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refactor: Last batch of multi-goal proofs

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jstoobysmith 2024-08-21 06:40:58 -04:00
parent b9479c904d
commit c0499483a8
43 changed files with 910 additions and 955 deletions
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61
HepLean/FeynmanDiagrams/Basic.lean
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@ -217,14 +217,14 @@ def External {P : PreFeynmanRule} (v : P.VertexLabel) : Prop :=
lemma external_iff_exists_bijection {P : PreFeynmanRule} (v : P.VertexLabel) :
External v ↔ ∃ (κ : (P.vertexLabelMap v).left → Fin 1), Function.Bijective κ := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
obtain ⟨κ, κm1, h1, h2⟩ := h
let f : (P.vertexLabelMap v).left ≅ (Fin 1) := ⟨κ, κm1, h1, h2⟩
exact ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f⟩
obtain ⟨κ, h1⟩ := h
let f : (P.vertexLabelMap v).left ⟶ (Fin 1) := κ
have ft : IsIso f := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := ft
exact ⟨f, fm, hf1, hf2⟩
· obtain ⟨κ, κm1, h1, h2⟩ := h
let f : (P.vertexLabelMap v).left ≅ (Fin 1) := ⟨κ, κm1, h1, h2⟩
exact ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f⟩
· obtain ⟨κ, h1⟩ := h
let f : (P.vertexLabelMap v).left ⟶ (Fin 1) := κ
have ft : IsIso f := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := ft
exact ⟨f, fm, hf1, hf2⟩
instance externalDecidable [IsFinitePreFeynmanRule P] (v : P.VertexLabel) :
Decidable (External v) :=
@ -258,14 +258,14 @@ lemma diagramVertexProp_iff {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel ×
Function.Bijective κ
∧ ((P.preimageVertex v).obj F).hom ∘ κ = (P.vertexLabelMap (f v)).hom := by
refine Iff.intro (fun h v => ?_) (fun h v => ?_)
obtain ⟨κ, κm1, h1, h2⟩ := h v
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
refine ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f, κ.w⟩
obtain ⟨κ, h1, h2⟩ := h v
let f : (P.vertexLabelMap (f v)) ⟶ ((P.preimageVertex v).obj F) := Over.homMk κ h2
have ft : IsIso ((Over.forget P.HalfEdgeLabel).map f) := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := (isIso_of_reflects_iso _ (Over.forget P.HalfEdgeLabel) : IsIso f)
exact ⟨f, fm, hf1, hf2⟩
· obtain ⟨κ, κm1, h1, h2⟩ := h v
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
exact ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f, κ.w⟩
· obtain ⟨κ, h1, h2⟩ := h v
let f : (P.vertexLabelMap (f v)) ⟶ ((P.preimageVertex v).obj F) := Over.homMk κ h2
have ft : IsIso ((Over.forget P.HalfEdgeLabel).map f) := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := (isIso_of_reflects_iso _ (Over.forget P.HalfEdgeLabel) : IsIso f)
exact ⟨f, fm, hf1, hf2⟩
lemma diagramEdgeProp_iff {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
(f : 𝓔 ⟶ P.EdgeLabel) : P.DiagramEdgeProp F f ↔
@ -273,14 +273,14 @@ lemma diagramEdgeProp_iff {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔
Function.Bijective κ
∧ ((P.preimageEdge v).obj F).hom ∘ κ = (P.edgeLabelMap (f v)).hom := by
refine Iff.intro (fun h v => ?_) (fun h v => ?_)
obtain ⟨κ, κm1, h1, h2⟩ := h v
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
refine ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f, κ.w⟩
obtain ⟨κ, h1, h2⟩ := h v
let f : (P.edgeLabelMap (f v)) ⟶ ((P.preimageEdge v).obj F) := Over.homMk κ h2
have ft : IsIso ((Over.forget P.HalfEdgeLabel).map f) := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := (isIso_of_reflects_iso _ (Over.forget P.HalfEdgeLabel) : IsIso f)
exact ⟨f, fm, hf1, hf2⟩
· obtain ⟨κ, κm1, h1, h2⟩ := h v
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
exact ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f, κ.w⟩
· obtain ⟨κ, h1, h2⟩ := h v
let f : (P.edgeLabelMap (f v)) ⟶ ((P.preimageEdge v).obj F) := Over.homMk κ h2
have ft : IsIso ((Over.forget P.HalfEdgeLabel).map f) := (isIso_iff_bijective κ).mpr h1
obtain ⟨fm, hf1, hf2⟩ := (isIso_of_reflects_iso _ (Over.forget P.HalfEdgeLabel) : IsIso f)
exact ⟨f, fm, hf1, hf2⟩
instance diagramVertexPropDecidable
[IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type} [Fintype 𝓥] [DecidableEq 𝓥]
@ -547,13 +547,12 @@ lemma cond_satisfied {F G : FeynmanDiagram P} (f : Hom F G) :
lemma cond_symm {F G : FeynmanDiagram P} (𝓔 : F.𝓔 ≃ G.𝓔) (𝓥 : F.𝓥 ≃ G.𝓥) (𝓱𝓔 : F.𝓱𝓔 ≃ G.𝓱𝓔)
(h : Cond 𝓔 𝓥 𝓱𝓔) : Cond 𝓔.symm 𝓥.symm 𝓱𝓔.symm := by
refine ⟨?_, ?_, ?_⟩
simpa using fun x => (h.1 (𝓔.symm x)).symm
simpa using fun x => (h.2.1 (𝓥.symm x)).symm
intro x
have h1 := h.2.2 (𝓱𝓔.symm x)
simp at h1
exact (edgeVertexEquiv 𝓔 𝓥).apply_eq_iff_eq_symm_apply.mp (h1).symm
refine ⟨?_, ?_, fun x => ?_⟩
· simpa using fun x => (h.1 (𝓔.symm x)).symm
· simpa using fun x => (h.2.1 (𝓥.symm x)).symm
· have h1 := h.2.2 (𝓱𝓔.symm x)
simp only [Functor.const_obj_obj, Equiv.apply_symm_apply] at h1
exact (edgeVertexEquiv 𝓔 𝓥).apply_eq_iff_eq_symm_apply.mp (h1).symm
instance {F G : FeynmanDiagram P} [IsFiniteDiagram F] [IsFinitePreFeynmanRule P]
(𝓔 : F.𝓔 → G.𝓔) : Decidable (∀ x, G.𝓔𝓞.hom (𝓔 x) = F.𝓔𝓞.hom x) :=