1. Physlib
  2. QFT
  3. PerturbationTheory
  4. FieldOpFreeAlgebra
  5. SuperCommute
Imports
/- Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joseph Tooby-Smith -/ module public import Physlib.QFT.PerturbationTheory.FieldOpFreeAlgebra.Grading public import Physlib.QFT.PerturbationTheory.FieldStatistics.ExchangeSign

Super Commute

@[expose] public section

The super commutator on the FieldOpFreeAlgebra.

@[inherit_doc superCommuteF] scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛF" => superCommuteF φs φs'

The super commutator of different types of elements

lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) : [ofCrAnListF φs, ofCrAnListF φs']ₛF = ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs) := 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') = ofCrAnListF (φs ++ φs') - (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp⊢ ofCrAnListF [φ] * ofCrAnListF [φ'] - (exchangeSign (ofList 𝓕.crAnStatistics [φ])) (ofList 𝓕.crAnStatistics [φ']) • (ofCrAnListF [φ'] * ofCrAnListF [φ]) = ofCrAnListF [φ] * ofCrAnListF [φ'] - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') • ofCrAnListF [φ'] * ofCrAnListF [φ] All goals completed! 🐙𝓕:FieldSpecificationφcas:List 𝓕.CrAnFieldOpφs:List 𝓕.FieldOp⊢ ofCrAnListF φcas * ofFieldOpListF φs - (exchangeSign (ofList 𝓕.crAnStatistics φcas)) (ofList 𝓕.fieldOpStatistic φs) • (ofFieldOpListF φs * ofCrAnListF φcas) = ofCrAnListF φcas * ofFieldOpListF φs - (exchangeSign (ofList 𝓕.crAnStatistics φcas)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * ofCrAnListF φcas All goals completed! 🐙All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφs:List 𝓕.FieldOp⊢ ofFieldOpF φ * ofFieldOpListF φs - (exchangeSign (ofList 𝓕.fieldOpStatistic [φ])) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * ofFieldOpF φ = ofFieldOpF φ * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * ofFieldOpF φ All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.FieldOpφ:𝓕.FieldOp⊢ ofFieldOpListF φs * ofFieldOpF φ - (exchangeSign (ofList 𝓕.fieldOpStatistic φs)) (ofList 𝓕.fieldOpStatistic [φ]) • ofFieldOpF φ * ofFieldOpListF φs = ofFieldOpListF φs * ofFieldOpF φ - (exchangeSign (ofList 𝓕.fieldOpStatistic φs)) (𝓕|>ₛφ) • ofFieldOpF φ * ofFieldOpListF φs All goals completed! 🐙lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) : [anPartF φ, crPartF φ']ₛF = anPartF φ * crPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ (superCommuteF (anPartF φ)) (crPartF φ') = anPartF φ * crPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * anPartF φ match φ, φ' with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ (superCommuteF (anPartF (FieldOp.inAsymp φ))) (crPartF x✝) = anPartF (FieldOp.inAsymp φ) * crPartF x✝ - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛx✝) • crPartF x✝ * anPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpx✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF x✝)) (crPartF (FieldOp.outAsymp φ)) = anPartF x✝ * crPartF (FieldOp.outAsymp φ) - (exchangeSign (𝓕|>ₛx✝)) (𝓕|>ₛFieldOp.outAsymp φ) • crPartF (FieldOp.outAsymp φ) * anPartF x✝ All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (anPartF (FieldOp.position φ))) (crPartF (FieldOp.position φ')) = anPartF (FieldOp.position φ) * crPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.position φ') • crPartF (FieldOp.position φ') * anPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (anPartF (FieldOp.outAsymp φ))) (crPartF (FieldOp.position φ')) = anPartF (FieldOp.outAsymp φ) * crPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛFieldOp.position φ') • crPartF (FieldOp.position φ') * anPartF (FieldOp.outAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.position φ))) (crPartF (FieldOp.inAsymp φ')) = anPartF (FieldOp.position φ) * crPartF (FieldOp.inAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.inAsymp φ') • crPartF (FieldOp.inAsymp φ') * anPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.outAsymp φ))) (crPartF (FieldOp.inAsymp φ')) = anPartF (FieldOp.outAsymp φ) * crPartF (FieldOp.inAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛFieldOp.inAsymp φ') • crPartF (FieldOp.inAsymp φ') * anPartF (FieldOp.outAsymp φ) All goals completed! 