FoxDifferential.Completed.Semidirect
This module develops Fox differentials and completed Fox coordinates for free, profinite, and pro-\(C\) group constructions.
import
structure ZCCompletedFoxSemidirect where
/-- The completed Fox-coordinate component. -/
left : ZCFreeFoxCoordinates C (X := X) (H := H)
/-- The target group component. -/
right : HThe completed Fox semidirect target \(\mathbb{Z}_C\llbracket H\rrbracket^{X} \rtimes H\).
theorem ext {a b : ZCCompletedFoxSemidirect C X H}
(hleft : a.left = b.left) (hright : a.right = b.right) : a = bExtensionality for the free crossed-differential semidirect product.
Show proof
by
cases a
cases b
simp_allProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□instance instOneZCCompletedFoxSemidirect : One (ZCCompletedFoxSemidirect C X H) where
one := ⟨0, 1⟩The identity element of the completed Fox semidirect product is \((0,1)\).
instance instMulZCCompletedFoxSemidirect : Mul (ZCCompletedFoxSemidirect C X H) where
mul a b :=
⟨a.left + zcGroupLike C H a.right • b.left, a.right * b.right⟩Multiplication in the completed Fox semidirect product is defined coordinatewise from the completed group-algebra action and group multiplication.
instance instInvZCCompletedFoxSemidirect : Inv (ZCCompletedFoxSemidirect C X H) where
inv a :=
⟨-(zcGroupLike C H a.right⁻¹ • a.left), a.right⁻¹⟩Inversion in the completed Fox semidirect product is computed from the completed coefficient action and the inverse in the base group.
theorem one_left :
(1 : ZCCompletedFoxSemidirect C X H).left = 0The left component of the identity semidirect element is zero.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem one_right :
(1 : ZCCompletedFoxSemidirect C X H).right = 1The right component of the identity semidirect element is the group identity.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem mul_left (a b : ZCCompletedFoxSemidirect C X H) :
(a * b).left = a.left + zcGroupLike C H a.right • b.leftThe left component of semidirect multiplication is computed by the Fox product rule.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums.
□theorem mul_right (a b : ZCCompletedFoxSemidirect C X H) :
(a * b).right = a.right * b.rightThe right component of semidirect multiplication is the product of right components.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums.
□theorem inv_left (a : ZCCompletedFoxSemidirect C X H) :
a⁻¹.left = -(zcGroupLike C H a.right⁻¹ • a.left)The left component of the semidirect inverse is computed by the Fox inverse formula.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums.
□theorem inv_right (a : ZCCompletedFoxSemidirect C X H) :
a⁻¹.right = a.right⁻¹The right component of the semidirect inverse is the inverse of the right component.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums.
□instance instGroupZCCompletedFoxSemidirect : Group (ZCCompletedFoxSemidirect C X H) where
one := 1
mul := (· * ·)
inv := Inv.inv
mul_assoc a b c := by
ext
· simp only [mul_left, mul_right, map_mul, Pi.add_apply, Pi.smul_apply, smul_eq_mul, add_assoc,
zcCompletedGroupAlgebraProjection_add, zcCompletedGroupAlgebraProjection_mul,
zcCompletedGroupAlgebraProjection_groupLike, MonoidAlgebra.of_apply, MonoidAlgebra.single_mul_single, mul_one,
MonoidAlgebra.coe_add, MonoidAlgebra.single_mul_apply, one_mul, mul_inv_rev, smul_add, smul_smul]
· simp only [mul_right, mul_assoc]
one_mul a := by
ext
· simp only [mul_left, one_left, one_right, map_one, one_smul, Pi.add_apply, Pi.zero_apply, zero_add]
· simp only [mul_right, one_right, one_mul]
mul_one a := by
ext
· simp only [mul_left, one_left, smul_zero, Pi.add_apply, Pi.zero_apply, add_zero]
· simp only [mul_right, one_right, mul_one]
inv_mul_cancel a := by
ext
· simp only [mul_left, inv_left, inv_right, Pi.add_apply, Pi.neg_apply, Pi.smul_apply, smul_eq_mul,
neg_add_cancel, zcCompletedGroupAlgebraProjection_zero, Finsupp.coe_zero, Pi.zero_apply, one_left]
· simp only [mul_right, inv_right, inv_mul_cancel, one_right]The completed Fox semidirect product carries the group structure induced by the completed coefficient action.
