228 lines
7.7 KiB
Text
228 lines
7.7 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Logic.Equiv.Fin
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import Mathlib.Tactic.FinCases
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import Mathlib.Data.Finset.Card
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import Mathlib.CategoryTheory.IsomorphismClasses
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import Mathlib.Data.Fintype.Pi
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import LeanCopilot
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/-!
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# Feynman diagrams in Phi^4 theory
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The aim of this file is to start building up the theory of Feynman diagrams in the context of
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Phi^4 theory.
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## References
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- The approach taking to defining Feynman diagrams is based on:
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Theo Johnson-Freyd (https://mathoverflow.net/users/78/theo-johnson-freyd), How to count symmetry
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factors of Feynman diagrams? , URL (version: 2010-06-03): https://mathoverflow.net/q/26938
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## TODO
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- Develop a way to display Feynman diagrams.
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- Define the symmetry factor and compute for examples of Feynman diagrams.
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- Define the Feynman rules, and perform an example calculation.
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-/
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namespace PhiFour
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open CategoryTheory
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/-- Edges in Φ^4 internal `0`.
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Here `Type` is the category in which half-edges live. In general `Type` will be e.g.
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`Type × Type` with more fields. -/
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def edgeType : Fin 1 → Type
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| 0 => Fin 2
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/-- Vertices in Φ^4, can either be `external` corresponding to `0`, or a `phi^4` interaction
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corresponding to `1`. -/
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def vertexType : Fin 2 → Type
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| 0 => Fin 1
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| 1 => Fin 4
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/-- The type of vacuum Feynman diagrams for Phi-4 theory. -/
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structure FeynmanDiagram where
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/-- The type of half edges in the Feynman diagram. Sometimes also called `flags`. -/
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𝓱𝓔 : Type
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/-- The type of edges in the Feynman diagram. -/
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𝓔 : Type
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/-- Maps each edge to a label. Labels `0` if it is an external edge,
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and labels `1` if an internal edge. -/
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𝓔Label : 𝓔 → Fin 1
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/-- Maps half-edges to edges. -/
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𝓱𝓔To𝓔 : 𝓱𝓔 → 𝓔
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/-- Requires that the fiber of the map `𝓱𝓔To𝓔` at `x ∈ 𝓔` agrees with the corresponding
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`edgeType`. -/
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𝓔Fiber : ∀ x, CategoryTheory.IsIsomorphic (𝓱𝓔To𝓔 ⁻¹' {x} : Type) $ (edgeType ∘ 𝓔Label) x
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/-- The type of vertices in the Feynman diagram. -/
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𝓥 : Type
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/-- Maps each vertex to a label. In this case this map contains no information since
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there is only one type of vertex.. -/
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𝓥Label : 𝓥 → Fin 2
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/-- Maps half-edges to vertices. -/
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𝓱𝓔To𝓥 : 𝓱𝓔 → 𝓥
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/-- Requires that the fiber of the map `𝓱𝓔To𝓥` at `x ∈ 𝓥` agrees with the corresponding
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`vertexType`. -/
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𝓥Fiber : ∀ x, CategoryTheory.IsIsomorphic (𝓱𝓔To𝓥 ⁻¹' {x} : Type) $ (vertexType ∘ 𝓥Label) x
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namespace FeynmanDiagram
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variable (F : FeynmanDiagram)
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section Decidability
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/-!
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## Decidability
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The aim of this section is to make it easy to prove the `𝓔Fiber` and `𝓥Fiber` conditions by
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showing that they are decidable in cases when everything is finite and nice
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(which in practice is always).
