mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case refine_1
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ ↑I⁻¹ * spanSingleton S w ≤ 1

pp: case refine_2
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ 1 ≤ ↑I⁻¹ * spanSingleton S w

Tactic:
·

Goal state after:
case refine_1
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ ↑I⁻¹ * spanSingleton S w ≤ 1
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case refine_2
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ 1 ≤ ↑I⁻¹ * spanSingleton S w

Tactic:
·

Goal state after:
case refine_2
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
hJ : IsLocalization.coeSubmodule A J = ↑↑I * (R ∙ v)
this : 1 ∈ ↑↑I * (R ∙ v)
w : A
hw : w ∈ ↑↑I
hvw : w * v = 1
⊢ 1 ≤ ↑I⁻¹ * spanSingleton S w
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
M : Ideal R
hM : M ∈ {I | I.IsMaximal}
M' : Ideal R
hne : M' ≠ M
hM' : M' ∈ {I | I.IsMaximal}
⊢ M ⊔ M' = ⊤

Tactic:
exact Ideal.IsMaximal.coprime_of_ne hM hM' hne.symm

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this✝ : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
hv : v ∈ I'
h : ¬Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) = ⊤
M : Ideal R
hJM : Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) ≤ M
hM : M ∈ hf.toFinset
this : ∀ a ∈ I, ∀ b ∈ I', ∃ c, (algebraMap R A) c = a * b
hmem : a M * v ∈ IsLocalization.coeSubmodule A M
⊢ False

Tactic:
v,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this✝ : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
hv : v ∈ I'
h : ¬Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) = ⊤
M : Ideal R
hJM : Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) ≤ M
hM : M ∈ hf.toFinset
this : ∀ a ∈ I, ∀ b ∈ I', ∃ c, (algebraMap R A) c = a * b
hmem : a M * ∑ M ∈ s, u M • b M ∈ IsLocalization.coeSubmodule A M
⊢ False
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case a
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this✝ : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
hv : v ∈ I'
h : ¬Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) = ⊤
M : Ideal R
hJM : Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) ≤ M
hM : M ∈ hf.toFinset
this : ∀ a ∈ I, ∀ b ∈ I', ∃ c, (algebraMap R A) c = a * b
hmem : u M • (a M * b M) + ∑ x ∈ s.erase M, u x • (a M * b x) ∈ IsLocalization.coeSubmodule A M
⊢ ∑ x ∈ s.erase M, u x • (a M * b x) ∈ IsLocalization.coeSubmodule A M

Tactic:
·

Goal state after:
case a
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this✝ : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
hv : v ∈ I'
h : ¬Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) = ⊤
M : Ideal R
hJM : Submodule.comap (Algebra.linearMap R A) (↑↑(Units.mkOfMulEqOne I I' hinv') * (R ∙ v)) ≤ M
hM : M ∈ hf.toFinset
this : ∀ a ∈ I, ∀ b ∈ I', ∃ c, (algebraMap R A) c = a * b
hmem : u M • (a M * b M) + ∑ x ∈ s.erase M, u x • (a M * b x) ∈ IsLocalization.coeSubmodule A M
⊢ ∑ x ∈ s.erase M, u x • (a M * b x) ∈ IsLocalization.coeSubmodule A M
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
h : {I | I.IsMaximal}.Finite
⊢ Submodule.IsPrincipal ⊥

pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
h : {I | I.IsMaximal}.Finite
I : Ideal R
hI : I ≠ ⊥
⊢ Submodule.IsPrincipal I

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
h : {I | I.IsMaximal}.Finite
⊢ Submodule.IsPrincipal ⊥
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ (IsLocalRing.maximalIdeal (Localization.AtPrime p)).IsPrime

pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ (IsLocalRing.maximalIdeal (Localization.AtPrime p)).IsPrime
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Ideal.map (algebraMap R Sₚ) p ≠ 0
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ ¬p = ⊥

pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ Function.Injective ⇑(algebraMap R Sₚ)

Tactic:
·

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module.IsTorsionFree R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ ¬p = ⊥
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Ring R
I : Ideal R
h : absNorm (under ℤ I) = 0
⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}

pp: case neg
R : Type u_1
inst✝ : Ring R
I : Ideal R
h : ¬absNorm (under ℤ I) = 0
⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : Ring R
I : Ideal R
h : absNorm (under ℤ I) = 0
⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Ring R
I : Ideal R
h : ¬absNorm (under ℤ I) = 0
⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : Ring R
I : Ideal R
h : ¬absNorm (under ℤ I) = 0
⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: case inl
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Finite (S ⧸ I)
⊢ absNorm (under ℤ I) ∣ absNorm I

pp: case inr
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Infinite (S ⧸ I)
⊢ absNorm (under ℤ I) ∣ absNorm I

Tactic:
·

Goal state after:
case inl
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Finite (S ⧸ I)
⊢ absNorm (under ℤ I) ∣ absNorm I
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: case inr
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Infinite (S ⧸ I)
⊢ absNorm (under ℤ I) ∣ absNorm I

Tactic:
·

Goal state after:
case inr
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Infinite (S ⧸ I)
⊢ absNorm (under ℤ I) ∣ absNorm I
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ (under ℤ P).IsPrime

pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ ↑(absNorm (under ℤ P)) ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ (under ℤ P).IsPrime
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ ↑(absNorm (under ℤ P)) ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ ↑(absNorm (under ℤ P)) ≠ 0
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: case refine_3
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ (↑(FractionalIdeal.dual ℤ ℚ 1)).toAddSubgroup = AddSubgroup.closure (Set.range ⇑b')

pp: case refine_4
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ |b'.det ⇑b| = |↑(discr K)|

Tactic:
·

Goal state after:
case refine_3
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ (↑(FractionalIdeal.dual ℤ ℚ 1)).toAddSubgroup = AddSubgroup.closure (Set.range ⇑b')
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: case refine_4
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ |b'.det ⇑b| = |↑(discr K)|

Tactic:
·

Goal state after:
case refine_4
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ |b'.det ⇑b| = |↑(discr K)|