l3lab/ntp-mathlib huggingface dataset:
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    

Proof state:
case inl.h
S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : IsDedekindDomain S
inst✝² : Module.Free ℤ S
inst✝¹ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : IsEmpty (Module.Free.ChooseBasisIndex ℤ S)
this : Subsingleton S
inst✝ : Nontrivial S
⊢ False

Next tactic:
exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  

Proof state:
case inr
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  

Proof state:
case inr
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
by_cases hI : I = ⊥
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · 

Proof state:
case pos
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
hI : I = ⊥
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
hI : ¬I = ⊥
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
by_cases hJ : J = ⊥
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · 

Proof state:
case pos
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
hI : ¬I = ⊥
hJ : J = ⊥
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
hIJ : I * J ≠ ⊥
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
letI := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
  letI := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this✝ : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
hIJ : I * J ≠ ⊥
this : DecidableEq (Module.Free.ChooseBasisIndex ℤ S) := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
letI := I.fintypeQuotientOfFreeOfNeBot hI
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
  letI := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
  letI := I.fintypeQuotientOfFreeOfNeBot hI
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this✝¹ : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
hIJ : I * J ≠ ⊥
this✝ : DecidableEq (Module.Free.ChooseBasisIndex ℤ S) := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
this : Fintype (S ⧸ I) := Ideal.fintypeQuotientOfFreeOfNeBot I hI
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
letI := J.fintypeQuotientOfFreeOfNeBot hJ
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
  letI := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
  letI := I.fintypeQuotientOfFreeOfNeBot hI
  letI := J.fintypeQuotientOfFreeOfNeBot hJ
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this✝² : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
hIJ : I * J ≠ ⊥
this✝¹ : DecidableEq (Module.Free.ChooseBasisIndex ℤ S) := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
this✝ : Fintype (S ⧸ I) := Ideal.fintypeQuotientOfFreeOfNeBot I hI
this : Fintype (S ⧸ J) := Ideal.fintypeQuotientOfFreeOfNeBot J hJ
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
letI := (I * J).fintypeQuotientOfFreeOfNeBot hIJ
/-- Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem.
-/
theorem cardQuot_mul_of_coprime [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S]
    {I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by
  let b := Module.Free.chooseBasis ℤ S
  cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
  · haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
    nontriviality S
    exfalso
    exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
  haveI : Infinite S := Infinite.of_surjective _ b.repr.toEquiv.surjective
  by_cases hI : I = ⊥
  · rw [hI, Submodule.bot_mul, cardQuot_bot, zero_mul]
  by_cases hJ : J = ⊥
  · rw [hJ, Submodule.mul_bot, cardQuot_bot, mul_zero]
  have hIJ : I * J ≠ ⊥ := mt Ideal.mul_eq_bot.mp (not_or_of_not hI hJ)
  letI := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
  letI := I.fintypeQuotientOfFreeOfNeBot hI
  letI := J.fintypeQuotientOfFreeOfNeBot hJ
  letI := (I * J).fintypeQuotientOfFreeOfNeBot hIJ
  

Proof state:
case neg
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
I J : Ideal S
coprime : IsCoprime I J
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty (Module.Free.ChooseBasisIndex ℤ S)
this✝³ : Infinite S
hI : ¬I = ⊥
hJ : ¬J = ⊥
hIJ : I * J ≠ ⊥
this✝² : DecidableEq (Module.Free.ChooseBasisIndex ℤ S) := Classical.decEq (Module.Free.ChooseBasisIndex ℤ S)
this✝¹ : Fintype (S ⧸ I) := Ideal.fintypeQuotientOfFreeOfNeBot I hI
this✝ : Fintype (S ⧸ J) := Ideal.fintypeQuotientOfFreeOfNeBot J hJ
this : Fintype (S ⧸ I * J) := Ideal.fintypeQuotientOfFreeOfNeBot (I * J) hIJ
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Next tactic:
rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Fintype.card_eq.mpr ⟨(Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv⟩,
    Fintype.card_prod]
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  

Proof state:
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ a * d + e - (a * d' + e') ∈ P ^ (i + 1)

Next tactic:
have : a * d - a * d' ∈ P ^ (i + 1) := by
    convert Ideal.mul_mem_mul a_mem h using 1 <;> simp [mul_sub, pow_succ, mul_comm]
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  have : a * d - a * d' ∈ P ^ (i + 1) := by
    

Proof state:
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ a * d - a * d' ∈ P ^ (i + 1)

Next tactic:
convert Ideal.mul_mem_mul a_mem h using 1
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  have : a * d - a * d' ∈ P ^ (i + 1) := by
    convert Ideal.mul_mem_mul a_mem h using 1 <;> 

Proof state:
case h.e'_4
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ a * d - a * d' = a * (d - d')

