ∧ p ∣ n

pp: case mpr
m n : ℕ
⊢ (∃ p, Prime p ∧ p ∣ m ∧ p ∣ n) → ¬m.Coprime n

Tactic:
·

Goal state after:
case mp
m n : ℕ
⊢ ¬m.Coprime n → ∃ p, Prime p ∧ p ∣ m ∧ p ∣ n
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case mpr
m n : ℕ
⊢ (∃ p, Prime p ∧ p ∣ m ∧ p ∣ n) → ¬m.Coprime n

Tactic:
·

Goal state after:
case mpr
m n : ℕ
⊢ (∃ p, Prime p ∧ p ∣ m ∧ p ∣ n) → ¬m.Coprime n
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: n m : ℕ
h₀ : m ≠ 0
h : m < n.minFac
p : ℕ
hp : Prime p
hn : p ∣ n
hm : p ∣ m
⊢ 0 < m

Tactic:
by lia

Goal state after:
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case mp
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
⊢ Bijective ⇑(mk I) → I = ⊥

pp: case mpr
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
⊢ I = ⊥ → Bijective ⇑(mk I)

Tactic:
·

Goal state after:
case mp
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
⊢ Bijective ⇑(mk I) → I = ⊥
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case mp
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
h : Bijective ⇑(mk I)
⊢ I = ⊥

Tactic:
← map_eq_bot_iff_of_injective h.1

Goal state after:
case mp
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
h : Bijective ⇑(mk I)
⊢ map (mk I) I = ⊥
Module: Mathlib.RingTheory.Ideal.Quotient.Operations

Goal state before:
pp: case mpr
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
⊢ I = ⊥ → Bijective ⇑(mk I)

Tactic:
·

Goal state after:
case mpr
A : Type u_3
inst✝¹ : Ring A
I : Ideal A
inst✝ : I.IsTwoSided
⊢ I = ⊥ → Bijective ⇑(mk I)
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inl
x✝ y✝ p : ℕ
hp : Prime p
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
h✝ : x✝ * y✝ = p ^ 2
pdvdxy : p ∣ x✝ * y✝
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
h : x * y = p ^ 2
ha : x = p * 1
this : 1 ∣ p
⊢ x = p ∧ y = p

pp: case inr
x✝ y✝ p : ℕ
hp : Prime p
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
h✝ : x✝ * y✝ = p ^ 2
pdvdxy : p ∣ x✝ * y✝
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
h : x * y = p ^ 2
a : ℕ
ha : x = p * a
this : a ∣ p
hap : a = p
⊢ x = p ∧ y = p

Tactic:
·

Goal state after:
case inl
x✝ y✝ p : ℕ
hp : Prime p
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
h✝ : x✝ * y✝ = p ^ 2
pdvdxy : p ∣ x✝ * y✝
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
h : x * y = p ^ 2
ha : x = p * 1
this : 1 ∣ p
⊢ x = p ∧ y = p
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inl
x✝ y✝ : ℕ
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
hp : Prime x
h✝ : x✝ * y✝ = x ^ 2
pdvdxy : x ∣ x✝ * y✝
h : x * y = x ^ 2
this : 1 ∣ x
⊢ x = x ∧ y = x

Tactic:
simp only [sq, mul_right_inj' hp.ne_zero] at h

Goal state after:
case inl
x✝ y✝ : ℕ
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
hp : Prime x
h✝ : x✝ * y✝ = x ^ 2
pdvdxy : x ∣ x✝ * y✝
this : 1 ∣ x
h : y = x
⊢ x = x ∧ y = x
Module: Mathlib.Data.Nat.Prime.Basic

Goal state before:
pp: case inr
x✝ y✝ p : ℕ
hp : Prime p
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
h✝ : x✝ * y✝ = p ^ 2
pdvdxy : p ∣ x✝ * y✝
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
h : x * y = p ^ 2
a : ℕ
ha : x = p * a
this : a ∣ p
hap : a = p
⊢ x = p ∧ y = p

