AdicValuation.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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37 declarations, 0 contain sorry, 0 declared as axioms

Adic Completions

If A is a valued ring with field of fractions K there are two different complete rings containing A one might define, the first is 𝒪_v = {x ∈ K_v | v x ≤ 1} (defined in Lean as adicCompletionIntegers K v) and the second is the v-adic completion of A. In the case when A is a Dedekind domain these definitions give isomorphic topological A-algebras. This file makes some progress towards this.

Main theorems/defs

IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_integers : The closure of A in K_v is 𝒪_v.
IsDedekindDomain.HeightOneSpectrum.ResidueFieldEquivCompletionResidueField : The canonical isomorphism A ⧸ v ≅ 𝓞ᵥ / v.
IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_prodIntegers : If s is a set of primes of A, then the closure of A in ∏_{v ∈ s} K_v is ∏_{v ∈ s} 𝒪_v.
IsDedekindDomain.HeightOneSpectrum.denseRange_of_prodAlgebraMap : If s is a finite set of primes of A, then K is dense in ∏_{v ∈ s} K_v.
We show (as an unnamed instance) IsDiscreteValuationRing (𝒪[v.adicCompletion K])

Imports (10)

Imported by 4 files

lemma IsDedekindDomain.HeightOneSpectrum.exists_ofAdd_natCast_of_le_one line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WithZero, Multiplicative, Int, Ne, OfNat.ofNat, LE.le, Exists, Nat, Eq, Equiv, Multiplicative.ofAdd, Neg.neg
Proof uses: CanLift.prf, Int.eq_ofNat_of_zero_le, Int.neg_nonneg_of_nonpos, Eq.mp, Exists.intro, Eq.mpr, id, congrArg, Eq.symm, Int.neg_neg, Eq.refl
42 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.exists_ofAdd_natCast_lt line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WithZero, Multiplicative, Int, Ne, OfNat.ofNat, Exists, Nat, LT.lt, Equiv, Multiplicative.ofAdd, Neg.neg
Proof uses: And, WithZero.exists_ne_zero_and_lt, Eq, CanLift.prf, Exists.intro, Int.natAbs, lt_of_le_of_lt, Eq.mpr, LE.le, abs, id, Eq.trans, congrFun', congrArg, Int.natCast_natAbs, inv_mabs_le
53 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.ne_zero_of_some_le_intValuation line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Multiplicative, Int, LE.le, WithZero, Valuation, IsDedekindDomain.HeightOneSpectrum.intValuation, Ne, OfNat.ofNat
Proof uses: Eq, False, False.elim, Eq.mp, Eq.trans, congrArg, map_zero, Eq.symm
71 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.emultiplicity_eq_of_valuation_eq_ofAdd line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Nat, Eq, WithZero, Multiplicative, Int, Valuation, IsDedekindDomain.HeightOneSpectrum.intValuation, Equiv, Multiplicative.ofAdd, Neg.neg, ENat, emultiplicity, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, Ideal.span, Singleton.singleton, Set
Proof uses: Ne, OfNat.ofNat, IsDedekindDomain.HeightOneSpectrum.ne_zero_of_some_le_intValuation, le_of_eq, Eq.symm, Bot.bot, Eq.mpr, Not, id, congrArg, ne_eq, propext, Ideal.span_singleton_eq_bot, Associates.count, Associates.mk, Associates.factors, Eq.mp, Eq.trans, Inv.inv, congrFun' and 11 more
112 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.exists_adicValued_mul_sub_le Given a, b ∈ A and v b ≤ v a we can find y in A such that y is close to a / b by the valuation v line 94 source
Source documentation
Given a, b ∈ A and v b ≤ v a we can find y in A such that y is close to a / b by the valuation v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, WithZero, Multiplicative, Int, Ne, OfNat.ofNat, LE.le, Valuation, IsDedekindDomain.HeightOneSpectrum.intValuation, Exists, HSub.hSub, HMul.hMul
