Module

Exists_isUnit_det_evalAt_differentialCoeff.lean

Nyll.AlgebraicCurves.Differentials.Exists_isUnit_det_evalAt_differentialCoeff

33 declarations indexed from sourceView source ↗

Module documentation

Exists is Unit det eval At differential Coeff

Imports12

Imported by1

Nyll.AlgebraicCurves.Differentials.All

theorem

ord_pos_of_not_isUnit

L132 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ord_pos_of_not_isUnit

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem ord_pos_of_not_isUnit (v : Place K F) {a : v.toValuationSubring} (ha : a ≠ 0)
    (hu : ¬IsUnit a) : 0 < v.ord (a : F)

theorem

ord_neg_of_not_mem

L145 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ord_neg_of_not_mem

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem ord_neg_of_not_mem (v : Place K F) {f : F} (hf : f ∉ v.toValuationSubring) :
    v.ord f < 0

theorem

mem_of_ord_nonneg

L170 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.mem_of_ord_nonneg

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mem_of_ord_nonneg (v : Place K F) {f : F} (h : 0 ≤ v.ord f) :
    f ∈ v.toValuationSubring

theorem

ord_nonneg_of_mem

L175 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ord_nonneg_of_mem

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem ord_nonneg_of_mem (v : Place K F) {f : F} (hf : f ∈ v.toValuationSubring) :
    0 ≤ v.ord f

theorem

isUnit_algebraMap

L183 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.isUnit_algebraMap

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isUnit_algebraMap (v : Place K F) {c : K} (hc : c ≠ 0) :
    IsUnit (⟨algebraMap K F c, v.algebraMap_mem' c⟩ : v.toValuationSubring)

theorem

ord_algebraMap

L190 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ord_algebraMap

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem ord_algebraMap (v : Place K F) (c : K) : v.ord (algebraMap K F c) = 0

theorem

mem_maximalIdeal_of_ord_pos

L195 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.mem_maximalIdeal_of_ord_pos

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mem_maximalIdeal_of_ord_pos (v : Place K F) {f : F} (hf : f ∈ v.toValuationSubring)
    (h : 0 < v.ord f) :
    (⟨f, hf⟩ : v.toValuationSubring) ∈ IsLocalRing.maximalIdeal v.toValuationSubring

theorem

isRational_of_isAlgClosed

L205 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.isRational_of_isAlgClosed

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isRational_of_isAlgClosed [IsAlgClosed K] (w : Place K F) [w.FiniteResidue] :
    w.IsRational

theorem

residue_algebraMap

L211 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.residue_algebraMap

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem residue_algebraMap (w : Place K F) (a : K) :
    IsLocalRing.residue w.toValuationSubring
        (⟨algebraMap K F a, w.algebraMap_mem' a⟩ : w.toValuationSubring) =
      algebraMap K w.ResidueField a

theorem

evalAt_algebraMap'

L220 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalAt_algebraMap'

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem evalAt_algebraMap' (w : Place K F) (a : K) : w.evalAt (algebraMap K F a) = a

theorem

evalAt_mul

L223 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalAt_mul

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem evalAt_mul (w : Place K F) (hw : w.IsRational) {f g : F}
    (hf : f ∈ w.toValuationSubring) (hg : g ∈ w.toValuationSubring) :
    w.evalAt (f * g) = w.evalAt f * w.evalAt g

theorem

evalAt_add

L231 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalAt_add

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem evalAt_add (w : Place K F) (hw : w.IsRational) {f g : F}
    (hf : f ∈ w.toValuationSubring) (hg : g ∈ w.toValuationSubring) :
    w.evalAt (f + g) = w.evalAt f + w.evalAt g

theorem

ord_pos_of_evalAt_eq_zero

L239 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ord_pos_of_evalAt_eq_zero

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem ord_pos_of_evalAt_eq_zero (w : Place K F) (hw : w.IsRational) {f : F} (hf0 : f ≠ 0)
    (hf : f ∈ w.toValuationSubring) (h : w.evalAt f = 0) : 0 < w.ord f

theorem

evalAt_eq_zero_of_ord_pos

L245 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalAt_eq_zero_of_ord_pos