def rightMonoidHom : ZCCompletedFoxSemidirect C X H →* H where
toFun a := a.right
map_one' := rfl
map_mul' _ _ := rflThe right projection from the completed Fox semidirect product to the target group.
theorem rightMonoidHom_apply (a : ZCCompletedFoxSemidirect C X H) :
rightMonoidHom C X H a = a.rightThe right projection of the completed Fox semidirect monoid homomorphism is its right component.
Show proof
rflProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□def zcCompletedFoxSemidirectLift (ψ : FreeGroup X →* H) :
FreeGroup X →* ZCCompletedFoxSemidirect C X H :=
FreeGroup.lift fun x =>
{ left := Pi.single x (1 : ZCCompletedGroupAlgebra C H)
right := ψ (FreeGroup.of x) }The completed Fox semidirect lift of a free group homomorphism.
theorem zcCompletedFoxSemidirectLift_right
(ψ : FreeGroup X →* H) (w : FreeGroup X) :
(zcCompletedFoxSemidirectLift C ψ w).right = ψ wThe right component of the completed Fox semidirect lift is \(\psi\).
Show proof
by
induction w using FreeGroup.induction_on with
| C1 =>
simp only [zcCompletedFoxSemidirectLift, map_one, ZCCompletedFoxSemidirect.one_right]
| of x =>
simp only [zcCompletedFoxSemidirectLift, FreeGroup.lift_apply_of]
| inv_of x hx =>
simpa using congrArg Inv.inv hx
| mul u v hu hv =>
simp only [map_mul, ZCCompletedFoxSemidirect.mul_right, hu, hv]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□def zcCompletedFoxSemidirectDerivativeVector
(ψ : FreeGroup X →* H) (w : FreeGroup X) :
ZCFreeFoxCoordinates C (X := X) (H := H) :=
(zcCompletedFoxSemidirectLift C ψ w).leftThe left component of the completed Fox semidirect lift.
theorem zcCompletedFoxSemidirectDerivativeVector_one
(ψ : FreeGroup X →* H) :
zcCompletedFoxSemidirectDerivativeVector C ψ (1 : FreeGroup X) = 0The completed semidirect derivative vector sends the identity word to zero.
Show proof
by
simp only [zcCompletedFoxSemidirectDerivativeVector, zcCompletedFoxSemidirectLift, map_one,
ZCCompletedFoxSemidirect.one_left]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectDerivativeVector_of
(ψ : FreeGroup X →* H) (x : X) :
zcCompletedFoxSemidirectDerivativeVector C ψ (FreeGroup.of x) =
Pi.single x (1 : ZCCompletedGroupAlgebra C H)The completed semidirect derivative vector sends a free generator to the coordinate basis vector.
Show proof
by
simp only [zcCompletedFoxSemidirectDerivativeVector, zcCompletedFoxSemidirectLift, FreeGroup.lift_apply_of]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem zcCompletedFoxSemidirectDerivativeVector_mul
(ψ : FreeGroup X →* H) (u v : FreeGroup X) :
zcCompletedFoxSemidirectDerivativeVector C ψ (u * v) =
zcCompletedFoxSemidirectDerivativeVector C ψ u +
zcCompletedGroupAlgebraScalar C ψ u •
zcCompletedFoxSemidirectDerivativeVector C ψ vProduct rule for the completed semidirect derivative vector.
Show proof
by
simp only [zcCompletedFoxSemidirectDerivativeVector, map_mul, ZCCompletedFoxSemidirect.mul_left,
zcCompletedFoxSemidirectLift_right, zcCompletedGroupAlgebraScalar_apply]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem zcCompletedFoxSemidirectDerivativeVector_isCrossedDifferential
(ψ : FreeGroup X →* H) :
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C ψ)
(zcCompletedFoxSemidirectDerivativeVector C ψ)The left component of the completed semidirect lift is a crossed differential.