Next tactic:
simp [mul_sub, pow_succ, mul_comm]
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  have : a * d - a * d' ∈ P ^ (i + 1) := by
    convert Ideal.mul_mem_mul a_mem h using 1 <;> 

Proof state:
case h.e'_5
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ P ^ (i + 1) = P ^ i * P

Next tactic:
simp [mul_sub, pow_succ, mul_comm]
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  have : a * d - a * d' ∈ P ^ (i + 1) := by
    convert Ideal.mul_mem_mul a_mem h using 1 <;> simp [mul_sub, pow_succ, mul_comm]
  

Proof state:
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
this : a * d - a * d' ∈ P ^ (i + 1)
⊢ a * d + e - (a * d' + e') ∈ P ^ (i + 1)

Next tactic:
convert Ideal.add_mem _ this (Ideal.sub_mem _ e_mem e'_mem) using 1
/-- If the `d` from `Ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by
  have : a * d - a * d' ∈ P ^ (i + 1) := by
    convert Ideal.mul_mem_mul a_mem h using 1 <;> simp [mul_sub, pow_succ, mul_comm]
  convert Ideal.add_mem _ this (Ideal.sub_mem _ e_mem e'_mem) using 1
  

Proof state:
case h.e'_4
S : Type u_1
inst✝¹ : CommRing S
inst✝ : IsDomain S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
this : a * d - a * d' ∈ P ^ (i + 1)
⊢ a * d + e - (a * d' + e') = a * d - a * d' + (e - e')

Next tactic:
ring
/-- If `a ∈ P^i \ P^(i+1)` and `c ∈ P^i`, then `a * d + e = c` for `e ∈ P^(i+1)`.
`Ideal.mul_add_mem_pow_succ_unique` shows the choice of `d` is unique, up to `P`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.exists_mul_add_mem_pow_succ [IsDedekindDomain S] {i : ℕ} (a c : S) (a_mem : a ∈ P ^ i)
    (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i) :
    ∃ d : S, ∃ e ∈ P ^ (i + 1), a * d + e = c := by
  

Proof state:
S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDomain S
P : Ideal S
P_prime : IsPrime P
hP : P ≠ ⊥
inst✝ : IsDedekindDomain S
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_not_mem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ ∃ d, ∃ e ∈ P ^ (i + 1), a * d + e = c

Next tactic:
suffices eq_b : P ^ i = Ideal.span {a} ⊔ P ^ (i + 1)
/-- If `a ∈ P^i \ P^(i+1)` and `c ∈ P^i`, then `a * d + e = c` for `e ∈ P^(i+1)`.
`Ideal.mul_add_mem_pow_succ_unique` shows the choice of `d` is unique, up to `P`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.exists_mul_add_mem_pow_succ [IsDedekindDomain S] {i : ℕ} (a c : S) (a_mem : a ∈ P ^ i)
    (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i) :
    ∃ d : S, ∃ e ∈ P ^ (i + 1), a * d + e = c := by
  suffices eq_b : P ^ i = Ideal.span {a} ⊔ P ^ (i + 1)
  · 

Proof state:
S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDomain S
P : Ideal S
P_prime : IsPrime P
hP : P ≠ ⊥
inst✝ : IsDedekindDomain S
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_not_mem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
eq_b : P ^ i = span {a} ⊔ P ^ (i + 1)
⊢ ∃ d, ∃ e ∈ P ^ (i + 1), a * d + e = c

Next tactic:
rw [eq_b] at c_mem
/-- If `a ∈ P^i \ P^(i+1)` and `c ∈ P^i`, then `a * d + e = c` for `e ∈ P^(i+1)`.
`Ideal.mul_add_mem_pow_succ_unique` shows the choice of `d` is unique, up to `P`.
Inspired by [Neukirch], proposition 6.1 -/
theorem Ideal.exists_mul_add_mem_pow_succ [IsDedekindDomain S] {i : ℕ} (a c : S) (a_mem : a ∈ P ^ i)
    (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i) :
    ∃ d : S, ∃ e ∈ P ^ (i + 1), a * d + e = c := by
  suffices eq_b : P ^ i = Ideal.span {a} ⊔ P ^ (i + 1)
  · rw [eq_b] at c_mem
    

Proof state:
S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDomain S
P : Ideal S
P_prime : IsPrime P
hP : P ≠ ⊥
inst✝ : IsDedekindDomain S
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_not_mem : a ∉ P ^ (i + 1)
c_mem : c ∈ span {a} ⊔ P ^ (i + 1)
eq_b : P ^ i = span {a} ⊔ P ^ (i + 1)
⊢ ∃ d, ∃ e ∈ P ^ (i + 1), a * d + e = c

Next tactic:
simp only [mul_comm a]