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem evalAt_eq_zero_of_ord_pos (w : Place K F) {f : F} (hf : f ∈ w.toValuationSubring)
    (h : 0 < w.ord f) : w.evalAt f = 0

theorem

differentialCoeff_add

L251 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.differentialCoeff_add

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem differentialCoeff_add (v : Place K F) [v.DCoordGenerates] [Nontrivial (Ω[F⁄K])]
    (θ θ' : Ω[F⁄K]) :
    v.differentialCoeff (θ + θ') = v.differentialCoeff θ + v.differentialCoeff θ'

theorem

differentialCoeff_mem

L257 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.differentialCoeff_mem

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem differentialCoeff_mem [Nontrivial (Ω[F⁄K])] {η : Ω[F⁄K]}
    (hη : η ∈ regularDifferentials K F) (v : Place K F) [v.DCoordGenerates] :
    v.differentialCoeff η ∈ v.toValuationSubring

theorem

differentialCoeff_ne_zero

L264 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.differentialCoeff_ne_zero

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem differentialCoeff_ne_zero (v : Place K F) [v.DCoordGenerates] {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    v.differentialCoeff ω ≠ 0

theorem

mem_regularDifferentials_of_ord_nonneg

L270 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.mem_regularDifferentials_of_ord_nonneg

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Algebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mem_regularDifferentials_of_ord_nonneg [Nontrivial (Ω[F⁄K])]
    [∀ v : Place K F, v.DCoordGenerates] {η : Ω[F⁄K]}
    (h : ∀ v : Place K F, 0 ≤ v.ord (v.differentialCoeff η)) : η ∈ regularDifferentials K F

theorem

isRational

L284 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.isRational

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
omit [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem isRational (w : Place K F) : w.IsRational

def

evalFunctional

L288 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalFunctional

Documentation

The linear functional evaluating the local coefficient of a regular differential at the place.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]

Lean source · full declaration

def evalFunctional (v : Place K F) : ↥(regularDifferentials K F) →ₗ[K] K where
  toFun θ := v.evalAt (v.differentialCoeff (θ : Ω[F⁄K]))
  map_add' θ θ' := by
    rw [Submodule.coe_add, differentialCoeff_add, evalAt_add v (isRational v)
      (differentialCoeff_mem θ.2 v) (differentialCoeff_mem θ'.2 v)]
  map_smul' c θ := by
    rw [Submodule.coe_smul, RingHom.id_apply, ← algebraMap_smul (A := F) c (θ : Ω[F⁄K]),
      v.differentialCoeff_smul, evalAt_mul v (isRational v) (v.algebraMap_mem' c)
        (differentialCoeff_mem θ.2 v), evalAt_algebraMap', smul_eq_mul]

theorem

evalFunctional_apply

L298 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.evalFunctional_apply

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]

Lean source · statement

theorem evalFunctional_apply (v : Place K F) (θ : ↥(regularDifferentials K F)) :
    evalFunctional v θ = v.evalAt (v.differentialCoeff (θ : Ω[F⁄K]))

theorem

eq_zero_of_forall_evalFunctional_eq_zero

L301 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.eq_zero_of_forall_evalFunctional_eq_zero

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]

Lean source · statement

theorem eq_zero_of_forall_evalFunctional_eq_zero [HasCanonicalDivisor (K := K) (F := F)]
    [Infinite (Place K F)] (θ : ↥(regularDifferentials K F))
    (h : ∀ v : Place K F, evalFunctional v θ = 0) : θ = 0

theorem

smul_mem_of_mem_lSpace

L317 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.smul_mem_of_mem_lSpace

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [IsAlgClosed K] in

Lean source · statement

theorem smul_mem_of_mem_lSpace {ω : Ω[F⁄K]} (hω : ω ≠ 0) {D : Divisor K F} (hD : ∀ v, 0 ≤ D v)
    {f : F} (hf : f ∈ LSpace (canonicalDivisorOf hω - D)) (hf0 : f ≠ 0) :
    f • ω ∈ regularDifferentials K F ∧
      ∀ v : Place K F, D v ≤ v.ord (v.differentialCoeff (f • ω))

theorem

ell_canonicalDivisorOf_eq

L331 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ell_canonicalDivisorOf_eq

