name stringlengths 1 203 | module stringlengths 7 90 | docstring stringlengths 0 11.2k | source_text stringlengths 6 20.6k | source_link stringlengths 106 200 | dependencies stringlengths 6 1.52k ⌀ | informalization stringlengths 8 98.5k |
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ABC.abc | FormalConjectures.Wikipedia.ABC | For every positive real number `ε`, there exist only finitely many triples `(a, b, c)` of coprime positive integers, with `a + b = c`, such that `c > rad(abc)^(1+ε)`
| /--
For every positive real number `ε`, there exist only finitely many triples `(a, b, c)` of coprime positive integers, with `a + b = c`, such that `c > rad(abc)^(1+ε)`
-/
@[category research open, AMS 11]
theorem abc (ε : ℝ) (hε : 0 < ε) :
{(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ ({a, b, c} : Set ℕ).Pairw... | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L56-L63 | ["Real", "LT.lt", "setOf", "Prod.mk", "Nat", "Prod", "And", "Insert.insert", "Set.Pairwise", "Nat.Coprime", "HAdd.hAdd", "Eq", "ABC.radical", "HMul.hMul", "HPow.hPow", "Set.Finite"] | **The abc Conjecture.** For every positive real number $\epsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and $c > \operatorname{rad}(abc)^{1+\epsilon}$, where $\operatorname{rad}(n)$ denotes the product of the distinct prime factors of $n$. |
ABC.abc.variants.lt_constant_mul | FormalConjectures.Wikipedia.ABC | For every positive real number ε, there exists a constant `K_ε` such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have `c < K_ε rad(abc)^(1+ε)`.
| /--
For every positive real number ε, there exists a constant `K_ε` such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have `c < K_ε rad(abc)^(1+ε)`.
-/
@[category research open, AMS 11]
theorem abc.variants.lt_constant_mul (ε : ℝ) (hε : 0 < ε) : ∃ K,
∀ (a b c : ℕ), 0 < a → 0 < b → ... | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L65-L72 | ["Real", "LT.lt", "Nat", "Insert.insert", "Set.Pairwise", "Nat.Coprime", "HAdd.hAdd", "Eq", "HMul.hMul", "ABC.radical", "HPow.hPow"] | **The $abc$ Conjecture.** For every positive real number $\varepsilon$, there exists a constant $K_\varepsilon$ such that for all triples of coprime positive integers $(a, b, c)$ satisfying $a + b = c$, the inequality $c < K_\varepsilon \operatorname{rad}(abc)^{1+\varepsilon}$ holds, where $\operatorname{rad}(abc)$ den... |
ABC.abc.variants.quality | FormalConjectures.Wikipedia.ABC | For every positive real number ε, there exist only finitely many triples `(a, b, c)` of coprime positive integers with `a + b = c` such that `q(a, b, c) > 1 + ε`.
| /--
For every positive real number ε, there exist only finitely many triples `(a, b, c)` of coprime positive integers with `a + b = c` such that `q(a, b, c) > 1 + ε`.
-/
@[category research open, AMS 11]
theorem abc.variants.quality (ε : ℝ) (hε : 0 < ε) :
{(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ ({a, b, c} ... | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L74-L81 | ["Real", "LT.lt", "setOf", "Prod.mk", "Nat", "Prod", "And", "Insert.insert", "Set.Pairwise", "Nat.Coprime", "HAdd.hAdd", "Eq", "ABC.quality", "GT.gt", "Set.Finite"] | **The abc Conjecture (Quality Variant).** For every positive real number $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers satisfying $a + b = c$ such that the quality $q(a, b, c)$ is greater than $1 + \varepsilon$. |
ABC.quality | FormalConjectures.Wikipedia.ABC | Quality `q(a, b, c)` of the triple `(a, b, c)` is defined as `q(a,b,c) = log (c) / log (rad(abc))`.
| /--
Quality `q(a, b, c)` of the triple `(a, b, c)` is defined as `q(a,b,c) = log (c) / log (rad(abc))`.
-/
noncomputable def quality (a b c : ℕ) : ℝ := (c : ℝ).log / (radical <| a * b * c : ℝ).log
| https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L51-L54 | ["Nat", "Real"] | **Quality of a Triple.** For any three natural numbers $a$, $b$, and $c$, the quality $q(a, b, c)$ is defined as the ratio of the natural logarithm of $c$ to the natural logarithm of the radical of the product $abc$. |
ABC.radical | FormalConjectures.Wikipedia.ABC | The radical of `n` denoted is the product of the distinct prime factors of `n`.
| /--
The radical of `n` denoted is the product of the distinct prime factors of `n`.
-/
def radical (n : ℕ) : ℕ := n.primeFactors.prod id
| https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L27-L30 | ["Nat"] | **Radical of a Natural Number.** The radical of a natural number $n$ is defined as the product of its distinct prime factors. |
ABC.radical_12 | FormalConjectures.Wikipedia.ABC | @[category test, AMS 11]
theorem radical_12 : radical 12 = 6 := by
rw [radical, show 12 = 2^2 * 3 by rfl, Nat.primeFactors_mul (by norm_num)
(by norm_num), Nat.primeFactors_pow _ (by norm_num),
Nat.Prime.primeFactors (by norm_num), Nat.Prime.primeFactors (by norm_num)]
rfl | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L44-L49 | ["ABC.radical", "Eq"] | **Radical of 12.** The radical of the natural number 12 is equal to 6. | |
ABC.radical_16 | FormalConjectures.Wikipedia.ABC | @[category test, AMS 11]
theorem radical_16 : radical 16 = 2 := by
have : Nat.primeFactors 16 = {2} := by
rw [show 16 = 2 ^ 4 by decide, Nat.primeFactors_pow]
· norm_num
· decide
norm_num [radical, this] | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L32-L38 | ["ABC.radical", "Eq"] | **Radical of 16.** The radical of the natural number 16 is equal to 2. | |
ABC.radical_17 | FormalConjectures.Wikipedia.ABC | @[category test, AMS 11]
theorem radical_17 : radical 17 = 17 := by
rw [radical, Nat.Prime.primeFactors (by norm_num), Finset.prod_singleton, id] | https://github.com/google-deepmind/formal-conjectures/blob/f01550cf4f910c7018f593f47979ab1cc9e8c70e/FormalConjectures/Wikipedia/ABC.lean#L40-L42 | ["ABC.radical", "Eq"] | **Radical of 17.** The radical of the natural number 17 is equal to 17. | |
ACCSystem | PhysLean.QFT.AnomalyCancellation.Basic | The type of charges plus the anomaly cancellation conditions.
In many physical settings these conditions are derived formally from the gauge group and the
fermionic representations. They arise from triangle Feynman diagrams, and can also be obtained
using index-theoretic or characteristic-class constructions.
In this... | /--
The type of charges plus the anomaly cancellation conditions.
In many physical settings these conditions are derived formally from the gauge group and the
fermionic representations. They arise from triangle Feynman diagrams, and can also be obtained
using index-theoretic or characteristic-class constructions.
In ... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L419-L431 | ["ACCSystemQuad"] | **Anomaly Cancellation Condition System.** An anomaly cancellation condition (ACC) system is a structure that extends a quadratic ACC system by incorporating a cubic anomaly cancellation condition, defined as a homogeneous cubic form on the space of rational charges. |
ACCSystem.Hom | PhysLean.QFT.AnomalyCancellation.Basic | The structure of a map between two ACCSystems. | /-- The structure of a map between two ACCSystems. -/
structure Hom (χ η : ACCSystem) where
/-- The linear map between vector spaces of charges. -/
charges : χ.Charges →ₗ[ℚ] η.Charges
/-- The map between solutions. -/
anomalyFree : χ.Sols → η.Sols
/-- The condition that the map commutes with the relevant incl... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L559-L566 | ["ACCSystem"] | **Morphism of Anomaly Cancellation Condition Systems.** A morphism between two anomaly cancellation condition systems, $\chi$ and $\eta$, consists of a $\mathbb{Q}$-linear map between their respective vector spaces of charges and a map between their spaces of anomaly-free solutions. These maps must be compatible such t... |
ACCSystem.Hom.anomalyFree | PhysLean.QFT.AnomalyCancellation.Basic | The map between solutions. | anomalyFree : χ.Sols → η.Sols
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L564-L564 | ["ACCSystem", "ACCSystem.Hom", "ACCSystem.Sols"] | **Map between solutions.** Given a morphism between two anomaly cancellation condition systems $\chi$ and $\eta$, there exists a corresponding map that sends every solution of the system $\chi$ to a solution of the system $\eta$. |
ACCSystem.Hom.charges | PhysLean.QFT.AnomalyCancellation.Basic | The linear map between vector spaces of charges. | charges : χ.Charges →ₗ[ℚ] η.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L562-L562 | ["ACCSystem", "ACCSystem.Hom", "ACCSystemCharges.Charges", "LinearMap", "Rat"] | **Morphism of Charge Spaces.** Given a morphism between two anomaly cancellation condition systems, there is an associated $\mathbb{Q}$-linear map from the vector space of charges of the domain system to the vector space of charges of the codomain system. |
ACCSystem.Hom.commute | PhysLean.QFT.AnomalyCancellation.Basic | The condition that the map commutes with the relevant inclusions. | commute : charges ∘ χ.solsIncl = η.solsIncl ∘ anomalyFree
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L566-L566 | ["ACCSystem", "ACCSystem.Hom", "ACCSystem.Hom.charges", "Function.comp", "ACCSystem.solsIncl", "Eq", "ACCSystem.Hom.anomalyFree"] | **Commutativity of the Morphism with Solution Inclusions.** For a morphism between two anomaly cancellation condition systems, the composition of the solution inclusion of the domain system with the linear map of charges is equal to the composition of the map between solutions with the solution inclusion of the codomai... |
ACCSystem.Hom.comp | PhysLean.QFT.AnomalyCancellation.Basic | The definition of composition between two ACCSystems. | /-- The definition of composition between two ACCSystems. -/
def Hom.comp {χ η ε : ACCSystem} (g : Hom η ε) (f : Hom χ η) : Hom χ ε where
charges := LinearMap.comp g.charges f.charges
anomalyFree := g.anomalyFree ∘ f.anomalyFree
commute := by rw [LinearMap.coe_comp, Function.comp_assoc, f.commute,
← Function.... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L574-L579 | ["ACCSystem", "ACCSystem.Hom"] | **Composition of Morphisms of Anomaly Cancellation Condition Systems.** Given two morphisms of anomaly cancellation condition systems, $f: \chi \to \eta$ and $g: \eta \to \varepsilon$, their composition $g \circ f: \chi \to \varepsilon$ is the morphism defined by the composition of their underlying linear maps on the s... |
ACCSystem.IsSolution | PhysLean.QFT.AnomalyCancellation.Basic | A charge `S` is a solution if it extends to a solution. | /-- A charge `S` is a solution if it extends to a solution. -/
def IsSolution (χ : ACCSystem) (S : χ.Charges) : Prop :=
∃ (sol : χ.Sols), sol.val = S
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L471-L473 | ["ACCSystem", "ACCSystemCharges.Charges"] | **Existence of a Solution Extension.** A set of charges $S$ is considered a solution for a given anomaly cancellation condition system if there exists a valid solution to the system's equations whose underlying charge assignment is equal to $S$. |
ACCSystem.Sols | PhysLean.QFT.AnomalyCancellation.Basic | The type of solutions to the anomaly cancellation conditions. | /-- The type of solutions to the anomaly cancellation conditions. -/
structure Sols (χ : ACCSystem) extends χ.QuadSols where
/-- The condition that the charge satisfies the cubic ACC. -/
cubicSol : χ.cubicACC val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L443-L446 | ["ACCSystem", "ACCSystemQuad.QuadSols"] | **Anomaly Cancellation Condition Solutions.** For a given anomaly cancellation condition system, a solution is a charge that satisfies all the quadratic and linear conditions of the system and, additionally, satisfies the cubic anomaly cancellation condition. |
ACCSystem.Sols.cubicSol | PhysLean.QFT.AnomalyCancellation.Basic | The condition that the charge satisfies the cubic ACC. | cubicSol : χ.cubicACC val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L446-L446 | ["ACCSystem", "ACCSystem.Sols", "ACCSystem.cubicACC", "ACCSystemLinear.LinSols.val", "Eq"] | **Cubic Anomaly Cancellation Condition.** The value of the cubic form associated with the anomaly cancellation system, evaluated at the given charge, is equal to zero. |
ACCSystem.Sols.ext | PhysLean.QFT.AnomalyCancellation.Basic | Two solutions are equal if the underlying charges are equal. | /-- Two solutions are equal if the underlying charges are equal. -/
lemma Sols.ext {χ : ACCSystem} {S T : χ.Sols} (h : S.val = T.val) :
S = T := by
have h := ACCSystemQuad.QuadSols.ext h
cases' S
simp_all only
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L456-L461 | ["ACCSystem", "ACCSystem.Sols", "ACCSystemLinear.LinSols.val", "Eq"] | **Extensionality of Solutions.** Two solutions to an anomaly cancellation condition system are equal if and only if their underlying charges are equal. |
ACCSystem.cubicACC | PhysLean.QFT.AnomalyCancellation.Basic | The cubic ACC. | cubicACC : HomogeneousCubic toACCSystemCharges.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L431-L431 | ["ACCSystem", "HomogeneousCubic", "ACCSystemCharges.Charges"] | **Cubic Anomaly Cancellation Condition.** The system includes a homogeneous cubic form defined on the space of rational charges. |
ACCSystem.solsIncl | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of `Sols` into `LinSols`. | /-- The inclusion of `Sols` into `LinSols`. -/
def solsIncl (χ : ACCSystem) : χ.Sols →[ℚ] χ.Charges :=
MulActionHom.comp χ.quadSolsIncl χ.solsInclQuadSols
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L535-L537 | ["ACCSystem", "ACCSystem.Sols", "MulActionHom", "id", "ACCSystemCharges.Charges"] | **Inclusion of Solutions into the Space of Charges.** For a given anomaly cancellation condition system, the map from the set of solutions to the space of rational charges is defined as the $\mathbb{Q}$-equivariant composition of the inclusion of solutions into the quadratic solution set and the subsequent inclusion of... |
ACCSystem.solsInclLinSols | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of `Sols` into `LinSols`. | /-- The inclusion of `Sols` into `LinSols`. -/
def solsInclLinSols (χ : ACCSystem) : χ.Sols →[ℚ] χ.LinSols :=
MulActionHom.comp χ.quadSolsInclLinSols χ.solsInclQuadSols
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L517-L519 | ["ACCSystem", "ACCSystem.Sols", "MulActionHom", "id", "ACCSystemLinear.LinSols"] | **Inclusion of Anomaly Cancellation Solutions into Linear Solutions.** For a given anomaly cancellation condition system, there exists a $\mathbb{Q}$-equivariant map from the set of solutions satisfying the full system (including cubic and quadratic conditions) to the set of solutions satisfying only the linear conditi... |
ACCSystem.solsInclLinSols_injective | PhysLean.QFT.AnomalyCancellation.Basic | lemma solsInclLinSols_injective (χ : ACCSystem) :
Function.Injective χ.solsInclLinSols := by
intro S T h
have h' : χ.solsInclQuadSols S = χ.solsInclQuadSols T := by
apply ACCSystemQuad.quadSolsInclLinSols_injective (χ := χ.toACCSystemQuad)
simpa [ACCSystem.solsInclLinSols, MulActionHom.comp_apply] using... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L521-L527 | ["ACCSystem", "Function.Injective", "ACCSystem.solsInclLinSols"] | **Injectivity of the Inclusion of ACC Solutions into Linear Solutions.** For any anomaly cancellation condition system, the map that includes the set of solutions satisfying the full system (comprising cubic, quadratic, and linear constraints) into the space of solutions satisfying only the linear constraints is inject... | |
ACCSystem.solsInclQuadSols | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of `Sols` into `QuadSols`. | /-- The inclusion of `Sols` into `QuadSols`. -/
def solsInclQuadSols (χ : ACCSystem) : χ.Sols →[ℚ] χ.QuadSols where
toFun := Sols.toQuadSols