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--/
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lemma fiber_cond_edge_iff_exists {𝓱𝓔 𝓔 : Type} (𝓱𝓔To𝓔 : 𝓱𝓔 → 𝓔) (𝓔Label : 𝓔 → Fin 1) (x : 𝓔) :
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(CategoryTheory.IsIsomorphic (𝓱𝓔To𝓔 ⁻¹' {x} : Type) $ (edgeType ∘ 𝓔Label) x)
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↔ ∃ (f : 𝓱𝓔To𝓔 ⁻¹' {x} → (edgeType ∘ 𝓔Label) x), Function.Bijective f :=
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Iff.intro
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(fun h ↦ match h with
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| ⟨f1, f2, h1, h2⟩ => ⟨f1, (isIso_iff_bijective f1).mp ⟨f2, h1, h2⟩⟩)
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(fun ⟨f1, hb⟩ ↦ match (isIso_iff_bijective f1).mpr hb with
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| ⟨f2, h1, h2⟩ => ⟨f1, f2, h1, h2⟩)
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lemma fiber_cond_vertex_iff_exists {𝓱𝓥 𝓥 : Type} (𝓱𝓥To𝓥 : 𝓱𝓥 → 𝓥) (𝓥Label : 𝓥 → Fin 2) (x : 𝓥) :
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(CategoryTheory.IsIsomorphic (𝓱𝓥To𝓥 ⁻¹' {x} : Type) $ (vertexType ∘ 𝓥Label) x)
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↔ ∃ (f : 𝓱𝓥To𝓥 ⁻¹' {x} → (vertexType ∘ 𝓥Label) x), Function.Bijective f :=
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Iff.intro
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(fun h ↦ match h with
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| ⟨f1, f2, h1, h2⟩ => ⟨f1, (isIso_iff_bijective f1).mp ⟨f2, h1, h2⟩⟩)
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(fun ⟨f1, hb⟩ ↦ match (isIso_iff_bijective f1).mpr hb with
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| ⟨f2, h1, h2⟩ => ⟨f1, f2, h1, h2⟩)
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instance {𝓱𝓔 𝓔 : Type} [Fintype 𝓱𝓔] [Fintype 𝓔] [DecidableEq 𝓔] (𝓱𝓔To𝓔 : 𝓱𝓔 → 𝓔) (x : 𝓔):
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DecidablePred (fun y => y ∈ 𝓱𝓔To𝓔 ⁻¹' {x}) := fun y =>
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match decEq (𝓱𝓔To𝓔 y) x with
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| isTrue h => isTrue (by simp [h])
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| isFalse h => isFalse (by simp [h])
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instance {𝓱𝓔 𝓔 : Type} [Fintype 𝓔] [DecidableEq 𝓔] [DecidableEq 𝓱𝓔] (𝓱𝓔To𝓔 : 𝓱𝓔 → 𝓔) (x : 𝓔) :
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DecidableEq $ (𝓱𝓔To𝓔 ⁻¹' {x}) := Subtype.instDecidableEq
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instance edgeTypeFintype (x : Fin 1) : Fintype (edgeType x) :=
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match x with
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| 0 => Fin.fintype 2
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instance edgeTypeDecidableEq (x : Fin 1) : DecidableEq (edgeType x) :=
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match x with
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| 0 => instDecidableEqFin 2
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instance vertexTypeFintype (x : Fin 2) : Fintype (vertexType x) :=
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match x with
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| 0 => Fin.fintype 1
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| 1 => Fin.fintype 4
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instance vertexTypeDecidableEq (x : Fin 2) : DecidableEq (vertexType x) :=
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match x with
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| 0 => instDecidableEqFin 1
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| 1 => instDecidableEqFin 4
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instance {𝓔 : Type} (𝓔Label : 𝓔 → Fin 1) (x : 𝓔) :
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DecidableEq ((edgeType ∘ 𝓔Label) x) := edgeTypeDecidableEq (𝓔Label x)
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instance {𝓔 : Type} (𝓔Label : 𝓔 → Fin 1) (x : 𝓔) :
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Fintype ((edgeType ∘ 𝓔Label) x) := edgeTypeFintype (𝓔Label x)
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instance {𝓥 : Type} (𝓥Label : 𝓥 → Fin 2) (x : 𝓥) :
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DecidableEq ((vertexType ∘ 𝓥Label) x) := vertexTypeDecidableEq (𝓥Label x)
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instance {𝓥 : Type} (𝓥Label : 𝓥 → Fin 2) (x : 𝓥) :
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Fintype ((vertexType ∘ 𝓥Label) x) := vertexTypeFintype (𝓥Label x)
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instance {𝓱𝓔 𝓔 : Type} [Fintype 𝓱𝓔] [DecidableEq 𝓱𝓔] [Fintype 𝓔] [DecidableEq 𝓔] (𝓱𝓔To𝓔 : 𝓱𝓔 → 𝓔) (𝓔Label : 𝓔 → Fin 1) (x : 𝓔) :