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [IsAlgClosed K] in

Lean source · statement

theorem ell_canonicalDivisorOf_eq {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    ell (canonicalDivisorOf hω) = Module.finrank K ↥(regularDifferentials K F)

theorem

ell_zero_eq_one

L371 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.ell_zero_eq_one

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]

Lean source · statement

theorem ell_zero_eq_one [Algebra.EssFiniteType K F] (hRR : FunctionFieldRiemannRoch K F) :
    ell (0 : Divisor K F) = 1

theorem

exists_eq_algebraMap_of_ell_eq_one

L381 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.exists_eq_algebraMap_of_ell_eq_one

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [IsAlgClosed K] [IsCurveOver K F] [∀ (v : Place K F), v.DCoordGenerates]
    [HasCanonicalDivisor (K := K) (F := F)] in

Lean source · statement

theorem exists_eq_algebraMap_of_ell_eq_one {D : Divisor K F} (hD : 0 ≤ D) (hell : ell D = 1)
    {f : F} (hf : f ∈ LSpace D) : ∃ c : K, f = algebraMap K F c

theorem

exists_ord_neg_of_forall_ne

L393 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.exists_ord_neg_of_forall_ne

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [IsAlgClosed K] [IsCurveOver K F] [∀ (v : Place K F), v.DCoordGenerates]
    [HasCanonicalDivisor (K := K) (F := F)] in

Lean source · statement

theorem exists_ord_neg_of_forall_ne (hell : ell (0 : Divisor K F) = 1) {f : F}
    (hf : ∀ c : K, f ≠ algebraMap K F c) : ∃ v : Place K F, v.ord f < 0

theorem

infinite_place

L404 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.infinite_place

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [IsCurveOver K F] [∀ (v : Place K F), v.DCoordGenerates]
    [HasCanonicalDivisor (K := K) (F := F)] in

Lean source · statement

theorem infinite_place (hell : ell (0 : Divisor K F) = 1) {x : F} (hx : Transcendental K x) :
    Infinite (Place K F)

theorem

degree_sum_single

L435 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.degree_sum_single

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [∀ v : Place K F, v.DCoordGenerates] [HasCanonicalDivisor (K := K) (F := F)] in

Lean source · statement

theorem degree_sum_single {n : ℕ} (v : Fin n → Place K F) :
    Divisor.degree (∑ j, Finsupp.single (v j) (1 : ℤ) : Divisor K F) = n

theorem

exists_isUnit_det

L439 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.exists_isUnit_det

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]

Lean source · full declaration

theorem exists_isUnit_det [Infinite (Place K F)] {n : ℕ}
    (b : Module.Basis (Fin n) K ↥(regularDifferentials K F)) :
    ∃ v : Fin n → Place K F,
      IsUnit (Matrix.det (Matrix.of fun i j : Fin n => evalFunctional (v j) (b i))) := by
  classical
  set W := ↥(regularDifferentials K F)
  have : Module.Finite K W := Module.Finite.of_basis b
  set e : Place K F → Module.Dual K W := evalFunctional with he
  have hspan : Submodule.span K (Set.range e) = ⊤ := by
    by_contra hlt
    obtain ⟨φ, hφ0, hker⟩ :=
      Submodule.exists_le_ker_of_lt_top _ (lt_top_iff_ne_top.2 hlt)
    set θ : W := (Module.evalEquiv K W).symm φ with hθ
    have hθ0 : θ = 0 := by
      refine eq_zero_of_forall_evalFunctional_eq_zero θ fun v => ?_
      have h1 : e v ∈ LinearMap.ker φ := hker (Submodule.subset_span ⟨v, rfl⟩)
      rw [LinearMap.mem_ker] at h1
      rw [hθ, Module.apply_evalEquiv_symm_apply]
      exact h1
    apply hφ0
    have : φ = Module.evalEquiv K W θ := by rw [hθ, LinearEquiv.apply_symm_apply]
    rw [this, hθ0, map_zero]
  obtain ⟨t, ht, htspan, hli⟩ := exists_linearIndependent K (Set.range e)
  rw [hspan] at htspan
  set Bt : Module.Basis t K (Module.Dual K W) :=
    Module.Basis.mk hli (by rw [Subtype.range_coe, htspan]) with hBt
  set σ : t ≃ Fin n := Bt.indexEquiv b.dualBasis with hσ
  set Bt' : Module.Basis (Fin n) K (Module.Dual K W) := Bt.reindex σ with hBt'
  have hBt'mem : ∀ j, Bt' j ∈ Set.range e := fun j => by
    rw [hBt', Module.Basis.reindex_apply, hBt, Module.Basis.mk_apply]
    exact ht (σ.symm j).2
  choose v hv using hBt'mem
  refine ⟨v, ?_⟩
  have hmat : (Matrix.of fun i j : Fin n => evalFunctional (v j) (b i)) = b.dualBasis.toMatrix Bt'
    := by
    ext i j
    rw [Matrix.of_apply, Module.Basis.toMatrix_apply, Module.Basis.dualBasis_repr, ← hv j]
  rw [hmat, ← Matrix.isUnit_iff_isUnit_det]
  let := b.dualBasis.invertibleToMatrix Bt'
  exact isUnit_of_invertible _