map_smul' _ _ := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L499-L502 | ["ACCSystem", "ACCSystem.Sols", "MulActionHom", "id", "ACCSystemQuad.QuadSols"] | **Inclusion of Solutions into Quadratic Solutions.** For any anomaly cancellation condition (ACC) system, there exists a $\mathbb{Q}$-linear scaling map that embeds the set of full solutions (which satisfy linear, quadratic, and cubic conditions) into the set of quadratic solutions (which satisfy only the linear and qu... |
ACCSystem.solsInclQuadSols_injective | PhysLean.QFT.AnomalyCancellation.Basic | lemma solsInclQuadSols_injective (χ : ACCSystem) :
Function.Injective χ.solsInclQuadSols := by
intro S T h
apply Sols.ext
have hv : (χ.solsInclQuadSols S).val = (χ.solsInclQuadSols T).val :=
congrArg (fun X => X.val) h
simpa [ACCSystem.solsInclQuadSols] using hv
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L504-L510 | ["ACCSystem", "Function.Injective", "ACCSystem.solsInclQuadSols"] | **Injectivity of the Inclusion of ACC Solutions into Quadratic Solutions.** For any anomaly cancellation condition (ACC) system, the map that includes the set of solutions satisfying the full system of equations (linear, quadratic, and cubic) into the set of solutions satisfying only the quadratic subsystem is injectiv... | |
ACCSystem.solsIncl_injective | PhysLean.QFT.AnomalyCancellation.Basic | @[sorryful]
lemma solsIncl_injective (χ : ACCSystem) :
Function.Injective χ.solsIncl := by
intro S T h
have h' : χ.solsInclQuadSols S = χ.solsInclQuadSols T := by
apply ACCSystemQuad.quadSolsIncl_injective (χ := χ.toACCSystemQuad)
simpa [ACCSystem.solsIncl, MulActionHom.comp_apply] using h
exact (sols... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L539-L546 | ["ACCSystem", "Function.Injective", "ACCSystem.solsIncl"] | **Injectivity of the Solution Inclusion Map.** For any anomaly cancellation condition system, the map that includes the set of solutions into the space of rational charges is injective. | |
ACCSystem.solsMulAction | PhysLean.QFT.AnomalyCancellation.Basic | An instance giving the properties and structures to define an action of `ℚ` on `Sols`. | /-- An instance giving the properties and structures to define an action of `ℚ` on `Sols`. -/
instance solsMulAction (χ : ACCSystem) : MulAction ℚ χ.Sols where
smul a S := ⟨a • S.toQuadSols, by
erw [(χ.cubicACC).map_smul, S.cubicSol]
exact Rat.mul_zero (a ^ 3)⟩
mul_smul a b S := Sols.ext (mul_smul _ _ _)
... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L485-L491 | ["ACCSystem", "MulAction", "Rat", "ACCSystem.Sols"] | **Rational Group Action on ACC System Solutions.** For a given anomaly cancellation condition system, the set of solutions to the system admits a group action by the rational numbers $\mathbb{Q}$ under multiplication. This action is defined by scaling a solution's underlying vector by a rational scalar $a$; since the c... |
ACCSystemCharges | PhysLean.QFT.AnomalyCancellation.Basic | A system of charges, specified by the number of charges. | /-- A system of charges, specified by the number of charges. -/
structure ACCSystemCharges where
/-- The number of charges. -/
numberCharges : ℕ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L96-L99 | null | **System of Charges.** A system of charges is characterized by a natural number representing the total count of charges within the system. |
ACCSystemCharges.Charges | PhysLean.QFT.AnomalyCancellation.Basic | The charges as functions from `Fin χ.numberCharges → ℚ`. | /-- The charges as functions from `Fin χ.numberCharges → ℚ`. -/
def Charges (χ : ACCSystemCharges) : Type := Fin χ.numberCharges → ℚ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L130-L131 | ["ACCSystemCharges"] | **Charges of a System.** For a given system of charges, the charges are defined as the set of functions mapping each index from $0$ to $n-1$ to a rational number, where $n$ is the total number of charges in the system. |
ACCSystemCharges.ChargesAddCommGroup | PhysLean.QFT.AnomalyCancellation.Basic | instance ChargesAddCommGroup (χ : ACCSystemCharges) : AddCommGroup χ.Charges :=
Module.addCommMonoidToAddCommGroup ℚ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L157-L158 | ["ACCSystemCharges", "AddCommGroup", "ACCSystemCharges.Charges"] | **Additive Commutative Group of Charges.** For any system of charges, the set of its charges forms an additive commutative group, with the group structure inherited from its representation as a vector space over the rational numbers $\mathbb{Q}$. | |
ACCSystemCharges.chargesAddCommMonoid | PhysLean.QFT.AnomalyCancellation.Basic | An instance to provide the necessary operations and properties for `charges` to form an additive
commutative monoid.
| /--
An instance to provide the necessary operations and properties for `charges` to form an additive
commutative monoid.
-/
@[simps!]
instance chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid χ.Charges :=
Pi.addCommMonoid
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L141-L147 | ["ACCSystemCharges", "AddCommMonoid", "ACCSystemCharges.Charges"] | **Additive Commutative Monoid of System Charges.** For any system of charges, the set of its charges forms an additive commutative monoid under pointwise addition, inheriting the structure of a function space mapping charge indices to rational numbers. |
ACCSystemCharges.chargesAddCommMonoid_add | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid χ.Charges := | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L145-L145 | ["ACCSystemCharges", "Fin", "ACCSystemCharges.numberCharges", "Rat", "HAdd.hAdd", "Eq"] | **Commutativity of Addition for System Charges.** For any system of charges, the set of its charges forms an additive commutative monoid under the addition operation. | |
ACCSystemCharges.chargesAddCommMonoid_nsmul | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid χ.Charges := | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L145-L145 | ["ACCSystemCharges", "Nat", "Fin", "ACCSystemCharges.numberCharges", "Rat", "HSMul.hSMul", "Eq", "HMul.hMul"] | **Additive Commutative Monoid of Charges.** For any system of charges, the set of its charges forms an additive commutative monoid. | |
ACCSystemCharges.chargesAddCommMonoid_zero_den | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid χ.Charges := | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L145-L145 | ["ACCSystemCharges", "Fin", "ACCSystemCharges.numberCharges", "Rat.den", "Eq"] | **Commutative Monoid Structure on System Charges.** For any system of charges, the set of its charges forms an additive commutative monoid. | |
ACCSystemCharges.chargesAddCommMonoid_zero_num | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid χ.Charges := | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L145-L145 | ["ACCSystemCharges", "Fin", "ACCSystemCharges.numberCharges", "Rat.num", "Eq"] | **Additive Commutative Monoid of Charges.** For any system of charges, the set of its charges forms an additive commutative monoid. | |
ACCSystemCharges.chargesModule | PhysLean.QFT.AnomalyCancellation.Basic | An instance to provide the necessary operations and properties for `charges` to form a module
over the field `ℚ`.
| /--
An instance to provide the necessary operations and properties for `charges` to form a module
over the field `ℚ`.
-/
@[simps!]
instance chargesModule (χ : ACCSystemCharges) : Module ℚ χ.Charges :=
Pi.module _ _ _
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L149-L155 | ["ACCSystemCharges", "Module", "Rat", "ACCSystemCharges.Charges"] | **Rational Module Structure on System Charges.** For any system of charges, the set of charges associated with that system forms a module over the field of rational numbers $\mathbb{Q}$, with operations defined pointwise. |
ACCSystemCharges.chargesModule_smul | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance chargesModule (χ : ACCSystemCharges) : Module ℚ χ.Charges := | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L153-L153 | ["ACCSystemCharges", "Rat", "Fin", "ACCSystemCharges.numberCharges", "SMul.smul", "Eq", "HMul.hMul"] | **Vector Space Structure on System Charges.** For any system of charges, the set of its charges is equipped with the structure of a module over the rational numbers $\mathbb{Q}$. | |
ACCSystemCharges.instFiniteRatCharges | PhysLean.QFT.AnomalyCancellation.Basic | The module `χ.Charges` over `ℚ` is finite. | /-- The module `χ.Charges` over `ℚ` is finite. -/
instance (χ : ACCSystemCharges) : Module.Finite ℚ χ.Charges :=
FiniteDimensional.finiteDimensional_pi ℚ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L168-L170 | ["ACCSystemCharges", "Module.Finite", "Rat", "ACCSystemCharges.Charges"] | **Finiteness of the Rational Module of Charges.** For any system of charges, the space of charges viewed as a module over the rational numbers is finitely generated. |
ACCSystemCharges.numberCharges | PhysLean.QFT.AnomalyCancellation.Basic | The number of charges. | numberCharges : ℕ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L99-L99 | ["ACCSystemCharges", "Nat"] | **Number of Charges.** The total number of charges in the system, represented as a natural number. |
ACCSystemChargesMk | PhysLean.QFT.AnomalyCancellation.Basic | Creates an `ACCSystemCharges` object with the specified number of charges.
| /--
Creates an `ACCSystemCharges` object with the specified number of charges.