Show proof
by
intro u v
exact zcCompletedFoxSemidirectDerivativeVector_mul C ψ u vProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem zcCompletedFoxSemidirectDerivativeVector_eq
(ψ : FreeGroup X →* H) :
zcCompletedFoxSemidirectDerivativeVector C ψ =
zcFreeGroupFoxDerivativeVector C ψThe completed semidirect derivative vector is the completed free-group Fox derivative vector.
Show proof
by
exact zcFreeGroupFoxDerivativeVector_unique C ψ
(zcCompletedFoxSemidirectDerivativeVector C ψ)
(zcCompletedFoxSemidirectDerivativeVector_isCrossedDifferential C ψ)
(zcCompletedFoxSemidirectDerivativeVector_of C ψ)Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectLift_eq
(ψ : FreeGroup X →* H) (w : FreeGroup X) :
zcCompletedFoxSemidirectLift C ψ w =
{ left := zcFreeGroupFoxDerivativeVector C ψ w
right := ψ w }The completed semidirect lift stores the completed Fox derivative vector and the target homomorphism.
Show proof
by
apply ZCCompletedFoxSemidirect.ext
· rw [← zcCompletedFoxSemidirectDerivativeVector_eq]
rfl
· exact zcCompletedFoxSemidirectLift_right C ψ wProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectLift_unique
(ψ : FreeGroup X →* H)
(φ : FreeGroup X →* ZCCompletedFoxSemidirect C X H)
(hright : ∀ w : FreeGroup X, (φ w).right = ψ w)
(hbasis :
∀ x : X, (φ (FreeGroup.of x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra C H)) :
φ = zcCompletedFoxSemidirectLift C ψUniqueness of the completed semidirect lift with prescribed right component and generator coordinate values.
Show proof
by
let delta : FreeGroup X → ZCFreeFoxCoordinates C (X := X) (H := H) :=
fun w => (φ w).left
have hdelta :
IsCrossedDifferential (zcCompletedGroupAlgebraScalar C ψ) delta := by
intro u v
have h := congrArg ZCCompletedFoxSemidirect.left (map_mul φ u v)
change (φ (u * v)).left =
(φ u).left + zcCompletedGroupAlgebraScalar C ψ u • (φ v).left
rw [h]
simp only [ZCCompletedFoxSemidirect.mul_left, hright u, zcCompletedGroupAlgebraScalar_apply]
have hdelta_eq : delta = zcFreeGroupFoxDerivativeVector C ψ :=
zcFreeGroupFoxDerivativeVector_unique C ψ delta hdelta hbasis
apply MonoidHom.ext
intro w
apply ZCCompletedFoxSemidirect.ext
· change delta w = zcCompletedFoxSemidirectDerivativeVector C ψ w
rw [hdelta_eq, zcCompletedFoxSemidirectDerivativeVector_eq]
· rw [hright w, zcCompletedFoxSemidirectLift_right]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem zcCompletedFoxSemidirectLift_left_isCrossedDifferential
(ψ : FreeGroup X →* H)
(φ : FreeGroup X →* ZCCompletedFoxSemidirect C X H)
(hright : ∀ w : FreeGroup X, (φ w).right = ψ w) :
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C ψ) (fun w : FreeGroup X => (φ w).left)The left component of any semidirect lift with prescribed right component is a completed crossed differential.
Show proof
by
intro u v
have h := congrArg ZCCompletedFoxSemidirect.left (map_mul φ u v)
change (φ (u * v)).left =
(φ u).left + zcCompletedGroupAlgebraScalar C ψ u • (φ v).left
rw [h]
simp only [ZCCompletedFoxSemidirect.mul_left, hright u, zcCompletedGroupAlgebraScalar_apply]Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem existsUnique_zcCompletedFoxSemidirectLift
(ψ : FreeGroup X →* H) :
∃! φ : FreeGroup X →* ZCCompletedFoxSemidirect C X H,
(∀ w : FreeGroup X, (φ w).right = ψ w) ∧
∀ x : X, (φ (FreeGroup.of x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra C H)Existence and uniqueness theorem for the completed semidirect lift.