🐙lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) : [crPartF φ, anPartF φ']ₛF = crPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ (superCommuteF (crPartF φ)) (anPartF φ') = crPartF φ * anPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * crPartF φ match φ, φ' with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ (superCommuteF (crPartF (FieldOp.outAsymp φ))) (anPartF x✝) = crPartF (FieldOp.outAsymp φ) * anPartF x✝ - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛx✝) • anPartF x✝ * crPartF (FieldOp.outAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpx✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF x✝)) (anPartF (FieldOp.inAsymp φ)) = crPartF x✝ * anPartF (FieldOp.inAsymp φ) - (exchangeSign (𝓕|>ₛx✝)) (𝓕|>ₛFieldOp.inAsymp φ) • anPartF (FieldOp.inAsymp φ) * crPartF x✝ All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (crPartF (FieldOp.position φ))) (anPartF (FieldOp.position φ')) = crPartF (FieldOp.position φ) * anPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.position φ') • anPartF (FieldOp.position φ') * crPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.position φ))) (anPartF (FieldOp.outAsymp φ')) = crPartF (FieldOp.position φ) * anPartF (FieldOp.outAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.outAsymp φ') • anPartF (FieldOp.outAsymp φ') * crPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (crPartF (FieldOp.inAsymp φ))) (anPartF (FieldOp.position φ')) = crPartF (FieldOp.inAsymp φ) * anPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛFieldOp.position φ') • anPartF (FieldOp.position φ') * crPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.inAsymp φ))) (anPartF (FieldOp.outAsymp φ')) = crPartF (FieldOp.inAsymp φ) * anPartF (FieldOp.outAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛFieldOp.outAsymp φ') • anPartF (FieldOp.outAsymp φ') * crPartF (FieldOp.inAsymp φ) All goals completed! 🐙lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.FieldOp) : [crPartF φ, crPartF φ']ₛF = crPartF φ * crPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ (superCommuteF (crPartF φ)) (crPartF φ') = crPartF φ * crPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * crPartF φ match φ, φ' with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ (superCommuteF (crPartF (FieldOp.outAsymp φ))) (crPartF x✝) = crPartF (FieldOp.outAsymp φ) * crPartF x✝ - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛx✝) • crPartF x✝ * crPartF (FieldOp.outAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpx✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF x✝)) (crPartF (FieldOp.outAsymp φ)) = crPartF x✝ * crPartF (FieldOp.outAsymp φ) - (exchangeSign (𝓕|>ₛx✝)) (𝓕|>ₛFieldOp.outAsymp φ) • crPartF (FieldOp.outAsymp φ) * crPartF x✝ All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (crPartF (FieldOp.position φ))) (crPartF (FieldOp.position φ')) = crPartF (FieldOp.position φ) * crPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.position φ') • crPartF (FieldOp.position φ') * crPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.position φ))) (crPartF (FieldOp.inAsymp φ')) = crPartF (FieldOp.position φ) * crPartF (FieldOp.inAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.inAsymp φ') • crPartF (FieldOp.inAsymp φ') * crPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (crPartF (FieldOp.inAsymp φ))) (crPartF (FieldOp.position φ')) = crPartF (FieldOp.inAsymp φ) * crPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛFieldOp.position φ') • crPartF (FieldOp.position φ') * crPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.inAsymp φ))) (crPartF (FieldOp.inAsymp φ')) = crPartF (FieldOp.inAsymp φ) * crPartF (FieldOp.inAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛFieldOp.inAsymp φ') • crPartF (FieldOp.inAsymp φ') * crPartF (FieldOp.inAsymp φ) All goals completed! 🐙lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.FieldOp) : [anPartF φ, anPartF φ']ₛF = anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ (superCommuteF (anPartF φ)) (anPartF φ') = anPartF φ * anPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * anPartF φ match φ, φ' with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumx✝:𝓕.FieldOp⊢ (superCommuteF (anPartF (FieldOp.inAsymp φ))) (anPartF x✝) = anPartF (FieldOp.inAsymp φ) * anPartF x✝ - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (𝓕|>ₛx✝) • anPartF x✝ * anPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ':𝓕.FieldOpx✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF x✝)) (anPartF (FieldOp.inAsymp φ)) = anPartF x✝ * anPartF (FieldOp.inAsymp φ) - (exchangeSign (𝓕|>ₛx✝)) (𝓕|>ₛFieldOp.inAsymp φ) • anPartF (FieldOp.inAsymp φ) * anPartF x✝ All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (anPartF (FieldOp.position φ))) (anPartF (FieldOp.position φ')) = anPartF (FieldOp.position φ) * anPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.position φ') • anPartF (FieldOp.position φ') * anPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTimeφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.position φ))) (anPartF (FieldOp.outAsymp φ')) = anPartF (FieldOp.position φ) * anPartF (FieldOp.outAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (𝓕|>ₛFieldOp.outAsymp φ') • anPartF (FieldOp.outAsymp φ') * anPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (anPartF (FieldOp.outAsymp φ))) (anPartF (FieldOp.position φ')) = anPartF (FieldOp.outAsymp φ) * anPartF (FieldOp.position φ') - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛFieldOp.position φ') • anPartF (FieldOp.position φ') * anPartF (FieldOp.outAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφ'✝:𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentumφ':((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.outAsymp φ))) (anPartF (FieldOp.outAsymp φ')) = anPartF (FieldOp.outAsymp φ) * anPartF (FieldOp.outAsymp φ') - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (𝓕|>ₛFieldOp.outAsymp φ') • anPartF (FieldOp.outAsymp φ') * anPartF (FieldOp.outAsymp φ) All goals completed! 🐙lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) : [crPartF φ, ofFieldOpListF φs]ₛF = crPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * crPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφs:List 𝓕.FieldOp⊢ (superCommuteF (crPartF φ)) (ofFieldOpListF φs) = crPartF φ * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * crPartF φ match φ with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.inAsymp φ))) (ofFieldOpListF φs) = crPartF (FieldOp.inAsymp φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * crPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (crPartF (FieldOp.position φ))) (ofFieldOpListF φs) = crPartF (FieldOp.position φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * crPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (crPartF (FieldOp.outAsymp φ))) (ofFieldOpListF φs) = crPartF (FieldOp.outAsymp φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * crPartF (FieldOp.outAsymp φ) All goals completed! 🐙lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) : [anPartF φ, ofFieldOpListF φs]ₛF = anPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * anPartF φ := 𝓕:FieldSpecificationφ:𝓕.FieldOpφs:List 𝓕.FieldOp⊢ (superCommuteF (anPartF φ)) (ofFieldOpListF φs) = anPartF φ * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * anPartF φ match φ with 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.inAsymp φ))) (ofFieldOpListF φs) = anPartF (FieldOp.inAsymp φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.inAsymp φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * anPartF (FieldOp.inAsymp φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.PositionLabel f) × SpaceTime⊢ (superCommuteF (anPartF (FieldOp.position φ))) (ofFieldOpListF φs) = anPartF (FieldOp.position φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.position φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * anPartF (FieldOp.position φ) All goals completed! 🐙 𝓕:FieldSpecificationφ✝:𝓕.FieldOpφs:List 𝓕.FieldOpφ:((f : 𝓕.Field) × 𝓕.AsymptoticLabel f) × Momentum⊢ (superCommuteF (anPartF (FieldOp.outAsymp φ))) (ofFieldOpListF φs) = anPartF (FieldOp.outAsymp φ) * ofFieldOpListF φs - (exchangeSign (𝓕|>ₛFieldOp.outAsymp φ)) (ofList 𝓕.fieldOpStatistic φs) • ofFieldOpListF φs * anPartF (FieldOp.outAsymp φ) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * ofFieldOpListF [φ'] - (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic [φ']) • ofFieldOpListF [φ'] * crPartF φ = crPartF φ * ofFieldOpListF [φ'] - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • ofFieldOpListF [φ'] * crPartF φ All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ anPartF φ * ofFieldOpListF [φ'] - (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic [φ']) • ofFieldOpListF [φ'] * anPartF φ = anPartF φ * ofFieldOpListF [φ'] - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • ofFieldOpListF [φ'] * anPartF φ All goals completed! 🐙