-/
def ACCSystemChargesMk (n : ℕ) : ACCSystemCharges where
numberCharges := n
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L110-L114 | ["Nat", "ACCSystemCharges"] | **System of Charges Construction.** Given a natural number $n$, we define a system of charges consisting of exactly $n$ charges. |
ACCSystemGroupAction | PhysLean.QFT.AnomalyCancellation.GroupActions | The type of a group action on a system of charges is defined as a representation on
the vector spaces of charges under which the anomaly equations are invariant.
| /-- The type of a group action on a system of charges is defined as a representation on
the vector spaces of charges under which the anomaly equations are invariant.
-/
structure ACCSystemGroupAction (χ : ACCSystem) where
/-- The underlying type of the group. -/
group : Type
/-- An instance given the `group` comp... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L21-L36 | ["ACCSystem"] | **Group Action on an Anomaly Cancellation System.** A group action on a system of charges consists of a group $G$ and a linear representation of $G$ on the rational vector space of charges such that the anomaly cancellation equations remain invariant under the action. Specifically, for every group element $g$ and every... |
ACCSystemGroupAction.cubicInvariant | PhysLean.QFT.AnomalyCancellation.GroupActions | The invariance of the cubic ACC under the group action. | cubicInvariant : ∀ g S, χ.cubicACC (rep g S) = χ.cubicACC S
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L36-L36 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemCharges.Charges", "ACCSystem.cubicACC", "ACCSystemGroupAction.rep", "Eq"] | **Invariance of the Cubic Anomaly Cancellation Condition.** For every element $g$ of the group and every charge assignment $S$, the cubic anomaly cancellation condition evaluated at the representation of $g$ acting on $S$ is equal to the cubic anomaly cancellation condition evaluated at $S$. |
ACCSystemGroupAction.group | PhysLean.QFT.AnomalyCancellation.GroupActions | The underlying type of the group. | group : Type
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L26-L26 | ["ACCSystem", "ACCSystemGroupAction"] | **Group of the ACC System Action.** The underlying group that acts on the space of charges within an anomaly cancellation condition system. |
ACCSystemGroupAction.groupInst | PhysLean.QFT.AnomalyCancellation.GroupActions | An instance given the `group` component the structure of a `Group`. | groupInst : Group group
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L28-L28 | ["ACCSystem", "ACCSystemGroupAction", "Group", "ACCSystemGroupAction.group"] | **Group Structure of the ACC System Action.** The underlying group component of a group action on an anomaly cancellation condition system is equipped with the structure of a group. |
ACCSystemGroupAction.instGroupGroup | PhysLean.QFT.AnomalyCancellation.GroupActions | The given instance of a group on the `group` field of a `ACCSystemGroupAction`. | /-- The given instance of a group on the `group` field of a `ACCSystemGroupAction`. -/
instance {χ : ACCSystem} (G : ACCSystemGroupAction χ) : Group G.group := G.groupInst
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L40-L41 | ["ACCSystem", "ACCSystemGroupAction", "Group", "ACCSystemGroupAction.group"] | **Group Structure of an ACC System Action.** For any group action on an anomaly cancellation condition system, the underlying set designated as the group of the action is equipped with a formal group structure. |
ACCSystemGroupAction.linSolMap | PhysLean.QFT.AnomalyCancellation.GroupActions | The action of a group element on the vector space of linear solutions. | /-- The action of a group element on the vector space of linear solutions. -/
def linSolMap {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) :
χ.LinSols →ₗ[ℚ] χ.LinSols where
toFun S := ⟨G.rep g S.val, by
intro i
rw [G.linearInvariant, S.linearSol]⟩
map_add' S T := ACCSystemLinear.LinSols.ext ... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L43-L50 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemLinear.LinSols", "LinearMap", "Rat"] | **Action on Linear Solutions.** For a given anomaly cancellation condition system and a group acting on it, every group element $g$ induces a linear map from the vector space of linear solutions to itself. This map is defined by applying the group's representation of $g$ to a solution vector, which remains a valid line... |
ACCSystemGroupAction.linSolRep | PhysLean.QFT.AnomalyCancellation.GroupActions | The representation acting on the vector space of solutions to the linear ACCs. | /-- The representation acting on the vector space of solutions to the linear ACCs. -/
@[simps!]
def linSolRep {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
Representation ℚ G.group χ.LinSols where
toFun := G.linSolMap
map_mul' g1 g2 := by
refine LinearMap.ext fun S ↦ ACCSystemLinear.LinSols.ext ?_
chan... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L52-L66 | ["ACCSystem", "ACCSystemGroupAction", "Representation", "Rat", "ACCSystemGroupAction.group", "ACCSystemLinear.LinSols"] | **Linear Solution Representation.** Given a group action on an anomaly cancellation condition system, there is a natural representation of the group $G$ on the rational vector space of solutions to the linear anomaly cancellation conditions. This representation is defined by the restriction of the group's action on the... |
ACCSystemGroupAction.linSolRep_apply_apply_val | PhysLean.QFT.AnomalyCancellation.GroupActions | @[simps!]
def linSolRep {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
Representation ℚ G.group χ.LinSols where
toFun := G.linSolMap | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L53-L53 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemLinear.LinSols", "ACCSystemGroupAction.linSolRep", "ACCSystemLinear.LinSols.val", "Eq", "ACCSystemGroupAction.rep"] | **Linear Solution Representation.** For a group action on an anomaly cancellation condition system $\chi$, the linear map that sends each group element to its corresponding action on the space of linear solutions $\chi.\text{LinSols}$ defines a representation of the group $G$ over $\mathbb{Q}$. | |
ACCSystemGroupAction.linSolRep_quadSolAction_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | lemma linSolRep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.QuadSols) : χ.quadSolsInclLinSols (G.quadSolAction.toFun _ _ S g) =
G.linSolRep g (χ.quadSolsInclLinSols S) := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L89-L91 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemQuad.QuadSols", "ACCSystemQuad.quadSolsInclLinSols", "MulAction.toFun", "Eq", "ACCSystemGroupAction.linSolRep"] | **Commutativity of Group Action and Solution Inclusion.** For an anomaly cancellation condition system $\chi$ and a group action $G$ on $\chi$, the inclusion of quadratic solutions into linear solutions is equivariant with respect to the action of $G$. Specifically, for any group element $g \in G$ and any quadratic sol... | |
ACCSystemGroupAction.linSolRep_solAction_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | lemma linSolRep_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.Sols) : χ.solsInclLinSols (G.solAction.toFun _ _ S g) =
G.linSolRep g (χ.solsInclLinSols S) := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L118-L120 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystem.Sols", "ACCSystem.solsInclLinSols", "MulAction.toFun", "Eq", "ACCSystemGroupAction.linSolRep"] | **Commutativity of Group Action and Solution Inclusion.** For a given anomaly cancellation condition system and a group action on it, the inclusion of the set of solutions into the space of linear solutions is equivariant with respect to the group action. Specifically, for any group element $g$ and any solution $S$, th... | |
ACCSystemGroupAction.linearInvariant | PhysLean.QFT.AnomalyCancellation.GroupActions | The invariance of the linear ACCs under the group action. | linearInvariant : ∀ i g S, χ.linearACCs i (rep g S) = χ.linearACCs i S
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L32-L32 | ["ACCSystem", "ACCSystemGroupAction", "Fin", "ACCSystemLinear.numberLinear", "ACCSystemGroupAction.group", "ACCSystemCharges.Charges", "ACCSystemLinear.linearACCs", "ACCSystemGroupAction.rep", "Eq"] | **Invariance of Linear Constraints.** For any index $i$, group element $g$, and charge configuration $S$, the $i$-th linear anomaly cancellation condition evaluated on the representation of $g$ acting on $S$ is equal to the $i$-th linear anomaly cancellation condition evaluated on $S$. |
ACCSystemGroupAction.quadInvariant | PhysLean.QFT.AnomalyCancellation.GroupActions | The invariance of the quadratic ACCs under the group action. | quadInvariant : ∀ i g S, (χ.quadraticACCs i) (rep g S) = (χ.quadraticACCs i) S
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L34-L34 | ["ACCSystem", "ACCSystemGroupAction", "Fin", "ACCSystemQuad.numberQuadratic", "ACCSystemGroupAction.group", "ACCSystemCharges.Charges", "ACCSystemQuad.quadraticACCs", "ACCSystemGroupAction.rep", "Eq"] | **Invariance of Quadratic Constraints.** For every index $i$ and every group element $g$, the $i$-th quadratic anomaly cancellation condition is invariant under the group representation; that is, the value of the $i$-th quadratic form evaluated at the image of a charge configuration $S$ under the representation of $g$ ... |
ACCSystemGroupAction.quadSolAction | PhysLean.QFT.AnomalyCancellation.GroupActions | A multiplicative action of `G.group` on `quadSols`. | /-- A multiplicative action of `G.group` on `quadSols`. -/
instance quadSolAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
MulAction G.group χ.QuadSols where
smul f S := ⟨G.linSolRep f S.1, by simp [linSolRep_apply_apply_val, G.quadInvariant, S.quadSol]⟩
mul_smul f1 f2 S := by