Show proof
by
refine ⟨zcCompletedFoxSemidirectLift C ψ, ?_, ?_⟩
· exact ⟨zcCompletedFoxSemidirectLift_right C ψ,
zcCompletedFoxSemidirectDerivativeVector_of C ψ⟩
· intro φ hφ
exact zcCompletedFoxSemidirectLift_unique C ψ φ hφ.1 hφ.2Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectDerivativeVector_foxBoundary
(ψ : FreeGroup X →* H) (w : FreeGroup X) :
zcFreeGroupFoxBoundary C ψ (zcCompletedFoxSemidirectDerivativeVector C ψ w) =
zcCompletedGroupAlgebraBoundary C ψ wBoundary-map form of the completed Fox fundamental formula for the semidirect derivative vector.
Show proof
by
exact zcFreeGroupFoxBoundary_of_crossedDifferential C ψ
(zcCompletedFoxSemidirectDerivativeVector C ψ)
(zcCompletedFoxSemidirectDerivativeVector_isCrossedDifferential C ψ)
(zcCompletedFoxSemidirectDerivativeVector_of C ψ) wProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectLift_euler_formula
(ψ : FreeGroup X →* H) (w : FreeGroup X) :
zcGroupLike C H (ψ w) - 1 =
∑ i : X,
(zcCompletedFoxSemidirectLift C ψ w).left i *
(zcGroupLike C H (ψ (FreeGroup.of i)) - 1)Completed Fox-Euler formula using the left component of the semidirect lift: \([\psi(w)] - 1 = \sum_i \varphi(w)_i * ([\psi(x_i)]-1)\).
Show proof
by
exact zcFreeGroupFoxDerivative_euler_formula_of_crossedDifferential C ψ
(zcCompletedFoxSemidirectDerivativeVector C ψ)
(zcCompletedFoxSemidirectDerivativeVector_isCrossedDifferential C ψ)
(zcCompletedFoxSemidirectDerivativeVector_of C ψ) wProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps. The Fox-specific step is the fundamental identity expressing an element minus its augmentation as the sum of its Fox derivatives multiplied by the corresponding generator increments. This identity identifies the relevant augmentation kernel with the image or closed span generated by the displayed differential coordinates.
□theorem zcCompletedFoxSemidirectLift_foxBoundary_of_generatorValues
(ψ : FreeGroup X →* H)
(φ : FreeGroup X →* ZCCompletedFoxSemidirect C X H)
(hright : ∀ w : FreeGroup X, (φ w).right = ψ w)
(hbasis :
∀ x : X, (φ (FreeGroup.of x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra C H))
(w : FreeGroup X) :
zcFreeGroupFoxBoundary C ψ (φ w).left =
zcCompletedGroupAlgebraBoundary C ψ wConditional semidirect Fox boundary formula. Any semidirect lift with right component \(\psi\) and standard generator coordinates satisfies the completed Fox boundary formula.
Show proof
by
exact zcFreeGroupFoxBoundary_of_crossedDifferential C ψ
(fun w : FreeGroup X => (φ w).left)
(zcCompletedFoxSemidirectLift_left_isCrossedDifferential C ψ φ hright)
hbasis wProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem zcCompletedFoxSemidirectLift_euler_formula_of_generatorValues
(ψ : FreeGroup X →* H)
(φ : FreeGroup X →* ZCCompletedFoxSemidirect C X H)
(hright : ∀ w : FreeGroup X, (φ w).right = ψ w)
(hbasis :
∀ x : X, (φ (FreeGroup.of x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra C H))
(w : FreeGroup X) :
zcGroupLike C H (ψ w) - 1 =
∑ i : X,
(φ w).left i * (zcGroupLike C H (ψ (FreeGroup.of i)) - 1)Conditional semidirect Fox-Euler formula. The left component of any semidirect lift with right component \(\psi\) and standard generator coordinates gives the completed Fox-Euler sum.
Show proof
by
exact zcFreeGroupFoxDerivative_euler_formula_of_crossedDifferential C ψ
(fun w : FreeGroup X => (φ w).left)
(zcCompletedFoxSemidirectLift_left_isCrossedDifferential C ψ φ hright)
hbasis wProof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□