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Decidable (CategoryTheory.IsIsomorphic (𝓱𝓔To𝓔 ⁻¹' {x} : Type) $ (edgeType ∘ 𝓔Label) x) :=
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decidable_of_decidable_of_iff (fiber_cond_edge_iff_exists 𝓱𝓔To𝓔 𝓔Label x).symm
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instance {𝓱𝓥 𝓥 : Type} [Fintype 𝓱𝓥] [DecidableEq 𝓱𝓥] [Fintype 𝓥] [DecidableEq 𝓥] (𝓱𝓥To𝓥 : 𝓱𝓥 → 𝓥) (𝓥Label : 𝓥 → Fin 2) (x : 𝓥) :
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Decidable (CategoryTheory.IsIsomorphic (𝓱𝓥To𝓥 ⁻¹' {x} : Type) $ (vertexType ∘ 𝓥Label) x) :=
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decidable_of_decidable_of_iff (fiber_cond_vertex_iff_exists 𝓱𝓥To𝓥 𝓥Label x).symm
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end Decidability
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section examples
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/-!
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## Examples
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In this section we give examples of Feynman diagrams in Phi^4 theory.
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--/
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/-- The propagator
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- - - - - -
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-/
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def propagator : FeynmanDiagram where
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𝓱𝓔 := Fin 2
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𝓔 := Fin 1
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𝓔Label := ![0]
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𝓱𝓔To𝓔 := ![0, 0]
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𝓔Fiber := by decide
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𝓥 := Fin 2
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𝓥Label := ![0, 0]
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𝓱𝓔To𝓥 := ![0, 1]
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𝓥Fiber := by decide
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/-- The figure 8 Feynman diagram
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_
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/ \
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/ \
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\ /
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\ /
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\ /
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/ \
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/ \
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\ /
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\ __ / -/
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def figureEight : FeynmanDiagram where
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𝓱𝓔 := Fin 4
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𝓔 := Fin 2
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𝓔Label := ![0, 0]
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𝓱𝓔To𝓔 := ![0, 0, 1, 1]
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𝓔Fiber := by decide
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𝓥 := Fin 1
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𝓥Label := ![1]
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𝓱𝓔To𝓥 := ![0, 0, 0, 0]
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𝓥Fiber := by decide
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/-- The feynman diagram
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_ _ _ _ _
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/ \
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/ \
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- - - - - - - - - - - -
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\ /
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\ _ _ _ _ _/
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-/
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def propagtor1 : FeynmanDiagram where
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𝓱𝓔 := Fin 10
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𝓔 := Fin 5
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𝓔Label := ![0, 0, 0, 0, 0]
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𝓱𝓔To𝓔 := ![0, 0, 1, 1, 2, 2, 3, 3, 4, 4]
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𝓔Fiber := by decide
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𝓥 := Fin 4
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𝓥Label := ![0, 1, 1, 0]
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𝓱𝓔To𝓥 := ![0, 1, 1, 2, 1, 2, 1, 2, 2, 3]
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𝓥Fiber := by decide
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end examples
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end FeynmanDiagram
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end PhiFour
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