apply ACCSystemQuad.Quad... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L74-L87 | ["ACCSystem", "ACCSystemGroupAction", "MulAction", "ACCSystemGroupAction.group", "ACCSystemQuad.QuadSols"] | **Group Action on Quadratic Solutions.** Given a group action on an anomaly cancellation system, the underlying group acts on the set of quadratic solutions. This action is defined by applying the group's linear representation to the charge vector of a solution, which preserves the quadratic and linear constraints of t... |
ACCSystemGroupAction.quadSolAction_solAction_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | lemma quadSolAction_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.Sols) : χ.solsInclQuadSols (G.solAction.toFun _ _ S g) =
G.quadSolAction.toFun _ _ (χ.solsInclQuadSols S) g := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L114-L116 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystem.Sols", "ACCSystem.solsInclQuadSols", "MulAction.toFun", "Eq", "ACCSystemQuad.QuadSols"] | **Commutativity of Group Actions and Solution Inclusion.** For an anomaly cancellation condition system $\chi$ and a group action $G$ on $\chi$, the inclusion map from the set of full solutions to the set of quadratic solutions is equivariant with respect to the action of $G$. Specifically, for any group element $g \in... | |
ACCSystemGroupAction.rep | PhysLean.QFT.AnomalyCancellation.GroupActions | The representation of group acting on the vector space of charges. | rep : Representation ℚ group χ.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L30-L30 | ["ACCSystem", "ACCSystemGroupAction", "Representation", "Rat", "ACCSystemGroupAction.group", "ACCSystemCharges.Charges"] | **Group Representation on the Space of Charges.** Given a group action on an anomaly cancellation system, there exists a linear representation of the group over the rational numbers on the vector space of charges. |
ACCSystemGroupAction.rep_linSolRep_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | The representation on the charges and anomaly free solutions
commutes with the inclusion. | /-- The representation on the charges and anomaly free solutions
commutes with the inclusion. -/
lemma rep_linSolRep_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.LinSols) : χ.linSolsIncl (G.linSolRep g S) =
G.rep g (χ.linSolsIncl S) := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L68-L72 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemLinear.LinSols", "ACCSystemLinear.linSolsIncl", "ACCSystemGroupAction.linSolRep", "Eq", "ACCSystemGroupAction.rep"] | **Commutativity of Group Action and Linear Solution Inclusion.** For any anomaly cancellation condition system and a group acting on it, the group representation on the space of linear anomaly-free solutions is compatible with the representation on the space of all charges. Specifically, for any group element $g$ and a... |
ACCSystemGroupAction.rep_quadSolAction_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | lemma rep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.QuadSols) : χ.quadSolsIncl (G.quadSolAction.toFun _ _ S g) =
G.rep g (χ.quadSolsIncl S) := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L93-L95 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystemQuad.QuadSols", "ACCSystemQuad.quadSolsIncl", "MulAction.toFun", "Eq", "ACCSystemGroupAction.rep"] | **Commutativity of Group Action and Quadratic Solution Inclusion.** For an anomaly cancellation condition system $\chi$ and a group action $G$ on it, the action of an element $g \in G$ on a quadratic solution $S$ is compatible with the representation of $g$ on the space of charges. Specifically, the charge correspondin... | |
ACCSystemGroupAction.rep_solAction_commute | PhysLean.QFT.AnomalyCancellation.GroupActions | lemma rep_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.Sols) : χ.solsIncl (G.solAction.toFun _ _ S g) = G.rep g (χ.solsIncl S) := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L122-L123 | ["ACCSystem", "ACCSystemGroupAction", "ACCSystemGroupAction.group", "ACCSystem.Sols", "ACCSystem.solsIncl", "MulAction.toFun", "Eq", "ACCSystemGroupAction.rep"] | **Commutativity of Group Action and Solution Inclusion.** For an anomaly cancellation condition system $\chi$ and a group action $G$ on it, the inclusion of solutions into the space of charges is equivariant with respect to the group action. Specifically, for any group element $g \in G$ and any solution $S$ to the syst... | |
ACCSystemGroupAction.solAction | PhysLean.QFT.AnomalyCancellation.GroupActions | The group action acting on solutions to the anomaly cancellation conditions. | /-- The group action acting on solutions to the anomaly cancellation conditions. -/
instance solAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) : MulAction G.group χ.Sols where
smul g S := ⟨G.quadSolAction.toFun _ _ S.1 g, by
simp only [MulAction.toFun_apply]
change χ.cubicACC (G.rep g S.val) = 0
rw [... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/GroupActions.lean#L97-L112 | ["ACCSystem", "ACCSystemGroupAction", "MulAction", "ACCSystemGroupAction.group", "ACCSystem.Sols"] | **Group Action on Anomaly Cancellation Solutions.** Given an anomaly cancellation condition system and a group action compatible with that system, there is a group action of the associated group on the set of solutions to the anomaly cancellation conditions. For any group element $g$ and solution $S$, the action is def... |
ACCSystemLinear | PhysLean.QFT.AnomalyCancellation.Basic | The type of charges plus the linear ACCs. | /-- The type of charges plus the linear ACCs. -/
structure ACCSystemLinear extends ACCSystemCharges where
/-- The number of linear ACCs. -/
numberLinear : ℕ
/-- The linear ACCs. -/
linearACCs : Fin numberLinear → (toACCSystemCharges.Charges →ₗ[ℚ] ℚ)
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L183-L188 | ["ACCSystemCharges"] | **Linear ACC System.** A linear ACC system is a system of charges equipped with a finite collection of linear constraints. It is defined by a natural number $n$, representing the number of linear ACCs, and a mapping that assigns to each index $i \in \{0, \dots, n-1\}$ a linear functional from the space of charges to th... |
ACCSystemLinear.LinSols | PhysLean.QFT.AnomalyCancellation.Basic | The type of solutions to the linear ACCs. | /-- The type of solutions to the linear ACCs. -/
structure LinSols (χ : ACCSystemLinear) where
/-- The underlying charge. -/
val : χ.1.Charges
/-- The condition that the charge satisfies the linear ACCs. -/
linearSol : ∀ i : Fin χ.numberLinear, χ.linearACCs i val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L201-L206 | ["ACCSystemLinear"] | **Linear ACC Solutions.** For a given linear ACC system $\chi$, a solution is a charge $v$ in the space of charges of $\chi$ such that $v$ satisfies all $n$ linear anomaly cancellation conditions; that is, for every index $i \in \{0, \dots, n-1\}$, the $i$-th linear functional of the system evaluates to zero at $v$. |
ACCSystemLinear.LinSols.ext | PhysLean.QFT.AnomalyCancellation.Basic | Two solutions are equal if the underlying charges are equal. | /-- Two solutions are equal if the underlying charges are equal. -/
@[ext]
lemma LinSols.ext {χ : ACCSystemLinear} {S T : χ.LinSols} (h : S.val = T.val) : S = T := by
cases' S
simp_all only
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L216-L220 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "ACCSystemLinear.LinSols.val", "Eq"] | **Extensionality of Linear Solutions.** Two solutions to a linear ACC system are equal if and only if their underlying charge vectors are equal. |
ACCSystemLinear.LinSols.ext_iff | PhysLean.QFT.AnomalyCancellation.Basic | @[ext]
lemma LinSols.ext {χ : ACCSystemLinear} {S T : χ.LinSols} (h : S.val = T.val) : S = T := by | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L217-L217 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "Eq", "Iff", "ACCSystemLinear.LinSols.val"] | **Equality of Linear ACC Solutions.** Two solutions to a linear ACC system are equal if and only if their underlying charge vectors are equal. | |
ACCSystemLinear.LinSols.linearSol | PhysLean.QFT.AnomalyCancellation.Basic | The condition that the charge satisfies the linear ACCs. | linearSol : ∀ i : Fin χ.numberLinear, χ.linearACCs i val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L206-L206 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "Fin", "ACCSystemLinear.numberLinear", "ACCSystemLinear.linearACCs", "ACCSystemLinear.LinSols.val", "Eq"] | **Linear Anomaly Cancellation Condition.** A charge satisfies the linear anomaly cancellation conditions if, for every index $i$ from the set of linear constraints, the $i$-th linear constraint evaluated at that charge is equal to zero. |
ACCSystemLinear.LinSols.val | PhysLean.QFT.AnomalyCancellation.Basic | The underlying charge. | val : χ.1.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L204-L204 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "ACCSystemCharges.Charges"] | **Underlying Charge of a Linear Solution.** For a given linear anomaly cancellation system, this represents the specific charge vector associated with a solution to the system's linear constraints. |
ACCSystemLinear.linSolsAddCommGroup | PhysLean.QFT.AnomalyCancellation.Basic | instance linSolsAddCommGroup (χ : ACCSystemLinear) : AddCommGroup χ.LinSols :=
Module.addCommMonoidToAddCommGroup ℚ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L258-L259 | ["ACCSystemLinear", "AddCommGroup", "ACCSystemLinear.LinSols"] | **Additive Commutative Group of Linear Solutions.** For any linear ACC system, the set of its linear solutions forms an additive commutative group. | |
ACCSystemLinear.linSolsAddCommMonoid | PhysLean.QFT.AnomalyCancellation.Basic | An instance providing the operations and properties for `LinSols` to form an
additive commutative monoid. | /-- An instance providing the operations and properties for `LinSols` to form an
additive commutative monoid. -/
@[simps!]
instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
AddCommMonoid χ.LinSols where
add S T := ⟨S.val + T.val, fun _ ↦ by simp [(χ.linearACCs _).map_add, S.linearSol _,
T.linearSol _]⟩
... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L230-L244 | ["ACCSystemLinear", "AddCommMonoid", "ACCSystemLinear.LinSols"] | **Additive Commutative Monoid of Linear ACC Solutions.** The set of solutions to a linear ACC system forms an additive commutative monoid. Addition and scalar multiplication by natural numbers are defined pointwise on the underlying space of charges, with the zero charge serving as the identity element. This structure ... |
ACCSystemLinear.linSolsAddCommMonoid_add_val | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
AddCommMonoid χ.LinSols where
add S T := ⟨S.val + T.val, fun _ ↦ by simp [(χ.linearACCs _).map_add, S.linearSol _, | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L232-L232 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "HAdd.hAdd", "ACCSystemLinear.LinSols.val", "Eq"] | **Additive Commutative Monoid of Linear Solutions.** For a linear ACC system, the set of its linear solutions forms an additive commutative monoid, where the addition of two solutions is defined by the pointwise sum of their underlying charge vectors. | |
ACCSystemLinear.linSolsAddCommMonoid_nsmul_val | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
AddCommMonoid χ.LinSols where
add S T := ⟨S.val + T.val, fun _ ↦ by simp [(χ.linearACCs _).map_add, S.linearSol _, | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L232-L232 | ["ACCSystemLinear", "Nat", "ACCSystemLinear.LinSols", "HSMul.hSMul", "ACCSystemLinear.LinSols.val", "Eq"] | **Scalar Multiplication of Linear ACC Solutions.** For a given linear anomaly cancellation system, the scalar multiplication of a natural number $n$ and a solution $S$ is defined such that the underlying charge of the resulting solution $n \cdot S$ is equal to $n$ times the underlying charge of $S$. | |
ACCSystemLinear.linSolsAddCommMonoid_zero_val | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
AddCommMonoid χ.LinSols where
add S T := ⟨S.val + T.val, fun _ ↦ by simp [(χ.linearACCs _).map_add, S.linearSol _, | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L232-L232 | ["ACCSystemLinear", "ACCSystemLinear.LinSols.val", "Eq", "Zero.zero"] | **Additive Commutative Monoid of Linear Solutions.** For a given linear ACC system, the set of its linear solutions forms an additive commutative monoid, where the addition of two solutions is defined by the pointwise sum of their underlying charge vectors. | |
ACCSystemLinear.linSolsIncl | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of `LinSols` into `charges`. | /-- The inclusion of `LinSols` into `charges`. -/
def linSolsIncl (χ : ACCSystemLinear) : χ.LinSols →ₗ[ℚ] χ.Charges where
toFun S := S.val
map_add' _ _ := rfl
map_smul' _ _ := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L270-L274 | ["ACCSystemLinear", "ACCSystemLinear.LinSols", "LinearMap", "Rat", "ACCSystemCharges.Charges"] | **Inclusion of Linear Solutions into Charges.** For a given linear ACC system, there is a $\mathbb{Q}$-linear map from the space of linear solutions to the space of charges that sends each solution to its underlying charge. |
ACCSystemLinear.linSolsIncl_injective | PhysLean.QFT.AnomalyCancellation.Basic | @[sorryful]
lemma linSolsIncl_injective (χ : ACCSystemLinear) :
Function.Injective χ.linSolsIncl := by
sorry | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L276-L279 | ["ACCSystemLinear", "Function.Injective", "ACCSystemLinear.linSolsIncl"] | **Injectivity of the Linear Solution Inclusion.** For any linear ACC system, the linear map that includes the space of linear solutions into the space of charges is injective. | |
ACCSystemLinear.linSolsModule | PhysLean.QFT.AnomalyCancellation.Basic | An instance providing the operations and properties for `LinSols` to form a
module over `ℚ`. | /-- An instance providing the operations and properties for `LinSols` to form a
module over `ℚ`. -/
@[simps!]
instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
smul a S := ⟨a • S.val, fun _ ↦ by simp [(χ.linearACCs _).map_smul, S.linearSol _]⟩
one_smul one_smul := LinSols.ext (χ.chargesModul... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L246-L256 | ["ACCSystemLinear", "Module", "Rat", "ACCSystemLinear.LinSols"] | **Vector Space of Linear ACC Solutions.** For any linear ACC system, the set of solutions to the linear anomaly cancellation conditions forms a module over the rational numbers $\mathbb{Q}$. Scalar multiplication is defined pointwise on the underlying charges, and the module axioms are satisfied by inheriting the prope... |
ACCSystemLinear.linSolsModule_smul_val | PhysLean.QFT.AnomalyCancellation.Basic | @[simps!]
instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
smul a S := ⟨a • S.val, fun _ ↦ by simp [(χ.linearACCs _).map_smul, S.linearSol _]⟩ | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L248-L248 | ["ACCSystemLinear", "Rat", "ACCSystemLinear.LinSols", "SMul.smul", "ACCSystemLinear.LinSols.val", "Eq", "HSMul.hSMul"] | **Scalar Multiplication of Linear ACC Solutions.** For a linear ACC system, the set of linear solutions forms a module over the rational numbers $\mathbb{Q}$. The scalar multiplication of a rational number $a$ and a solution $S$ is defined such that the underlying charge vector of the resulting solution is equal to $a$... | |
ACCSystemLinear.linearACCs | PhysLean.QFT.AnomalyCancellation.Basic | The linear ACCs. | linearACCs : Fin numberLinear → (toACCSystemCharges.Charges →ₗ[ℚ] ℚ)
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L188-L188 | ["ACCSystemLinear", "Fin", "ACCSystemLinear.numberLinear", "ACCSystemCharges.Charges", "LinearMap", "Rat"] | **Linear ACC Constraints.** For a linear ACC system, there exists a collection of linear functionals indexed by the set $\{0, \dots, n-1\}$, where $n$ is the number of linear constraints. Each such functional maps the space of charges to the rational numbers. |
ACCSystemLinear.numberLinear | PhysLean.QFT.AnomalyCancellation.Basic | The number of linear ACCs. | numberLinear : ℕ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L186-L186 | ["ACCSystemLinear", "Nat"] | **Number of Linear Constraints.** Within a linear ACC system, this value represents the total number of linear constraints imposed on the system of charges. |
ACCSystemQuad | PhysLean.QFT.AnomalyCancellation.Basic | The type of charges plus the linear ACCs plus the quadratic ACCs. | /-- The type of charges plus the linear ACCs plus the quadratic ACCs. -/
structure ACCSystemQuad extends ACCSystemLinear where
/-- The number of quadratic ACCs. -/
numberQuadratic : ℕ
/-- The quadratic ACCs. -/
quadraticACCs : Fin numberQuadratic → HomogeneousQuadratic toACCSystemCharges.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L295-L300 | ["ACCSystemLinear"] | **Quadratic ACC System.** A quadratic ACC system is a linear ACC system extended by a finite collection of quadratic constraints. It is defined by a natural number $n$, representing the number of quadratic constraints, and a mapping that assigns to each index $i \in \{0, \dots, n-1\}$ a homogeneous quadratic form on th... |
ACCSystemQuad.QuadSols | PhysLean.QFT.AnomalyCancellation.Basic | The type of solutions to the linear and quadratic ACCs. | /-- The type of solutions to the linear and quadratic ACCs. -/
structure QuadSols (χ : ACCSystemQuad) extends χ.LinSols where
/-- The condition that the charge satisfies the quadratic ACCs. -/
quadSol : ∀ i : Fin χ.numberQuadratic, (χ.quadraticACCs i) val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L312-L315 | ["ACCSystemQuad", "ACCSystemLinear.LinSols"] | **Quadratic ACC Solutions.** For a given quadratic anomaly cancellation system, a solution is a charge that satisfies all linear anomaly cancellation conditions of the system and, additionally, satisfies every quadratic anomaly cancellation condition, such that each quadratic form in the system evaluates to zero at tha... |
ACCSystemQuad.QuadSols.ext | PhysLean.QFT.AnomalyCancellation.Basic | Two `QuadSols` are equal if the underlying charges are equal. | /-- Two `QuadSols` are equal if the underlying charges are equal. -/
@[ext]
lemma QuadSols.ext {χ : ACCSystemQuad} {S T : χ.QuadSols} (h : S.val = T.val) :
S = T := by
have h := ACCSystemLinear.LinSols.ext h
cases' S
simp_all only
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L325-L331 | ["ACCSystemQuad", "ACCSystemQuad.QuadSols", "ACCSystemLinear.LinSols.val", "Eq"] | **Extensionality of Quadratic ACC Solutions.** Two solutions to a quadratic anomaly cancellation system are equal if and only if their underlying charge vectors are equal. |
ACCSystemQuad.QuadSols.ext_iff | PhysLean.QFT.AnomalyCancellation.Basic | @[ext]
lemma QuadSols.ext {χ : ACCSystemQuad} {S T : χ.QuadSols} (h : S.val = T.val) :
S = T := by | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L326-L326 | ["ACCSystemQuad", "ACCSystemQuad.QuadSols", "Eq", "Iff", "ACCSystemLinear.LinSols.val"] | **Equality of Quadratic ACC Solutions.** Two solutions to a quadratic anomaly cancellation system are equal if and only if their underlying charge vectors are equal. | |
ACCSystemQuad.QuadSols.quadSol | PhysLean.QFT.AnomalyCancellation.Basic | The condition that the charge satisfies the quadratic ACCs. | quadSol : ∀ i : Fin χ.numberQuadratic, (χ.quadraticACCs i) val = 0
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L315-L315 | ["ACCSystemQuad", "ACCSystemQuad.QuadSols", "Fin", "ACCSystemQuad.numberQuadratic", "ACCSystemQuad.quadraticACCs", "ACCSystemLinear.LinSols.val", "Eq"] | **Quadratic Anomaly Cancellation Condition.** A charge vector satisfies the quadratic anomaly cancellation conditions if every quadratic constraint in the system, when evaluated at that charge, vanishes. |
ACCSystemQuad.linSolsInclQuadSolsZero | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of the linear solutions into the quadratic solutions, where there is
no quadratic equations (i.e. no U(1)'s in the underlying gauge group). | /-- The inclusion of the linear solutions into the quadratic solutions, where there is
no quadratic equations (i.e. no U(1)'s in the underlying gauge group). -/
def linSolsInclQuadSolsZero (χ : ACCSystemQuad) (h : χ.numberQuadratic = 0) :
χ.LinSols →[ℚ] χ.QuadSols where
toFun S := ⟨S, by intro i; rw [h] at i; e... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L379-L384 | ["ACCSystemQuad", "ACCSystemQuad.numberQuadratic", "Eq", "ACCSystemLinear.LinSols", "MulActionHom", "id", "ACCSystemQuad.QuadSols"] | **Inclusion of Linear Solutions into Quadratic Solutions for Systems with No Quadratic Constraints.** For a quadratic anomaly cancellation system with zero quadratic constraints, every solution to the linear constraints is vacuously a solution to the quadratic constraints. This induces a $\mathbb{Q}$-linear map from th... |
ACCSystemQuad.numberQuadratic | PhysLean.QFT.AnomalyCancellation.Basic | The number of quadratic ACCs. | numberQuadratic : ℕ
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L298-L298 | ["ACCSystemQuad", "Nat"] | **Number of Quadratic Constraints.** The natural number representing the total count of quadratic constraints within a quadratic ACC system. |
ACCSystemQuad.quadSolsIncl | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of quadratic solutions into all charges. | /-- The inclusion of quadratic solutions into all charges. -/
def quadSolsIncl (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.Charges :=
MulActionHom.comp χ.linSolsIncl.toMulActionHom χ.quadSolsInclLinSols
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L395-L397 | ["ACCSystemQuad", "ACCSystemQuad.QuadSols", "MulActionHom", "id", "ACCSystemCharges.Charges"] | **Inclusion of Quadratic Solutions into Charges.** For a quadratic anomaly cancellation system, the inclusion of the set of quadratic solutions into the space of all charges is defined as the composition of the inclusion of quadratic solutions into linear solutions with the inclusion of linear solutions into the space ... |
ACCSystemQuad.quadSolsInclLinSols | PhysLean.QFT.AnomalyCancellation.Basic | The inclusion of quadratic solutions into linear solutions. | /-- The inclusion of quadratic solutions into linear solutions. -/
def quadSolsInclLinSols (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.LinSols where
toFun := QuadSols.toLinSols
map_smul' _ _ := rfl
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L358-L361 | ["ACCSystemQuad", "ACCSystemQuad.QuadSols", "MulActionHom", "id", "ACCSystemLinear.LinSols"] | **Inclusion of Quadratic Solutions into Linear Solutions.** For a given quadratic anomaly cancellation system, there exists a $\mathbb{Q}$-equivariant map from the space of quadratic solutions to the space of linear solutions, defined by the natural inclusion that views every solution satisfying both linear and quadrat... |
ACCSystemQuad.quadSolsInclLinSols_injective | PhysLean.QFT.AnomalyCancellation.Basic | lemma quadSolsInclLinSols_injective (χ : ACCSystemQuad) :
Function.Injective χ.quadSolsInclLinSols := by
intro S T h
ext
simpa [ACCSystemQuad.quadSolsInclLinSols] using
congrArg (fun X => X.val) h
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L363-L368 | ["ACCSystemQuad", "Function.Injective", "ACCSystemQuad.quadSolsInclLinSols"] | **Injectivity of the Inclusion from Quadratic to Linear Solutions.** For any quadratic anomaly cancellation system, the map that includes the space of quadratic solutions into the space of linear solutions is injective. | |
ACCSystemQuad.quadSolsIncl_injective | PhysLean.QFT.AnomalyCancellation.Basic | @[sorryful]
lemma quadSolsIncl_injective (χ : ACCSystemQuad) :
Function.Injective χ.quadSolsIncl := by
intro S T h
have h' : χ.quadSolsInclLinSols S = χ.quadSolsInclLinSols T := by
apply ACCSystemLinear.linSolsIncl_injective (χ := χ.toACCSystemLinear)
