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feat: make informal_definition and informal_lemma commands (#300)

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* make informal_definition and informal_lemma commands
* drop the fields "math", "physics", and "proof" from InformalDefinition/InformalLemma and use docstrings instead
* render informal docstring in dependency graph
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HepLean/BeyondTheStandardModel/PatiSalam/Basic.lean
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@ -22,54 +22,62 @@ namespace PatiSalam
-/
/-- The gauge group of the Pati-Salam model (unquotiented by ℤ₂), i.e., `SU(4) × SU(2) × SU(2)`. -/
informal_definition GaugeGroupI where
math :≈ "The group `SU(4) x SU(2) x SU(2)`."
physics :≈ "The gauge group of the Pati-Salam model (unquotiented by ℤ₂)."
deps := []
/-- The homomorphism of the Standard Model gauge group into the Pati-Salam gauge group, i.e., the
group homomorphism `SU(3) × SU(2) × U(1) → SU(4) × SU(2) × SU(2)` taking `(h, g, α)` to
`(blockdiag (α h, α ^ (-3)), g, diag (α ^ 3, α ^(-3))`.
See page 54 of https://math.ucr.edu/home/baez/guts.pdf
-/
informal_definition inclSM where
physics :≈ "The homomorphism of the Standard Model gauge group into the Pati-Salam gauge group."
math :≈ "The group homomorphism `SU(3) x SU(2) x U(1) -> SU(4) x SU(2) x SU(2)`
taking (h, g, α) to (blockdiag (α h, α ^ (-3)), g, diag(α ^ (3), α ^(-3)))."
ref :≈ "Page 54 of https://math.ucr.edu/home/baez/guts.pdf"
deps :≈ [``GaugeGroupI, ``StandardModel.GaugeGroupI]
deps := [``GaugeGroupI, ``StandardModel.GaugeGroupI]
/-- The kernel of the map `inclSM` is equal to the subgroup `StandardModel.gaugeGroupℤ₃SubGroup`.
See footnote 10 of https://arxiv.org/pdf/2201.07245
-/
informal_lemma inclSM_ker where
math :≈ "The kernel of the map ``inclSM is equal to the subgroup
``StandardModel.gaugeGroupℤ₃SubGroup."
ref :≈ "Footnote 10 of https://arxiv.org/pdf/2201.07245"
deps :≈ [``inclSM, ``StandardModel.gaugeGroupℤ₃SubGroup]
deps := [``inclSM, ``StandardModel.gaugeGroupℤ₃SubGroup]
/-- The group embedding from `StandardModel.GaugeGroupℤ₃` to `GaugeGroupI` induced by `inclSM` by
quotienting by the kernal `inclSM_ker`.
-/
informal_definition embedSMℤ₃ where
math :≈ "The group embedding from ``StandardModel.GaugeGroupℤ₃ to ``GaugeGroupI
induced by ``inclSM by quotienting by the kernal ``inclSM_ker."
deps :≈ [``inclSM, ``StandardModel.GaugeGroupℤ₃, ``GaugeGroupI, ``inclSM_ker]
deps := [``inclSM, ``StandardModel.GaugeGroupℤ₃, ``GaugeGroupI, ``inclSM_ker]
/-- The equivalence between `GaugeGroupI` and `Spin(6) × Spin(4)`. -/
informal_definition gaugeGroupISpinEquiv where
math :≈ "The equivalence between `GaugeGroupI` and `Spin(6) × Spin(4)`."
deps :≈ [``GaugeGroupI]
deps := [``GaugeGroupI]
/-- The ℤ₂-subgroup of the un-quotiented gauge group which acts trivially on all particles in the
standard model, i.e., the ℤ₂-subgroup of `GaugeGroupI` with the non-trivial element `(-1, -1, -1)`.
See https://math.ucr.edu/home/baez/guts.pdf
-/
informal_definition gaugeGroupℤ₂SubGroup where
physics :≈ "The ℤ₂-subgroup of the un-quotiented gauge group which acts trivially on
all particles in the standard model."
math :≈ "The ℤ₂-subgroup of ``GaugeGroupI with the non-trivial element (-1, -1, -1)."
ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
deps :≈ [``GaugeGroupI]
deps := [``GaugeGroupI]
/-- The gauge group of the Pati-Salam model with a ℤ₂ quotient, i.e., the quotient of `GaugeGroupI`
by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`.
See https://math.ucr.edu/home/baez/guts.pdf
-/
informal_definition GaugeGroupℤ₂ where
physics :≈ "The gauge group of the Pati-Salam model with a ℤ₂ quotient."
math :≈ "The quotient of ``GaugeGroupI by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`."
ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
deps :≈ [``GaugeGroupI, ``gaugeGroupℤ₂SubGroup]
deps := [``GaugeGroupI, ``gaugeGroupℤ₂SubGroup]
/-- The group `StandardModel.gaugeGroupℤ₆SubGroup` under the homomorphism `embedSM` factors through
the subgroup `gaugeGroupℤ₂SubGroup`.
-/
informal_lemma sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup where
math :≈ "The group ``StandardModel.gaugeGroupℤ₆SubGroup under the homomorphism ``embedSM factors
through the subgroup ``gaugeGroupℤ₂SubGroup."
deps :≈ [``inclSM, ``StandardModel.gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup]
deps := [``inclSM, ``StandardModel.gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup]
/-- The group homomorphism from `StandardModel.GaugeGroupℤ₆` to `GaugeGroupℤ₂` induced by `embedSM`.
-/
informal_definition embedSMℤ₆Toℤ₂ where
math :≈ "The group homomorphism from ``StandardModel.GaugeGroupℤ₆ to ``GaugeGroupℤ₂
induced by ``embedSM."
deps :≈ [``inclSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
deps := [``inclSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
``sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup]
end PatiSalam