Mul equal superCommuteF

Lemmas which rewrite a multiplication of two elements of the algebra as their commuted multiplication with a sign plus the super commutator.

𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF φs * ofCrAnListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF φs' * ofCrAnListF φs + (ofCrAnListF (φs ++ φs') - (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs)) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics [φ])) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF φs' * ofCrAnListF [φ] + (superCommuteF (ofCrAnListF [φ])) (ofCrAnListF φs') = (exchangeSign (𝓕.crAnStatistics φ)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF φs' * ofCrAnListF [φ] + (superCommuteF (ofCrAnListF [φ])) (ofCrAnListF φs') All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.FieldOpφs':List 𝓕.FieldOp⊢ ofFieldOpListF φs * ofFieldOpListF φs' = (exchangeSign (ofList 𝓕.fieldOpStatistic φs)) (ofList 𝓕.fieldOpStatistic φs') • ofFieldOpListF φs' * ofFieldOpListF φs + (ofFieldOpListF φs * ofFieldOpListF φs' - (exchangeSign (ofList 𝓕.fieldOpStatistic φs)) (ofList 𝓕.fieldOpStatistic φs') • ofFieldOpListF φs' * ofFieldOpListF φs) All goals completed! 🐙lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) : ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ + [ofFieldOpF φ, ofFieldOpListF φs']ₛF := 𝓕:FieldSpecificationφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ ofFieldOpF φ * ofFieldOpListF φs' = (exchangeSign (𝓕|>ₛφ)) (ofList 𝓕.fieldOpStatistic φs') • ofFieldOpListF φs' * ofFieldOpF φ + (superCommuteF (ofFieldOpF φ)) (ofFieldOpListF φs') All goals completed! 🐙lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) : ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs + [ofFieldOpListF φs, ofFieldOpF φ]ₛF := 𝓕:FieldSpecificationφs:List 𝓕.FieldOpφ:𝓕.FieldOp⊢ ofFieldOpListF φs * ofFieldOpF φ = (exchangeSign (ofList 𝓕.fieldOpStatistic φs)) (𝓕|>ₛφ) • ofFieldOpF φ * ofFieldOpListF φs + (superCommuteF (ofFieldOpListF φs)) (ofFieldOpF φ) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * anPartF φ' = (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * crPartF φ + (crPartF φ * anPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * crPartF φ) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ anPartF φ * crPartF φ' = (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * anPartF φ + (anPartF φ * crPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * anPartF φ) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ crPartF φ * crPartF φ' = (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * crPartF φ + (crPartF φ * crPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • crPartF φ' * crPartF φ) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφ':𝓕.FieldOp⊢ anPartF φ * anPartF φ' = (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * anPartF φ + (anPartF φ * anPartF φ' - (exchangeSign (𝓕|>ₛφ)) (𝓕|>ₛφ') • anPartF φ' * anPartF φ) All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ ofCrAnListF φs * ofFieldOpListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • ofFieldOpListF φs' * ofCrAnListF φs + (ofCrAnListF φs * ofFieldOpListF φs' - (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') • ofFieldOpListF φs' * ofCrAnListF φs) All goals completed! 🐙

Symmetry of the super commutator.

𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (φs ++ φs') - (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs) = -((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs)) + ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') * (exchangeSign (ofList 𝓕.crAnStatistics φs')) (ofList 𝓕.crAnStatistics φs)) • ofCrAnListF (φs ++ φs') conv_rhs => 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') * (exchangeSign (ofList 𝓕.crAnStatistics φs')) (ofList 𝓕.crAnStatistics φs)) • ofCrAnListF (φs ++ φs') 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| 1 • ofCrAnListF (φs ++ φs') 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ ofCrAnListF (φs ++ φs') - (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs) = -((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φs' ++ φs)) + ofCrAnListF (φs ++ φs') All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp⊢ ofCrAnOpF φ * ofCrAnOpF φ' - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') • (ofCrAnOpF φ' * ofCrAnOpF φ) = -((exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') • (ofCrAnOpF φ' * ofCrAnOpF φ)) + ((exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') * (exchangeSign (𝓕.crAnStatistics φ')) (𝓕.crAnStatistics φ)) • (ofCrAnOpF φ * ofCrAnOpF φ') conv_rhs => 𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp| ((exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') * (exchangeSign (𝓕.crAnStatistics φ')) (𝓕.crAnStatistics φ)) • (ofCrAnOpF φ * ofCrAnOpF φ') 𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp| 1 • (ofCrAnOpF φ * ofCrAnOpF φ') 𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp⊢ ofCrAnOpF φ * ofCrAnOpF φ' - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') • (ofCrAnOpF φ' * ofCrAnOpF φ) = -((exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics φ') • (ofCrAnOpF φ' * ofCrAnOpF φ)) + ofCrAnOpF φ * ofCrAnOpF φ' All goals completed! 🐙

Splitting the super commutator on lists into sums.

𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕.crAnStatistics φ * ofList 𝓕.crAnStatistics φs') • ofCrAnListF (φ :: (φs' ++ φs)) = ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics φs') * (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕.crAnStatistics φ)) • ofCrAnListF (φ :: (φs' ++ φs)) All goals completed! 🐙𝓕:FieldSpecificationφ:𝓕.FieldOpφs:List 𝓕.CrAnFieldOpφs':List 𝓕.FieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) ((𝓕|>ₛφ) * ofList 𝓕.fieldOpStatistic φs') • (ofFieldOpF φ * (ofFieldOpListF φs' * ofCrAnListF φs)) = ((exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic φs') * (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕|>ₛφ)) • (ofFieldOpF φ * (ofFieldOpListF φs' * ofCrAnListF φs)) All goals completed! 🐙

For a field specification 𝓕, and two lists φs = φ₀…φₙ and φs' of 𝓕.CrAnFieldOp the following super commutation relation holds:

[φs', φ₀…φₙ]ₛF = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛF * φᵢ₊₁ … φₙ

The proof of this relation is via induction on the length of φs.

𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF φ) * ofCrAnListF φs' + (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕.crAnStatistics φ) • ofCrAnOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑n) φs')) • ofCrAnListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF (φs'.get n)) * ofCrAnListF (List.drop (↑n + 1) φs') = ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑n) (φ :: φs'))) • ofCrAnListF (List.take (↑n) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get n)) * ofCrAnListF (List.drop (↑n + 1) (φ :: φs')) conv_rhs => 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp| (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑0) (φ :: φs'))) • ofCrAnListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get 0)) * ofCrAnListF (List.drop (↑0 + 1) (φ :: φs')) + ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑i.succ) (φ :: φs'))) • ofCrAnListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get i.succ)) * ofCrAnListF (List.drop (↑i.succ + 1) (φ :: φs')) 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF φ) * ofCrAnListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑0) (φ :: φs'))) • ofCrAnListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get 0)) * ofCrAnListF (List.drop (↑0 + 1) (φ :: φs'))𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕.crAnStatistics φ) • ofCrAnOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑n) φs')) • ofCrAnListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF (φs'.get n)) * ofCrAnListF (List.drop (↑n + 1) φs') = ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑i.succ) (φ :: φs'))) • ofCrAnListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get i.succ)) * ofCrAnListF (List.drop (↑i.succ + 1) (φ :: φs')) 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF φ) * ofCrAnListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑0) (φ :: φs'))) • ofCrAnListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get 0)) * ofCrAnListF (List.drop (↑0 + 1) (φ :: φs')) All goals completed! 🐙 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕.crAnStatistics φ) • ofCrAnOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑n) φs')) • ofCrAnListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF (φs'.get n)) * ofCrAnListF (List.drop (↑n + 1) φs') = ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.crAnStatistics (List.take (↑i.succ) (φ :: φs'))) • ofCrAnListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofCrAnOpF ((φ :: φs').get i.succ)) * ofCrAnListF (List.drop (↑i.succ + 1) (φ :: φs')) All goals completed! 🐙
𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofFieldOpF φ) * ofFieldOpListF φs' + (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕|>ₛφ) • ofFieldOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑n) φs')) • ofFieldOpListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF (φs'.get n)) * ofFieldOpListF (List.drop (↑n + 1) φs') = ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑n) (φ :: φs'))) • ofFieldOpListF (List.take (↑n) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get n)) * ofFieldOpListF (List.drop (↑n + 1) (φ :: φs')) conv_rhs => 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp| (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑0) (φ :: φs'))) • ofFieldOpListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get 0)) * ofFieldOpListF (List.drop (↑0 + 1) (φ :: φs')) + ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑i.succ) (φ :: φs'))) • ofFieldOpListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get i.succ)) * ofFieldOpListF (List.drop (↑i.succ + 1) (φ :: φs')) 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofFieldOpF φ) * ofFieldOpListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑0) (φ :: φs'))) • ofFieldOpListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get 0)) * ofFieldOpListF (List.drop (↑0 + 1) (φ :: φs'))𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕|>ₛφ) • ofFieldOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑n) φs')) • ofFieldOpListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF (φs'.get n)) * ofFieldOpListF (List.drop (↑n + 1) φs') = ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑i.succ) (φ :: φs'))) • ofFieldOpListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get i.succ)) * ofFieldOpListF (List.drop (↑i.succ + 1) (φ :: φs')) 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ (superCommuteF (ofCrAnListF φs)) (ofFieldOpF φ) * ofFieldOpListF φs' = (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑0) (φ :: φs'))) • ofFieldOpListF (List.take (↑0) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get 0)) * ofFieldOpListF (List.drop (↑0 + 1) (φ :: φs')) All goals completed! 