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refactor: Change case of type and props

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jstoobysmith 2024-06-26 11:54:02 -04:00
parent 18b83f582e
commit f7a638d32e
58 changed files with 695 additions and 696 deletions
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10
HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
View file

@ -148,7 +148,7 @@ end theDeltas
section theBasisVectors
/-- The first part of the basis as charge assignments. -/
def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).charges :=
def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
fun i =>
if i = δ₁ j then
1
@ -159,7 +159,7 @@ def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).charges :=
0
/-- The second part of the basis as charge assignments. -/
def basis!AsCharges (j : Fin n) : (PureU1 (2 * n + 1)).charges :=
def basis!AsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
fun i =>
if i = δ!₁ j then
1
@ -351,13 +351,13 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
simp
/-- A point in the span of the first part of the basis as a charge. -/
def P (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := ∑ i, f i • basisAsCharges i
def P (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := ∑ i, f i • basisAsCharges i
/-- A point in the span of the second part of the basis as a charge. -/
def P! (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := ∑ i, f i • basis!AsCharges i
def P! (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := ∑ i, f i • basis!AsCharges i
/-- A point in the span of the basis as a charge. -/
def Pa (f : Fin n → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := P f + P! g
def Pa (f : Fin n → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := P f + P! g
lemma P_δ₁ (f : Fin n → ℚ) (j : Fin n) : P f (δ₁ j) = f j := by
rw [P, sum_of_charges]

34
HepLean/AnomalyCancellation/PureU1/Odd/LineInCubic.lean
View file

@ -36,11 +36,11 @@ open VectorLikeOddPlane
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
in the basis through that point is in the cubic. -/
def lineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
def LineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
∀ (g f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
accCube (2 * n + 1) (a • P g + b • P! f) = 0
lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S) :
lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) :
∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) (a b : ℚ) ,
3 * a * b * (a * accCubeTriLinSymm (P g) (P g) (P! f)
+ b * accCubeTriLinSymm (P! f) (P! f) (P g)) = 0 := by
@ -55,7 +55,7 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S)
ring
lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S) :
lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) :
∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
intro g f hS
@ -65,19 +65,19 @@ lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
def lineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
def LineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
∀ (M : (FamilyPermutations (2 * n + 1)).group ),
lineInCubic ((FamilyPermutations (2 * n + 1)).linSolRep M S)
LineInCubic ((FamilyPermutations (2 * n + 1)).linSolRep M S)
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
lemma lineInCubicPerm_self {S : (PureU1 (2 * n + 1)).LinSols} (hS : lineInCubicPerm S) :
lineInCubic S := hS 1
lemma lineInCubicPerm_self {S : (PureU1 (2 * n + 1)).LinSols} (hS : LineInCubicPerm S) :
LineInCubic S := hS 1
/-- If `lineInCubicPerm S` then `lineInCubicPerm (M S)` for all permutations `M`. -/
lemma lineInCubicPerm_permute {S : (PureU1 (2 * n + 1)).LinSols}
(hS : lineInCubicPerm S) (M' : (FamilyPermutations (2 * n + 1)).group) :
lineInCubicPerm ((FamilyPermutations (2 * n + 1)).linSolRep M' S) := by
rw [lineInCubicPerm]
(hS : LineInCubicPerm S) (M' : (FamilyPermutations (2 * n + 1)).group) :
LineInCubicPerm ((FamilyPermutations (2 * n + 1)).linSolRep M' S) := by
rw [LineInCubicPerm]
intro M
have ht : ((FamilyPermutations (2 * n + 1)).linSolRep M)
((FamilyPermutations (2 * n + 1)).linSolRep M' S)
@ -88,7 +88,7 @@ lemma lineInCubicPerm_permute {S : (PureU1 (2 * n + 1)).LinSols}
lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols}
(LIC : lineInCubicPerm S) :
(LIC : LineInCubicPerm S) :
∀ (j : Fin n.succ) (g f : Fin n.succ → ℚ) (_ : S.val = Pa g f) ,
(S.val (δ!₂ j) - S.val (δ!₁ j))
* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
@ -134,8 +134,8 @@ lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
ring
lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols}
(LIC : lineInCubicPerm S) :
lineInPlaneProp ((S.val (δ!₂ 0)), ((S.val (δ!₁ 0)), (S.val δ!₃))) := by
(LIC : LineInCubicPerm S) :
LineInPlaneProp ((S.val (δ!₂ 0)), ((S.val (δ!₁ 0)), (S.val δ!₃))) := by
obtain ⟨g, f, hfg⟩ := span_basis S
have h1 := lineInCubicPerm_swap LIC 0 g f hfg
rw [P_P_P!_accCube' g f hfg] at h1
@ -152,8 +152,8 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols}
linear_combination h1
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : lineInCubicPerm S) : lineInPlaneCond S := by
refine @Prop_three (2 * n.succ.succ + 1) lineInPlaneProp S (δ!₂ 0) (δ!₁ 0)
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
refine @Prop_three (2 * n.succ.succ + 1) LineInPlaneProp S (δ!₂ 0) (δ!₁ 0)
δ!₃ ?_ ?_ ?_ ?_
simp [Fin.ext_iff, δ!₂, δ!₁]
simp [Fin.ext_iff, δ!₂, δ!₃]
@ -162,11 +162,11 @@ lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : lineInCubicPerm S) : constAbs S.val :=
(LIC : LineInCubicPerm S) : ConstAbs S.val :=
linesInPlane_constAbs (lineInCubicPerm_last_perm LIC)
theorem lineInCubicPerm_zero {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : lineInCubicPerm S) : S = 0 :=
(LIC : LineInCubicPerm S) : S = 0 :=
ConstAbs.boundary_value_odd S (lineInCubicPerm_constAbs LIC)
end Odd