Tactic:
·

Goal state after:
case inr
x✝ y✝ p : ℕ
hp : Prime p
hx✝ : x✝ ≠ 1
hy✝ : y✝ ≠ 1
h✝ : x✝ * y✝ = p ^ 2
pdvdxy : p ∣ x✝ * y✝
x y : ℕ
hx : x ≠ 1
hy : y ≠ 1
h : x * y = p ^ 2
a : ℕ
ha : x = p * a
this : a ∣ p
hap : a = p
⊢ x = p ∧ y = p
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : Fintype α
⊢ Finset (List α)

pp: case refine_2
α : Type u_1
inst✝ : Fintype α
⊢ ∀ (x : List α), x ∈ ?refine_1 ↔ x.Nodup

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : Fintype α
⊢ Finset (List α)
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case h.left
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
⊢ univSubsets = ↑Finset.univ.powerset.toList

pp: case h.right
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
⊢ List.Pairwise (Function.onFun _root_.Disjoint fun s ↦ s.val.lists) Finset.univ.powerset.toList

Tactic:
·

Goal state after:
case h.left
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
⊢ univSubsets = ↑Finset.univ.powerset.toList
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case h.right
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
⊢ List.Pairwise (Function.onFun _root_.Disjoint fun s ↦ s.val.lists) Finset.univ.powerset.toList

Tactic:
·

Goal state after:
case h.right
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
⊢ List.Pairwise (Function.onFun _root_.Disjoint fun s ↦ s.val.lists) Finset.univ.powerset.toList
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case mp
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
m n : Finset α
this :
  (∀ ⦃x : Multiset (List α)⦄, x ≤ ↑m.toList.permutations → x ≤ ↑n.toList.permutations → x ≤ ⊥) ↔
    m.toList.permutations.Disjoint n.toList.permutations
⊢ (∀ a ∈ m.toList.permutations, ∀ b ∈ n.toList.permutations, a ≠ b) → ¬m = n

pp: case mpr
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
m n : Finset α
this :
  (∀ ⦃x : Multiset (List α)⦄, x ≤ ↑m.toList.permutations → x ≤ ↑n.toList.permutations → x ≤ ⊥) ↔
    m.toList.permutations.Disjoint n.toList.permutations
⊢ ¬m = n → ∀ a ∈ m.toList.permutations, ∀ b ∈ n.toList.permutations, a ≠ b

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
m n : Finset α
this :
  (∀ ⦃x : Multiset (List α)⦄, x ≤ ↑m.toList.permutations → x ≤ ↑n.toList.permutations → x ≤ ⊥) ↔
    m.toList.permutations.Disjoint n.toList.permutations
⊢ (∀ a ∈ m.toList.permutations, ∀ b ∈ n.toList.permutations, a ≠ b) → ¬m = n
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case mpr
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
m n : Finset α
this :
  (∀ ⦃x : Multiset (List α)⦄, x ≤ ↑m.toList.permutations → x ≤ ↑n.toList.permutations → x ≤ ⊥) ↔
    m.toList.permutations.Disjoint n.toList.permutations
⊢ ¬m = n → ∀ a ∈ m.toList.permutations, ∀ b ∈ n.toList.permutations, a ≠ b

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝ : Fintype α
univSubsets : Multiset (Finset α) := ⋯
allPerms : Multiset (List α) := ⋯
m n : Finset α
this :
  (∀ ⦃x : Multiset (List α)⦄, x ≤ ↑m.toList.permutations → x ≤ ↑n.toList.permutations → x ≤ ⊥) ↔
    m.toList.permutations.Disjoint n.toList.permutations
⊢ ¬m = n → ∀ a ∈ m.toList.permutations, ∀ b ∈ n.toList.permutations, a ≠ b
Module: Mathlib.Data.Fintype.List

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : Fintype α
⊢ ∀ (x : List α),
    (x ∈
        let univSubsets := Finset.univ.powerset.val;
        let allPerms := univSubsets.bind fun s ↦ s.val.lists;
        { val := allPerms, nodup := ⋯ }) ↔
      x.Nodup

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : Fintype α
⊢ ∀ (x : List α),
    (x ∈
        let univSubsets := Finset.univ.powerset.val;
        let allPerms := univSubsets.bind fun s ↦ s.val.lists;
        { val := allPerms, nodup := ⋯ }) ↔
      x.Nodup