theorem

injective_of_isUnit_det

L480 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.injective_of_isUnit_det

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [HasCanonicalDivisor (K := K) (F := F)] in

Lean source · full declaration

theorem injective_of_isUnit_det {n : ℕ} (b : Module.Basis (Fin n) K ↥(regularDifferentials K F))
    {v : Fin n → Place K F}
    (h : IsUnit (Matrix.det (Matrix.of fun i j : Fin n => evalFunctional (v j) (b i)))) :
    Function.Injective v := by
  intro j k hjk
  by_contra hne
  apply h.ne_zero
  rw [← Matrix.det_transpose]
  exact Matrix.det_zero_of_row_eq hne (funext fun i => by simp [Matrix.transpose_apply, hjk])

theorem

lSpace_eq_bot_of_isUnit_det

L490 ↗

AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff.AlgebraicCurve.GeneralPositionProof.lSpace_eq_bot_of_isUnit_det

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace GeneralPositionProof
open _root_.AlgebraicCurve
section Curve
variable {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
variable [IsCurveOver K F] [∀ v : Place K F, v.DCoordGenerates]
variable [HasCanonicalDivisor (K := K) (F := F)]

Lean source · full declaration

theorem lSpace_eq_bot_of_isUnit_det {n : ℕ} (b : Module.Basis (Fin n) K ↥(regularDifferentials K F))
    {v : Fin n → Place K F}
    (h : IsUnit (Matrix.det (Matrix.of fun i j : Fin n => evalFunctional (v j) (b i))))
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    LSpace (canonicalDivisorOf hω - ∑ j, Finsupp.single (v j) (1 : ℤ)) = ⊥ := by
  classical
  rw [Submodule.eq_bot_iff]
  intro f hf
  by_contra hf0
  set D : Divisor K F := ∑ j, Finsupp.single (v j) (1 : ℤ) with hD
  have hDnn : ∀ w, 0 ≤ D w := fun w => by
    rw [hD, Finsupp.finsetSum_apply]
    exact Finset.sum_nonneg fun k _ => by rw [Finsupp.single_apply]; split_ifs <;> norm_num
  obtain ⟨hreg, hord⟩ := smul_mem_of_mem_lSpace hω hDnn hf hf0
  set η : ↥(regularDifferentials K F) := ⟨f • ω, hreg⟩ with hη
  have hkill : ∀ j, evalFunctional (v j) η = 0 := fun j => by
    rw [evalFunctional_apply]
    refine evalAt_eq_zero_of_ord_pos (v j) (differentialCoeff_mem hreg (v j)) ?_
    refine lt_of_lt_of_le ?_ (hord (v j))
    have h1 : (1 : ℤ) ≤ D (v j) := by
      rw [hD, Finsupp.finsetSum_apply]
      calc (1 : ℤ) = Finsupp.single (v j) (1 : ℤ) (v j) := Finsupp.single_eq_same.symm
        _ ≤ ∑ k, Finsupp.single (v k) (1 : ℤ) (v j) :=
          Finset.single_le_sum (f := fun k => Finsupp.single (v k) (1 : ℤ) (v j))
            (fun k _ => by simp only [Finsupp.single_apply]; split_ifs <;> norm_num)
            (Finset.mem_univ j)
    exact Int.lt_of_lt_of_le zero_lt_one h1
  set a : Fin n → K := fun i => b.repr η i with ha
  have hvec : Matrix.vecMul a (Matrix.of fun i j : Fin n => evalFunctional (v j) (b i)) = 0 := by
    funext j
    have hsum : η = ∑ i, a i • b i := (b.sum_repr η).symm