Proof uses: LE.le.trans, IsDedekindDomain.HeightOneSpectrum.intValuation_le_one, Nat, Eq, Equiv, Multiplicative.ofAdd, Neg.neg, IsDedekindDomain.HeightOneSpectrum.exists_ofAdd_natCast_of_le_one, Eq.mpr, id, congrArg, Eq.symm, WithZero.exp, WithZero.exp.eq_1, IsDedekindDomain.HeightOneSpectrum.ne_zero_of_some_le_intValuation, Eq.mp, Ideal, Ideal.span, Singleton.singleton, Set and 32 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
lemma exists_adicValued_mul_sub_le {a b : A} {γ : WithZero (Multiplicative ℤ)} (hγ : γ ≠ 0)
    (hle : γ ≤ v.intValuation a)
    (hle' : v.intValuation b ≤ v.intValuation a) :
    ∃ y, v.intValuation (y * a - b) ≤ γ := by
…
```
lemma IsDedekindDomain.HeightOneSpectrum.exists_adicValued_sub_lt_of_adicValued_le_one line 128 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, WithVal, WithZero, Multiplicative, Int, IsDedekindDomain.HeightOneSpectrum.valuation, Units, LE.le, Valuation, Valued.v, OfNat.ofNat, Exists, LT.lt, IsDedekindDomain.HeightOneSpectrum.adicCompletion, HSub.hSub and 4 more
Proof uses: Prod, Subtype, Membership.mem, Submonoid, nonZeroDivisors, Eq, HMul.hMul, Prod.snd, Prod.fst, IsLocalization.surj, Prod.mk, Subtype.mk, congr_arg, GE.ge, Trans.trans, WithVal.ofVal, mul_le_of_le_one_left', Eq.mp, congrFun', congrArg and 39 more
202 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
instance IsDedekindDomain.HeightOneSpectrum.(instance at line 176) line 176 source
Source comment
could go in mathlib
```
instance : Valuation.IsRankOneDiscrete vK where
…
```
theorem IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_integers The closure of A in K_v is 𝒪_v line 184 source
Source documentation
The closure of A in K_v is 𝒪_v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Eq, Set, IsDedekindDomain.HeightOneSpectrum.adicCompletion, closure, Subring, RingHom.range, Algebra.algebraMap, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers
Proof uses: subset_antisymm, closure_minimal, WithZero, Multiplicative, Int, Valuation.valuationSubring, Valued.v, Membership.mem, RingHom, IsDedekindDomain.HeightOneSpectrum.coe_mem_adicCompletionIntegers, Valued.isClosed_valuationSubring, WithVal, IsDedekindDomain.HeightOneSpectrum.valuation, IsDedekindDomain.HeightOneSpectrum.adicCompletion.ofCompletion, LE.le, Set.image, Set.preimage, And, Eq.mpr, Filter and 53 more
141 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
```
theorem closureAlgebraMapIntegers_eq_integers :
    closure (algebraMap A (v.adicCompletion K)).range =
    SetLike.coe (v.adicCompletionIntegers K) := by
…
```
theorem IsDedekindDomain.HeightOneSpectrum.denseRange_of_integerAlgebraMap A is dense in 𝒪_v line 226 source
Source documentation
A is dense in 𝒪_v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, DenseRange, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, RingHom, Algebra.algebraMap
Proof uses: Eq.mpr, Eq, Set, closure, Set.range, Set.univ, id, congrArg, propext, denseRange_iff_closure_range, Set.eq_univ_iff_forall, Set.image, closure_subtype, Subring, RingHom.range, congrFun', Eq.refl, IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_integers, Subtype.coe_prop
47 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem denseRange_of_integerAlgebraMap :
    DenseRange (algebraMap A (v.adicCompletionIntegers K)) := by
…
```
theorem IsDedekindDomain.HeightOneSpectrum.exists_adicValued_sub_lt_of_adicCompletionInteger An element of 𝒪_v can be approximated by an element of A line 241 source
Source documentation
An element of 𝒪_v can be approximated by an element of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Units, WithZero, Multiplicative, Int, Exists, LT.lt, Valuation, Valued.v, HSub.hSub and 6 more