🐙 𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφ:𝓕.FieldOpφs':List 𝓕.FieldOp⊢ (exchangeSign (ofList 𝓕.crAnStatistics φs)) (𝓕|>ₛφ) • ofFieldOpF φ * ∑ n, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑n) φs')) • ofFieldOpListF (List.take (↑n) φs') * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF (φs'.get n)) * ofFieldOpListF (List.drop (↑n + 1) φs') = ∑ i, (exchangeSign (ofList 𝓕.crAnStatistics φs)) (ofList 𝓕.fieldOpStatistic (List.take (↑i.succ) (φ :: φs'))) • ofFieldOpListF (List.take (↑i.succ) (φ :: φs')) * (superCommuteF (ofCrAnListF φs)) (ofFieldOpF ((φ :: φs').get i.succ)) * ofFieldOpListF (List.drop (↑i.succ + 1) (φ :: φs')) All goals completed! 🐙𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics (φs2 ++ φs3)) • ofCrAnListF (φs2 ++ φs3 ++ φs1) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics (φs3 ++ φs2)) • ofCrAnListF (φs3 ++ φs2 ++ φs1)) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics (φs1 ++ φs2)) • ofCrAnListF (φs1 ++ φs2 ++ φs3) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics (φs2 ++ φs1)) • ofCrAnListF (φs2 ++ φs1 ++ φs3)))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics (φs3 ++ φs1)) • ofCrAnListF (φs3 ++ φs1 ++ φs2) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics (φs1 ++ φs3)) • ofCrAnListF (φs1 ++ φs3 ++ φs2)))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOp⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:¬ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:¬ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:¬ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:¬ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2)))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs1 = bosonich2:¬ofList 𝓕.crAnStatistics φs2 = bosonich3:¬ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs3) • (-((exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3) • (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs2) • ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs2 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs2 ++ (φs1 ++ φs3))))) - (exchangeSign (ofList 𝓕.crAnStatistics φs1)) (ofList 𝓕.crAnStatistics φs2) • (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs3 * ofList 𝓕.crAnStatistics φs1) • ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (exchangeSign (ofList 𝓕.crAnStatistics φs3)) (ofList 𝓕.crAnStatistics φs1) • (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (exchangeSign (ofList 𝓕.crAnStatistics φs2)) (ofList 𝓕.crAnStatistics φs1 * ofList 𝓕.crAnStatistics φs3) • ofCrAnListF (φs1 ++ (φs3 ++ φs2))))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = fermionich2:ofList 𝓕.crAnStatistics φs2 = fermionich3:ofList 𝓕.crAnStatistics φs3 = fermionic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))) = ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2))) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1)))) 𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = fermionic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = fermionich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = bosonich2:ofList 𝓕.crAnStatistics φs2 = fermionich3:ofList 𝓕.crAnStatistics φs3 = fermionic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))) = ofCrAnListF (φs3 ++ (φs1 ++ φs2)) + ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) + ofCrAnListF (φs2 ++ (φs1 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) + ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) + ofCrAnListF (φs1 ++ (φs3 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = fermionich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = fermionich2:ofList 𝓕.crAnStatistics φs2 = bosonich3:ofList 𝓕.crAnStatistics φs3 = fermionic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) + ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) + ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3))) - (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) + ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) + ofCrAnListF (φs1 ++ (φs2 ++ φs3))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = fermionich2:ofList 𝓕.crAnStatistics φs2 = fermionich3:ofList 𝓕.crAnStatistics φs3 = bosonic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) + ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs1 ++ (φs3 ++ φs2)) + ofCrAnListF (φs3 ++ (φs2 ++ φs1))) = ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3))) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) + ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) + ofCrAnListF (φs3 ++ (φs1 ++ φs2))))𝓕:FieldSpecificationφs1:List 𝓕.CrAnFieldOpφs2:List 𝓕.CrAnFieldOpφs3:List 𝓕.CrAnFieldOph1:ofList 𝓕.crAnStatistics φs1 = fermionich2:ofList 𝓕.crAnStatistics φs2 = fermionich3:ofList 𝓕.crAnStatistics φs3 = fermionic⊢ ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - (ofCrAnListF (φs3 ++ (φs2 ++ φs1)) - ofCrAnListF (φs1 ++ (φs3 ++ φs2))) = ofCrAnListF (φs1 ++ (φs3 ++ φs2)) - ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - (ofCrAnListF (φs2 ++ (φs3 ++ φs1)) - ofCrAnListF (φs3 ++ (φs1 ++ φs2))) - (ofCrAnListF (φs3 ++ (φs1 ++ φs2)) - ofCrAnListF (φs1 ++ (φs2 ++ φs3)) - (ofCrAnListF (φs2 ++ (φs1 ++ φs3)) - ofCrAnListF (φs3 ++ (φs2 ++ φs1)))) All goals completed! 🐙