    have := hkill j
    rw [hsum, map_sum] at this
    simp only [map_smul, smul_eq_mul] at this
    rw [Matrix.vecMul, dotProduct]
    simpa [Matrix.of_apply] using this
  have ha0 : a = 0 := by
    have hinv := h
    rw [← Matrix.isUnit_iff_isUnit_det] at hinv
    obtain ⟨u, hu⟩ := hinv
    have : Matrix.vecMul (Matrix.vecMul a (u : Matrix (Fin n) (Fin n) K))
        ((u⁻¹ : (Matrix (Fin n) (Fin n) K)ˣ) : Matrix (Fin n) (Fin n) K) = 0 := by
      rw [hu, hvec, Matrix.zero_vecMul]
    rwa [Matrix.vecMul_vecMul, Units.mul_inv, Matrix.vecMul_one] at this
  have hη0 : η = 0 := by
    rw [← b.linearCombination_repr η]
    have : b.repr η = 0 := by
      ext i; exact congrFun ha0 i
    rw [this, map_zero]
  have : f • ω = 0 := congrArg Subtype.val hη0
  rcases smul_eq_zero.1 this with h0 | h0
  · exact hf0 h0
  · exact hω h0

theorem

exists_isUnit_det_evalAt_differentialCoeff

L549 ↗

AlgebraicCurve.exists_isUnit_det_evalAt_differentialCoeff

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Proofs.Exists_isUnit_det_evalAt_differentialCoeff
attribute [-instance]
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instDistribMulActionSubtypeProdRingAutMemSubgroupDivisor
  AlgebraicCurve.Pic0.instModuleZModTorsion
  AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instDistribMulActionTorsion
attribute [-instance] AlgebraicCurve.SemilinearAut.instSMulSubtypeProdRingAutMemSubgroupPic0
  AlgebraicCurve.SemilinearAut.instSMulTorsion
  AlgebraicCurve.SemilinearAut.instMulActionSubtypeProdRingAutMemSubgroupPlace
  AlgebraicCurve.SemilinearAut.instSMulCommClassZModTorsion
  AlgebraicCurve.SemilinearAut.instMulSemiringActionSubtypeProdRingAutMemSubgroup
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply
attribute [-simp] AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.Divisor.degree_pushforwardAlong AlgebraicCurve.Pic0.coe_degZeroCorrespondence
  AlgebraicCurve.Place.mem_fiberAlong AlgebraicCurve.SemilinearAut.toRingAut_inv
  AlgebraicCurve.SemilinearAut.smul_def AlgebraicCurve.SemilinearAut.smul_single
  AlgebraicCurve.SemilinearAut.smul_toValuationSubring AlgebraicCurve.SemilinearAut.baseAut_inv
  AlgebraicCurve.SemilinearAut.baseAut_ofAlgAut AlgebraicCurve.SemilinearAut.toRingAut_ofAlgAut
  AlgebraicCurve.SemilinearAut.toRingAut_one AlgebraicCurve.SemilinearAut.deg_smul
  AlgebraicCurve.SemilinearAut.degree_smul AlgebraicCurve.SemilinearAut.coe_degZeroSMulHom
  AlgebraicCurve.SemilinearAut.baseAut_mul
  AlgebraicCurve.SemilinearAut.coe_smulValuationSubringEquiv_apply
  AlgebraicCurve.SemilinearAut.baseAut_one AlgebraicCurve.SemilinearAut.ofAlgAut_smul
  AlgebraicCurve.SemilinearAut.coe_torsion_smul AlgebraicCurve.SemilinearAut.toRingAut_mul
noncomputable section
open _root_.AlgebraicCurve
open _root_.Module
open AlgebraicCurve.GeneralPositionProof in