Proof uses: Eq, Set, closure, Subring, RingHom.range, IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_integers, Filter, nhds, Set.ofPred, Eq.mpr, MonoidWithZeroHom.ValueGroup₀, Subgroup, MonoidWithZeroHom.valueGroup, LE.le, Valuation.restrict, id, congrArg, propext, Valued.mem_nhds, Exists.intro and 25 more
142 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem exists_adicValued_sub_lt_of_adicCompletionInteger
    (x : v.adicCompletionIntegers K) (γ : ℤᵐ⁰ˣ) :
    ∃a, Valued.v ((algebraMap A K a) - (x : v.adicCompletion K)) < γ.val := by
…
```
abbrev IsDedekindDomain.HeightOneSpectrum.completionIdeal The maximal ideal of the integers of the completion of v line 263 source
Source documentation
The maximal ideal of the integers of the completion of v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Ideal, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers
Proof uses: IsLocalRing.maximalIdeal
23 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev completionIdeal : Ideal (v.adicCompletionIntegers K) :=
…
```
lemma IsDedekindDomain.HeightOneSpectrum.mem_completionIdeal_iff line 266 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Iff, Ideal, IsDedekindDomain.HeightOneSpectrum.completionIdeal, LT.lt, WithZero, Multiplicative, Int, Valuation, Valued.v and 1 more
Proof uses: Valuation.mem_maximalIdeal_iff
45 instances, coercions and auxiliary constants left out.
lemma IsDedekindDomain.HeightOneSpectrum.algebraMap_completionIntegers line 270 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Eq, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, RingHom, Subtype, Algebra.algebraMap
Proof uses: rfl
37 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
instance IsDedekindDomain.HeightOneSpectrum.(instance at line 274) line 274 source
def IsDedekindDomain.HeightOneSpectrum.ResidueFieldToCompletionResidueField The canonical ring homomorphism from A / v to 𝓞ᵥ / v, where 𝓞ᵥ is the integers of the completion Kᵥ of the field of fractions of A line 284 source
Source documentation
The canonical ring homomorphism from A / v to 𝓞ᵥ / v, where 𝓞ᵥ is the integers of the completion Kᵥ of the field of fractions of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, RingHom, HasQuotient.Quotient, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, IsLocalRing.ResidueField, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers
Proof uses: Ideal.quotientMap, IsLocalRing.maximalIdeal, Algebra.algebraMap
29 instances, coercions and auxiliary constants left out.
```
noncomputable def ResidueFieldToCompletionResidueField :
    A ⧸ v.asIdeal →+* ResidueField (v.adicCompletionIntegers K) :=
…
```
instance IsDedekindDomain.HeightOneSpectrum.(instance at line 289) line 289 source
Source comment
shortcut instances for next def: needed after mathlib #34045
```
noncomputable instance : CommSemiring ↥(adicCompletionIntegers K v) := inferInstance
```
instance IsDedekindDomain.HeightOneSpectrum.(instance at line 290) line 290 source
def IsDedekindDomain.HeightOneSpectrum.ResidueFieldEquivCompletionResidueField The canonical isomorphism from A / v to 𝓞ᵥ / v, where 𝓞ᵥ is the integers of the completion Kᵥ of the field of fractions K of A line 296 source
Source documentation
The canonical isomorphism from A / v to 𝓞ᵥ / v, where 𝓞ᵥ is the integers of the completion Kᵥ of the field of fractions K of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, RingEquiv, HasQuotient.Quotient, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, IsLocalRing.ResidueField, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers
Proof uses: RingEquiv.ofBijective, RingHom, IsLocalRing.maximalIdeal, IsDedekindDomain.HeightOneSpectrum.ResidueFieldToCompletionResidueField
42 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