Interaction with grading.

All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hp1:p x hxhp2:p y hy⊢ (superCommuteF x) (ofCrAnListF φs) + (superCommuteF y) (ofCrAnListF φs) ∈ statisticSubmodule (f1 * f2) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hp1:p x hx⊢ p (c • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1}hp1:p x hx⊢ c • (superCommuteF x) (ofCrAnListF φs) ∈ statisticSubmodule (f1 * f2) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)φs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f1 → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) ∈ statisticSubmodule (f1 + f2)⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f1} All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hp1:p x hxhp2:p y hy⊢ (superCommuteF a) x + (superCommuteF a) y ∈ statisticSubmodule (f1 * f2) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hp1:p x hx⊢ p (c • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2}hp1:p x hx⊢ c • (superCommuteF a) x ∈ statisticSubmodule (f1 * f2) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraf1:FieldStatisticf2:FieldStatisticha:a ∈ statisticSubmodule f1hb:b ∈ statisticSubmodule f2p:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule f2 → Prop := fun a2 hx => (superCommuteF a) a2 ∈ statisticSubmodule (f1 + f2)⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = f2} All goals completed! 🐙lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) : [a, b]ₛF = a * b - b * a := 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonic⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p b hb 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonic⊢ p (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p✝ (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p a ha 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2φs':List 𝓕.CrAnFieldOphφs':ofList 𝓕.crAnStatistics φs' = bosonic⊢ p (ofCrAnListF φs') ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:(superCommuteF x) (ofCrAnListF φs) = x * ofCrAnListF φs - ofCrAnListF φs * xhp2:(superCommuteF y) (ofCrAnListF φs) = y * ofCrAnListF φs - ofCrAnListF φs * y⊢ x * ofCrAnListF φs - ofCrAnListF φs * x + (y * ofCrAnListF φs - ofCrAnListF φs * y) = x * ofCrAnListF φs + y * ofCrAnListF φs - (ofCrAnListF φs * x + ofCrAnListF φs * y) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:(superCommuteF a) x = a * x - x * ahp2:(superCommuteF a) y = a * y - y * a⊢ a * x - x * a + (a * y - y * a) = a * x + a * y - (x * a + y * a) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ac:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} All goals completed! 🐙lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) : [a, b]ₛF = a * b - b * a := 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionic⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p b hb 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionic⊢ p (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p✝ (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p a ha 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2φs':List 𝓕.CrAnFieldOphφs':ofList 𝓕.crAnStatistics φs' = bosonic⊢ p (ofCrAnListF φs') ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:(superCommuteF x) (ofCrAnListF φs) = x * ofCrAnListF φs - ofCrAnListF φs * xhp2:(superCommuteF y) (ofCrAnListF φs) = y * ofCrAnListF φs - ofCrAnListF φs * y⊢ x * ofCrAnListF φs - ofCrAnListF φs * x + (y * ofCrAnListF φs - ofCrAnListF φs * y) = x * ofCrAnListF φs + y * ofCrAnListF φs - (ofCrAnListF φs * x + ofCrAnListF φs * y) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:(superCommuteF a) x = a * x - x * ahp2:(superCommuteF a) y = a * y - y * a⊢ a * x - x * a + (a * y - y * a) = a * x + a * y - (x * a + y * a) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ac:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} All goals completed! 