Lean source · full declaration

theorem _root_.AlgebraicCurve.exists_isUnit_det_evalAt_differentialCoeff
    {K F : Type*} [Field K] [IsAlgClosed K] [Field F] [Algebra K F]
    (hfg : ∃ x : F, Transcendental K x ∧
      FiniteDimensional (IntermediateField.adjoin K ({x} : Set F)) F)
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
    {n : ℕ} (b : Module.Basis (Fin n) K ↥(regularDifferentials K F)) :
    ∃ v : Fin n → Place K F, Function.Injective v ∧
      IsUnit (Matrix.det (Matrix.of fun i j : Fin n =>
        (v j).evalAt ((v j).differentialCoeff (b i : Ω[F⁄K])))) ∧
      ∀ f : F,
        (∀ w : Place K F, -((∑ j, Finsupp.single (v j) (1 : ℤ) : Divisor K F) w) ≤ w.ord f) →
          ∃ c : K, f = algebraMap K F c := by
  classical
  obtain ⟨x, hx, hxfd⟩ := hfg
  have : Algebra.EssFiniteType K F := essFiniteType_of_transcendental_of_finiteDimensional hx hxfd
  have : ∀ v : Place K F, v.DCoordGenerates := dCoordGenerates_of_isCurveOver
  have hRR : FunctionFieldRiemannRoch K F :=
    functionFieldRiemannRoch_of_isAlgClosed_of_transcendental hx hxfd
  have hell0 : ell (0 : Divisor K F) = 1 := ell_zero_eq_one hRR
  have : Infinite (Place K F) := infinite_place hell0 hx
  obtain ⟨v, hv⟩ := exists_isUnit_det b
  have hv' : IsUnit (Matrix.det (Matrix.of fun i j : Fin n =>
      (v j).evalAt ((v j).differentialCoeff (b i : Ω[F⁄K])))) := hv
  refine ⟨v, injective_of_isUnit_det b hv, hv', ?_⟩
  intro f hf
  obtain ⟨ω, hω⟩ := exists_ne (0 : Ω[F⁄K])
  have hDnn : (0 : Divisor K F) ≤ ∑ j, Finsupp.single (v j) (1 : ℤ) := fun w => by
    rw [Finsupp.finsetSum_apply]
    exact Finset.sum_nonneg fun k _ => by rw [Finsupp.single_apply]; split_ifs <;> norm_num
  have hgenus : (genus K F : ℤ) = n := by
    rw [← (finite_and_finrank_regularDifferentials_eq_genus (K := K) (F := F)).2,
      Module.finrank_eq_card_basis b, Fintype.card_fin]
  have hell' : ell (canonicalDivisorOf hω - ∑ j, Finsupp.single (v j) (1 : ℤ)) = 0 := by
    change Module.finrank K ↥(LSpace (canonicalDivisorOf hω - ∑ j, Finsupp.single (v j) (1 : ℤ))) =
      0
    rw [lSpace_eq_bot_of_isUnit_det b hv hω, finrank_bot]
  have hRRD := hRR hω (∑ j, Finsupp.single (v j) (1 : ℤ))
  rw [hell', hgenus, degree_sum_single] at hRRD
  have hell : ell (∑ j, Finsupp.single (v j) (1 : ℤ) : Divisor K F) = 1 := by
    have : (ell (∑ j, Finsupp.single (v j) (1 : ℤ) : Divisor K F) : ℤ) = 1 := by
      push_cast at hRRD; linarith
    exact_mod_cast this
  have hfL : f ∈ LSpace (∑ j, Finsupp.single (v j) (1 : ℤ) : Divisor K F) := by
    rw [mem_lSpace_iff_ord]
    by_cases hf0 : f = 0
    · exact Or.inl hf0
    · exact Or.inr hf
  exact exists_eq_algebraMap_of_ell_eq_one hDnn hell hfL

Module documentation

Exists is Unit det eval At differential Coeff

Declarations