noncomputable def ResidueFieldEquivCompletionResidueField :
    A ⧸ v.asIdeal ≃+* ResidueField (v.adicCompletionIntegers K) := by
…
```
theorem IsDedekindDomain.HeightOneSpectrum.inertiaDeg_asIdeal_completionIdeal line 316 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Eq, Nat, Ideal.inertiaDeg, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, IsDedekindDomain.HeightOneSpectrum.completionIdeal, OfNat.ofNat
Proof uses: Ideal.IsMaximal, IsLocalRing.maximalIdeal, Eq.mpr, Module.finrank, HasQuotient.Quotient, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, id, congrArg, Ideal.inertiaDeg_eq_of_isMaximal, LinearEquiv, RingHom.id, RingEquiv, IsLocalRing.ResidueField, IsDedekindDomain.HeightOneSpectrum.ResidueFieldEquivCompletionResidueField, LinearEquiv.mk, LinearMap.mk, AddHom.mk, Equiv.toFun, RingEquiv.toEquiv and 13 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem IsDedekindDomain.HeightOneSpectrum.exists_forall_adicValued_sub_lt An element of ∏_{v ∈ s} 𝒪_v, with s finite, can be approximated by an element of A line 333 source
Source documentation
An element of ∏_{v ∈ s} 𝒪_v, with s finite, can be approximated by an element of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, Finset, Units, WithZero, Multiplicative, Int, IsDedekindDomain.HeightOneSpectrum, Function.Injective, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Exists, LT.lt, Valuation and 8 more
Proof uses: Nat, Equiv, Multiplicative.ofAdd, Neg.neg, IsDedekindDomain.HeightOneSpectrum.exists_ofAdd_natCast_lt, Units.ne_zero, Ne, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, mt, Eq, IsDedekindDomain.HeightOneSpectrum.ext, HPow.hPow, Subtype.mk, IsDedekindDomain.exists_forall_sub_mem_ideal, IsDedekindDomain.HeightOneSpectrum.prime, Exists.intro, lt_of_le_of_lt, HAdd.hAdd, le_of_eq and 29 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem exists_forall_adicValued_sub_lt {ι : Type*} (s : Finset ι)
    (e : ι → (WithZero (Multiplicative ℤ))ˣ) (valuation : ι → HeightOneSpectrum A)
    (injective : Function.Injective valuation)
    (x : (i : ι) → (valuation i).adicCompletionIntegers K) :
    ∃ a, ∀ i ∈ s, Valued.v ((algebraMap A K a) - (x i).val) < (e i).val := by
…
```
theorem IsDedekindDomain.HeightOneSpectrum.closureAlgebraMapIntegers_eq_prodIntegers The closure of A in ∏_{v ∈ s} K_v is ∏_{v ∈ s} 𝒪_v line 369 source
Source documentation
The closure of A in ∏_{v ∈ s} K_v is ∏_{v ∈ s} 𝒪_v. s may be infinite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Function.Injective, Eq, Set, IsDedekindDomain.HeightOneSpectrum.adicCompletion, closure, Subring, RingHom.range, Algebra.algebraMap, Set.pi, Set.univ, Subsemigroup.carrier, Submonoid.toSubsemigroup, Subsemiring.toSubmonoid, Subring.toSubsemiring and 2 more
Proof uses: Set.Subset.antisymm, closure_minimal, Membership.mem, RingHom, Eq.mpr, id, congrArg, Eq.symm, IsDedekindDomain.HeightOneSpectrum.coe_mem_adicCompletionIntegers, isClosed_set_pi, Valued.isClosed_valuationSubring, WithZero, Multiplicative, Int, Filter, nhds, OfNat.ofNat, Exists, And, HSub.hSub and 46 more
149 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
```
theorem closureAlgebraMapIntegers_eq_prodIntegers {ι : Type*}
    (v : ι → HeightOneSpectrum A)
    (injective : Function.Injective v) :
    closure (SetLike.coe (algebraMap A ((i : ι) → (v i).adicCompletion K)).range) =
    (Set.pi Set.univ (fun (i : ι) ↦ ((v i).adicCompletionIntegers K).carrier)) := by
…
```
lemma IsDedekindDomain.HeightOneSpectrum.adicCompletion.eq_mul_nonZeroDivisor_inv_adicCompletionIntegers line 402 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Exists, And, Membership.mem, Submonoid, nonZeroDivisors, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Eq, HSMul.hSMul, Inv.inv, RingHom, Algebra.algebraMap
