feat: make informal_definition and informal_lemma commands (#300)

* 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

HepLean/StandardModel/HiggsBoson/Basic.lean

@@ -171,11 +171,11 @@ def ofReal (a : ℝ) : HiggsField := (HiggsVec.ofReal a).toField
 /-- The higgs field which is all zero. -/
 def zero : HiggsField := ofReal 0
 
+/-- The zero Higgs field is the zero section of the Higgs bundle, i.e., the HiggsField `zero`
+defined by `ofReal 0` is the constant zero-section of the bundle `HiggsBundle`.
+-/
 informal_lemma zero_is_zero_section where
-  physics :≈ "The zero Higgs field is the zero section of the Higgs bundle."
-  math :≈ "The HiggsField `zero` defined by `ofReal 0`
-    is the constant zero-section of the bundle `HiggsBundle`."
-  deps :≈ [`StandardModel.HiggsField.zero]
+  deps := [`StandardModel.HiggsField.zero]
 
 end HiggsField
 

HepLean/StandardModel/HiggsBoson/GaugeAction.lean

@@ -215,25 +215,24 @@ theorem rotate_fst_real_snd_zero (φ : HiggsVec) :
   · simp only [Fin.mk_one, Fin.isValue, Pi.smul_apply, Function.comp_apply, cons_val_one, head_cons,
       tail_cons, smul_zero]
 
+/-- There exists a `g` in `GaugeGroupI` such that `rep g φ = φ'` iff `‖φ‖ = ‖φ'‖`. -/
 informal_lemma guage_orbit where
-  math :≈ "There exists a `g` in ``GaugeGroupI such that `rep g φ = φ'` if and only if
-    ‖φ‖ = ‖φ'‖."
-  deps :≈ [``rotate_fst_zero_snd_real]
+  deps := [``rotate_fst_zero_snd_real]
 
+/-- The Higgs boson breaks electroweak symmetry down to the electromagnetic force, i.e., the
+stablity group of the action of `rep` on `![0, Complex.ofReal ‖φ‖]`, for non-zero `‖φ‖`, is the
+`SU(3) × U(1)` subgroup of `gaugeGroup := SU(3) × SU(2) × U(1)` with the embedding given by
+`(g, e^{i θ}) ↦ (g, diag (e ^ {3 * i θ}, e ^ {- 3 * i θ}), e^{i θ})`.
+-/
 informal_lemma stability_group_single where
-  physics :≈ "The Higgs boson breaks electroweak symmetry down to the electromagnetic force."
-  math :≈ "The stablity group of the action of `rep` on `![0, Complex.ofReal ‖φ‖]`,
-    for non-zero `‖φ‖` is the `SU(3) x U(1)` subgroup of
-    `gaugeGroup := SU(3) x SU(2) x U(1)` with the embedding given by
-    `(g, e^{i θ}) ↦ (g, diag (e ^ {3 * i θ}, e ^ {- 3 * i θ}), e^{i θ})`."
-  deps :≈ [``StandardModel.HiggsVec, ``StandardModel.HiggsVec.rep]
+  deps := [``StandardModel.HiggsVec, ``StandardModel.HiggsVec.rep]
 
+/-- The subgroup of `gaugeGroup := SU(3) × SU(2) × U(1)` which preserves every `HiggsVec` by the
+action of `StandardModel.HiggsVec.rep` is given by `SU(3) × ℤ₆` where `ℤ₆` is the subgroup of
+`SU(2) × U(1)` with elements `(α^(-3) * I₂, α)` where `α` is a sixth root of unity.
+-/
 informal_lemma stability_group where
-  math :≈ "The subgroup of `gaugeGroup := SU(3) x SU(2) x U(1)` which preserves every `HiggsVec`
-    by the action of ``StandardModel.HiggsVec.rep is given by `SU(3) x ℤ₆` where ℤ₆
-    is the subgroup of `SU(2) x U(1)` with elements `(α^(-3) * I₂, α)` where
-    α is a sixth root of unity."
-  deps :≈ [``HiggsVec, ``rep]
+  deps := [``HiggsVec, ``rep]
 
 end HiggsVec
 
@@ -249,20 +248,21 @@ namespace HiggsField
 
 -/
 
+/-- The action of `gaugeTransformI` on `HiggsField` acting pointwise through `HiggsVec.rep`. -/
 informal_definition gaugeAction where
-  math :≈ "The action of ``gaugeTransformI on ``HiggsField acting pointwise through
-    ``HiggsVec.rep."
-  deps :≈ [``HiggsVec.rep, ``gaugeTransformI]
+  deps := [``HiggsVec.rep, ``gaugeTransformI]
 
+/-- There exists a `g` in `gaugeTransformI` such that `gaugeAction g φ = φ'` iff
+`φ(x)^† φ(x) = φ'(x)^† φ'(x)`.
+-/
 informal_lemma guage_orbit where
-  math :≈ "There exists a `g` in ``gaugeTransformI such that `gaugeAction g φ = φ'` if and only if
-    φ(x)^† φ(x) = φ'(x)^† φ'(x)."
-  deps :≈ [``gaugeAction]
+  deps := [``gaugeAction]
 
+/-- For every smooth map `f` from `SpaceTime` to `ℝ` such that `f` is positive semidefinite, there
+exists a Higgs field `φ` such that `f = φ^† φ`.
+-/
 informal_lemma gauge_orbit_surject where
-  math :≈ "For every smooth map f from ``SpaceTime to ℝ such that `f` is positive semidefinite,
-    there exists a Higgs field φ such that `f = φ^† φ`."
-  deps :≈ [``HiggsField, ``SpaceTime]
+  deps := [``HiggsField, ``SpaceTime]
 
 end HiggsField
 

HepLean/StandardModel/HiggsBoson/Potential.lean

@@ -323,12 +323,12 @@ lemma isBounded_of_𝓵_pos (h : 0 < P.𝓵) : P.IsBounded := by
   have h2' := h2 φ x
   linarith
 
+/-- When there is no quartic coupling, the potential is bounded iff the mass squared is
+non-positive, i.e., for `P : Potential` then `P.IsBounded` iff `P.μ2 ≤ 0`. That is to say
+`- P.μ2 * ‖φ‖_H^2 x` is bounded below ifff `P.μ2 ≤ 0`.-/
 informal_lemma isBounded_iff_of_𝓵_zero where
-  physics :≈ "When there is no quartic coupling, the potential is bounded iff the mass squared is
-    non-positive."
-  math :≈ "For `P : Potential` then P.IsBounded if and only if P.μ2 ≤ 0.
-    That is to say `- P.μ2 * ‖φ‖_H^2 x` is bounded below if and only if `P.μ2 ≤ 0`."
-  deps :≈ [`StandardModel.HiggsField.Potential.IsBounded, `StandardModel.HiggsField.Potential]
+  deps := [`StandardModel.HiggsField.Potential.IsBounded, `StandardModel.HiggsField.Potential]
+
 /-!
 
 ## Minimum and maximum