22
HepLean/AnomalyCancellation/PureU1/Odd/Parameterization.lean
View file

@ -77,13 +77,13 @@ lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case our parameterization above will be able to recover this point. -/
def genericCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
def GenericCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : genericCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
intro g f hS hC
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -93,13 +93,13 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case we will show that S is zero if it is true for all permutations. -/
def specialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
def SpecialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) = 0
lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : specialCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
intro g f hS
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -108,7 +108,7 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
exact hC'
lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
genericCase S ∨ specialCase S := by
GenericCase S ∨ SpecialCase S := by
obtain ⟨g, f, h⟩ := span_basis S.1.1
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
@ -117,7 +117,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : genericCase S) :
theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : GenericCase S) :
∃ g f a, S = parameterization g f a := by
obtain ⟨g, f, hS⟩ := span_basis S.1.1
use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
@ -135,8 +135,8 @@ theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : genericCase S) :
lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ + 1)).Sols}
(h : specialCase S) :
lineInCubic S.1.1 := by
(h : SpecialCase S) :
LineInCubic S.1.1 := by
intro g f hS a b
erw [TriLinearSymm.toCubic_add]
rw [HomogeneousCubic.map_smul, HomogeneousCubic.map_smul]
@ -155,15 +155,15 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ + 1)).Sols}
lemma special_case_lineInCubic_perm {S : (PureU1 (2 * n.succ + 1)).Sols}
(h : ∀ (M : (FamilyPermutations (2 * n.succ + 1)).group),
specialCase ((FamilyPermutations (2 * n.succ + 1)).solAction.toFun S M)) :
lineInCubicPerm S.1.1 := by
SpecialCase ((FamilyPermutations (2 * n.succ + 1)).solAction.toFun S M)) :
LineInCubicPerm S.1.1 := by
intro M
have hM := special_case_lineInCubic (h M)
exact hM
theorem special_case {S : (PureU1 (2 * n.succ.succ + 1)).Sols}
(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ + 1)).group),
specialCase ((FamilyPermutations (2 * n.succ.succ + 1)).solAction.toFun S M)) :
SpecialCase ((FamilyPermutations (2 * n.succ.succ + 1)).solAction.toFun S M)) :
S.1.1 = 0 := by
have ht := special_case_lineInCubic_perm h
exact lineInCubicPerm_zero ht