Proof uses: HMul.hMul, IsDedekindDomain.HeightOneSpectrum.adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers, Exists.intro, And.intro, Eq.mpr, id, congrArg, Algebra.smul_def, Eq.symm, IsScalarTower.algebraMap_apply, mul_comm, smul_smul, OfNat.ofNat, inv_mul_cancel₀, IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors, one_smul, Eq.refl
87 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.adicCompletion.eq_mul_pi_adicCompletionIntegers line 413 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, Finite, IsDedekindDomain.HeightOneSpectrum, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Exists, And, Membership.mem, Set, Set.pi, Set.univ, Subsemigroup.carrier, Submonoid.toSubsemigroup, Subsemiring.toSubmonoid, Subring.toSubsemiring, ValuationSubring.toSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers and 2 more
Proof uses: Fintype, Fintype.ofFinite, Submonoid, nonZeroDivisors, ValuationSubring, Inv.inv, RingHom, Algebra.algebraMap, Exists.intro, Finset.prod, Finset.univ, Ne, OfNat.ofNat, IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors, id, And.left, And.intro, And.right, Eq.mpr, congrArg and 41 more
128 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem IsDedekindDomain.HeightOneSpectrum.denseRange_of_prodAlgebraMap If s is finite then K in dense in ∏_{v ∈ s} K_v line 438 source
Source documentation
If s is finite then K in dense in ∏_{v ∈ s} K_v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, Finite, IsDedekindDomain.HeightOneSpectrum, Function.Injective, DenseRange, IsDedekindDomain.HeightOneSpectrum.adicCompletion, RingHom, Algebra.algebraMap
Proof uses: Eq.mpr, Eq, Set, closure, Set.range, Set.univ, id, congrArg, propext, denseRange_iff_closure_range, Membership.mem, Set.eq_univ_iff_forall, LE.le, Set.pi, Subsemigroup.carrier, Submonoid.toSubsemigroup, Subsemiring.toSubmonoid, Subring.toSubsemiring, ValuationSubring.toSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers and 35 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem denseRange_of_prodAlgebraMap {ι : Type*} [Finite ι]
    {valuation : ι → HeightOneSpectrum A} (injective : Function.Injective valuation) :
    DenseRange (algebraMap K ((i : ι) → (valuation i).adicCompletion K)) := by
…
```
theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.exists_uniformizer line 470 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Exists, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Eq, WithZero, Multiplicative, Int, Valuation, Valued.v, Equiv, Multiplicative.ofAdd and 2 more
Proof uses: IsDedekindDomain.HeightOneSpectrum.intValuation, WithZero.exp, IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizer, Exists.intro, Eq.mpr, id, congrArg, Eq.symm, WithZero.exp.eq_1, RingHom, Algebra.algebraMap, ValuationSubring.algebraMap_apply, IsScalarTower.algebraMap_apply, IsDedekindDomain.HeightOneSpectrum.valuation, IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation, IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap, Eq.refl
71 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.uniformizer_ne_zero line 478 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Eq, WithZero, Multiplicative, Int, Valuation, Valued.v, Equiv, Multiplicative.ofAdd, Neg.neg and 2 more
Proof uses: Mathlib.Tactic.Contrapose.contrapose₃, id, Not, of_eq_true, Inv.inv, Eq.trans, True, congrFun', congrArg, map_zero, False, not_false_eq_true
96 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.(instance at line 485) line 485 source
Source comment
shortcut instance for next theorem: needed after mathlib #34045
```
noncomputable instance : Ring (adicCompletion K v) := inferInstance
…
```
theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.uniformizer_not_isUnit line 490 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Eq, WithZero, Multiplicative, Int, Valuation, Valued.v, Equiv, Multiplicative.ofAdd, Neg.neg and 3 more
Proof uses: Eq.mpr, Valuation.valuationSubring, id, congrArg, propext, ValuationSubring.isUnit_iff_valued_eq_one, Eq.symm, WithZero.coe_one, ofAdd_zero, ne_of_lt, LT.lt, WithZero.coe_lt_coe, Multiplicative.ofAdd_lt, Decidable.byContradiction
91 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.eq_pow_uniformizer_mul_unit line 499 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Ne, OfNat.ofNat, Eq, WithZero, Multiplicative, Int, Valuation, Valued.v, Equiv and 7 more
Proof uses: of_eq_true, Not, Eq.trans, False, True, congrArg, eq_false, not_false_eq_true, Multiplicative.toAdd, WithZero.unzero, LE.le, id, Eq.mpr, Eq.symm, propext, WithZero.coe_le_coe, Eq.rec, Subtype.property, WithZero.coe_unzero, Valuation.map_mul and 29 more
194 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.maximalIdeal_eq_span_uniformizer line 518 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Eq, WithZero, Multiplicative, Int, Valuation, Valued.v, Equiv, Multiplicative.ofAdd, Neg.neg and 6 more
Proof uses: Ideal.IsMaximal.eq_of_le, Ideal.span_singleton_ne_top, IsDedekindDomain.HeightOneSpectrum.adicCompletion.uniformizer_not_isUnit, Submodule, dite, Classical.propDecidable, of_eq_true, Eq.trans, True, congrArg, Not, Nat, Exists, Units, HMul.hMul, HPow.hPow, IsDedekindDomain.HeightOneSpectrum.adicCompletion.eq_pow_uniformizer_mul_unit, IsUnit, Ideal.IsMaximal.ne_top, Ideal.eq_top_of_isUnit_mem and 10 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.(instance at line 532) line 532 source
instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.(instance at line 547) line 547 source
instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.(instance at line 557) line 557 source
instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.(instance at line 566) line 566 source
lemma IsDedekindDomain.HeightOneSpectrum.adicCompletion.mem_completionIdeal_pow line 569 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Nat, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Iff, Ideal, HPow.hPow, IsDedekindDomain.HeightOneSpectrum.completionIdeal, LE.le, WithZero, Multiplicative, Int and 5 more
Proof uses: Eq, OfNat.ofNat, Exists, IsDedekindDomain.HeightOneSpectrum.adicCompletion.exists_uniformizer, id, Eq.mpr, IsLocalRing.maximalIdeal, Ideal.span, Singleton.singleton, Set, congrArg, IsDedekindDomain.HeightOneSpectrum.adicCompletion.maximalIdeal_eq_span_uniformizer, Ideal.span_singleton_pow, HMul.hMul, propext, Ideal.mem_span_singleton', Eq.trans, Inv.inv, congrFun', inv_pow and 30 more
230 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.mem_completionIdeal_iff' line 599 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Iff, Ideal, IsDedekindDomain.HeightOneSpectrum.completionIdeal, LE.le, WithZero, Multiplicative, Int, Valuation, Valued.v and 4 more
Proof uses: Eq.mpr, HPow.hPow, Submodule, Nat, id, Eq, congrArg, Eq.symm, Submodule.pow_one, propext, IsDedekindDomain.HeightOneSpectrum.adicCompletion.mem_completionIdeal_pow, Int.natCast_one, Iff.rfl
76 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.HeightOneSpectrum.completionIdeal_ne_bot line 604 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, IsFractionRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Ne, Ideal, Subtype, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, IsDedekindDomain.HeightOneSpectrum.completionIdeal, Bot.bot
Proof uses: IsDiscreteValuationRing.not_a_field
26 instances, coercions and auxiliary constants left out.