simpa [ACCSystemQuad.quadSolsIncl, MulActionHom.comp... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L399-L406 | ["ACCSystemQuad", "Function.Injective", "ACCSystemQuad.quadSolsIncl"] | **Injectivity of the Inclusion of Quadratic Solutions.** For any quadratic anomaly cancellation system, the map that includes the set of quadratic solutions into the space of charges is injective. | |
ACCSystemQuad.quadSolsMulAction | PhysLean.QFT.AnomalyCancellation.Basic | An instance giving the properties and structures to define an action of `ℚ` on `QuadSols`. | /-- An instance giving the properties and structures to define an action of `ℚ` on `QuadSols`. -/
instance quadSolsMulAction (χ : ACCSystemQuad) : MulAction ℚ χ.QuadSols where
smul a S := ⟨a • S.toLinSols, fun _ ↦ by erw [(χ.quadraticACCs _).map_smul, S.quadSol _,
mul_zero]⟩
mul_smul a b S := QuadSols.ext (mul_... | https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L342-L347 | ["ACCSystemQuad", "MulAction", "Rat", "ACCSystemQuad.QuadSols"] | **Rational Action on Quadratic ACC Solutions.** For a given quadratic anomaly cancellation system, the set of its solutions admits a scalar action by the rational numbers $\mathbb{Q}$. This action is defined by scaling the underlying linear solution by a rational constant $a \in \mathbb{Q}$, which preserves the quadrat... |
ACCSystemQuad.quadraticACCs | PhysLean.QFT.AnomalyCancellation.Basic | The quadratic ACCs. | quadraticACCs : Fin numberQuadratic → HomogeneousQuadratic toACCSystemCharges.Charges
| https://github.com/HEPLean/PhysLean/blob/7ba876c3f5fcd9f67fa2ee9326dd5d0b1e70079a/PhysLean/QFT/AnomalyCancellation/Basic.lean#L300-L300 | ["ACCSystemQuad", "Fin", "ACCSystemQuad.numberQuadratic", "HomogeneousQuadratic", "ACCSystemCharges.Charges"] | **Quadratic ACC Constraints.** In a quadratic ACC system, there is a collection of homogeneous quadratic forms on the space of system charges, indexed by the set of integers $\{0, 1, \dots, n-1\}$, where $n$ is the total number of quadratic constraints in the system. |
ADEInequality.A | Mathlib.NumberTheory.ADEInequality | `A r := {1,1,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $A_r$. | /-- `A r := {1,1,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $A_r$. -/
def A (r : ℕ+) : Multiset ℕ+ :=
A' 1 r
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L54-L60 | ["PNat", "Multiset"] | **Dynkin Diagram Type $A_r$ Inequality Solution.** For any positive integer $r$, let $A$ be the multiset of positive integers $\{1, 1, r\}$. This multiset represents a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$, where $p, q, r$ are the elements of the multiset. These solutions correspond t... |
ADEInequality.A' | Mathlib.NumberTheory.ADEInequality | `A' q r := {1,q,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`. | /-- `A' q r := {1,q,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`. -/
def A' (q r : ℕ+) : Multiset ℕ+ :=
{1, q, r}
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L48-L52 | ["PNat", "Multiset"] | **The $A'_n$ Family of Solutions.** For any two positive integers $q$ and $r$, the multiset $\{1, q, r\}$ is a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$ where $p=1$. |
ADEInequality.Admissible | Mathlib.NumberTheory.ADEInequality | A multiset `pqr` of positive natural numbers is `admissible`
if it is equal to `A' q r`, or `D' r`, or one of `E6`, `E7`, or `E8`. | /-- A multiset `pqr` of positive natural numbers is `admissible`
if it is equal to `A' q r`, or `D' r`, or one of `E6`, `E7`, or `E8`. -/
def Admissible (pqr : Multiset ℕ+) : Prop :=
(∃ q r, A' q r = pqr) ∨ (∃ r, D' r = pqr) ∨ E' 3 = pqr ∨ E' 4 = pqr ∨ E' 5 = pqr
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L113-L116 | ["Multiset", "PNat"] | **Admissible Multisets.** A multiset of positive natural numbers is considered admissible if it is equal to $A'(q, r)$ for some $q$ and $r$, $D'(r)$ for some $r$, or one of the specific multisets $E'(3)$, $E'(4)$, or $E'(5)$. |
ADEInequality.Admissible.one_lt_sumInv | Mathlib.NumberTheory.ADEInequality | theorem Admissible.one_lt_sumInv {pqr : Multiset ℕ+} : Admissible pqr → 1 < sumInv pqr := by
rw [Admissible]
rintro (⟨p', q', H⟩ | ⟨n, H⟩ | H | H | H)
· rw [← H, A', sumInv_pqr, add_assoc]
simp only [lt_add_iff_pos_right, PNat.one_coe, inv_one, Nat.cast_one]
apply add_pos <;> simp only [PNat.pos, Nat.cast... | https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L142-L152 | ["Multiset", "PNat", "ADEInequality.Admissible", "LT.lt", "ADEInequality.sumInv"] | **Sum of Inverses for Admissible Multisets.** If a multiset of positive integers $\{p, q, r\}$ is admissible, then the sum of the reciprocals of its elements is strictly greater than 1; that is, $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$. | |
ADEInequality.D' | Mathlib.NumberTheory.ADEInequality | `D' r := {2,2,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $D_{r+2}$. | /-- `D' r := {2,2,r}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $D_{r+2}$. -/
def D' (r : ℕ+) : Multiset ℕ+ :=
{2, 2, r}
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L62-L68 | ["PNat", "Multiset"] | **Dynkin Diagram Type $D_{r+2}$ Solutions.** For any positive integer $r$, we define a multiset of positive integers $\{2, 2, r\}$. This multiset provides a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$ and is associated with the Dynkin diagrams of type $D_{r+2}$. |
ADEInequality.E' | Mathlib.NumberTheory.ADEInequality | `E' r := {2,3,r}` is a `Multiset ℕ+`.
For `r ∈ {3,4,5}` is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $E_{r+3}$. | /-- `E' r := {2,3,r}` is a `Multiset ℕ+`.
For `r ∈ {3,4,5}` is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
These solutions are related to the Dynkin diagrams $E_{r+3}$. -/
def E' (r : ℕ+) : Multiset ℕ+ :=
{2, 3, r}
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L70-L76 | ["PNat", "Multiset"] | **Exceptional ADE Triple.** For any positive integer $r$, let $E'(r)$ be the multiset of positive integers $\{2, 3, r\}$. For $r \in \{3, 4, 5\}$, these multisets provide the triples $(p, q, r)$ that satisfy the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$, corresponding to the Dynkin diagrams $E_6, E_7,$ a... |
ADEInequality.E6 | Mathlib.NumberTheory.ADEInequality | `E6 := {2,3,3}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_6$. | /-- `E6 := {2,3,3}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_6$. -/
def E6 : Multiset ℕ+ :=
E' 3
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L78-L84 | ["Multiset", "PNat"] | **The $E_6$ solution to the ADE inequality.** The multiset of positive integers $\{2, 3, 3\}$ is a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$, where $p, q, r \in \mathbb{N}^+$. This specific solution corresponds to the Dynkin diagram $E_6$. |
ADEInequality.E7 | Mathlib.NumberTheory.ADEInequality | `E7 := {2,3,4}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_7$. | /-- `E7 := {2,3,4}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_7$. -/
def E7 : Multiset ℕ+ :=
E' 4
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L86-L92 | ["Multiset", "PNat"] | **The $E_7$ Solution.** The multiset of positive integers $\{2, 3, 4\}$ is a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$, where $p, q, r \in \mathbb{N}^+$. This specific solution corresponds to the $E_7$ Dynkin diagram. |
ADEInequality.E8 | Mathlib.NumberTheory.ADEInequality | `E8 := {2,3,5}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_8$. | /-- `E8 := {2,3,5}` is a `Multiset ℕ+`
that is a solution to the inequality
`(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`.
This solution is related to the Dynkin diagrams $E_8$. -/
def E8 : Multiset ℕ+ :=
E' 5
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L94-L100 | ["Multiset", "PNat"] | **The $E_8$ ADE Inequality Solution.** The multiset of positive integers $\{2, 3, 5\}$ is a solution to the inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$, corresponding to the $E_8$ Dynkin diagram. |
ADEInequality.admissible_A' | Mathlib.NumberTheory.ADEInequality | theorem admissible_A' (q r : ℕ+) : Admissible (A' q r) :=
Or.inl ⟨q, r, rfl⟩
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L118-L119 | ["PNat", "ADEInequality.Admissible", "ADEInequality.A'"] | **Admissibility of the $A'$ family.** For any two positive integers $q$ and $r$, the multiset $\{1, q, r\}$ is admissible. | |
ADEInequality.admissible_D' | Mathlib.NumberTheory.ADEInequality | theorem admissible_D' (n : ℕ+) : Admissible (D' n) :=
Or.inr <| Or.inl ⟨n, rfl⟩
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L121-L122 | ["PNat", "ADEInequality.Admissible", "ADEInequality.D'"] | **Admissibility of Type $D$ Multisets.** For any positive integer $n$, the multiset $\{2, 2, n\}$ is admissible. | |
ADEInequality.admissible_E'3 | Mathlib.NumberTheory.ADEInequality | theorem admissible_E'3 : Admissible (E' 3) :=
Or.inr <| Or.inr <| Or.inl rfl
| https://github.com/leanprover-community/mathlib4/blob/0e16bffea0a1b2664111fb1cfeb46c94e4a92604/Mathlib/NumberTheory/ADEInequality.lean#L124-L125 | ["ADEInequality.Admissible", "ADEInequality.E'"] | **Admissibility of the Exceptional Triple $E'(3)$.** The multiset $E'(3) = \{2, 3, 3\}$ is admissible. |
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