🐙lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) : [a, b]ₛF = a * b - b * a := 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonic⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ (superCommuteF a) b = a * b - b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p b hb 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonic⊢ p (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p✝ (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p a ha 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2φs':List 𝓕.CrAnFieldOphφs':ofList 𝓕.crAnStatistics φs' = fermionic⊢ p (ofCrAnListF φs') ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:(superCommuteF x) (ofCrAnListF φs) = x * ofCrAnListF φs - ofCrAnListF φs * xhp2:(superCommuteF y) (ofCrAnListF φs) = y * ofCrAnListF φs - ofCrAnListF φs * y⊢ x * ofCrAnListF φs - ofCrAnListF φs * x + (y * ofCrAnListF φs - ofCrAnListF φs * y) = x * ofCrAnListF φs + y * ofCrAnListF φs - (ofCrAnListF φs * x + ofCrAnListF φs * y) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs - ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:(superCommuteF a) x = a * x - x * ahp2:(superCommuteF a) y = a * y - y * a⊢ a * x - x * a + (a * y - y * a) = a * x + a * y - (x * a + y * a) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * ac:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule bosonicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule bosonic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 - a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrahb:b ∈ statisticSubmodule bosonic⊢ ↑(bosonicProjF a) * b - b * ↑(bosonicProjF a) + (↑(fermionicProjF a) * b - b * ↑(fermionicProjF a)) = (↑(bosonicProjF a) + ↑(fermionicProjF a)) * b - b * (↑(bosonicProjF a) + ↑(fermionicProjF a)) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrahb:b ∈ statisticSubmodule bosonic⊢ ↑(bosonicProjF a) * b - b * ↑(bosonicProjF a) + (↑(fermionicProjF a) * b - b * ↑(fermionicProjF a)) = ↑(bosonicProjF a) * b + ↑(fermionicProjF a) * b - (b * ↑(bosonicProjF a) + b * ↑(fermionicProjF a)) All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonic⊢ a * ↑(bosonicProjF b) - ↑(bosonicProjF b) * a + (a * ↑(fermionicProjF b) - ↑(fermionicProjF b) * a) = a * (↑(bosonicProjF b) + ↑(fermionicProjF b)) - (↑(bosonicProjF b) + ↑(fermionicProjF b)) * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonic⊢ a * ↑(bosonicProjF b) - ↑(bosonicProjF b) * a + (a * ↑(fermionicProjF b) - ↑(fermionicProjF b) * a) = a * ↑(bosonicProjF b) + a * ↑(fermionicProjF b) - (↑(bosonicProjF b) * a + ↑(fermionicProjF b) * a) All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebrahb:b ∈ statisticSubmodule bosonic⊢ a * b - b * a = -(b * a - a * b) All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule bosonic⊢ a * b - b * a = -(b * a - a * b) All goals completed! 🐙lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) : [a, b]ₛF = a * b + b * a := 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionic⊢ (superCommuteF a) b = a * b + b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ (superCommuteF a) b = a * b + b * a 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ p b hb 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * ax:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionic⊢ p (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ p✝ (ofCrAnListF φs) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ p a ha 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ p 0 ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (x : 𝓕.FieldOpFreeAlgebra) (h : x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebrahx:x ∈ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}⊢ p x ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2φs':List 𝓕.CrAnFieldOphφs':ofList 𝓕.crAnStatistics φs' = fermionic⊢ p (ofCrAnListF φs') ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:(superCommuteF x) (ofCrAnListF φs) = x * ofCrAnListF φs + ofCrAnListF φs * xhp2:(superCommuteF y) (ofCrAnListF φs) = y * ofCrAnListF φs + ofCrAnListF φs * y⊢ x * ofCrAnListF φs + ofCrAnListF φs * x + (y * ofCrAnListF φs + ofCrAnListF φs * y) = x * ofCrAnListF φs + y * ofCrAnListF φs + (ofCrAnListF φs * x + ofCrAnListF φs * y) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp✝:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * aφs:List 𝓕.CrAnFieldOphφs:ofList 𝓕.crAnStatistics φs = fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a2) (ofCrAnListF φs) = a2 * ofCrAnListF φs + ofCrAnListF φs * a2⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ p 0 ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (x y : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}) (hy : y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p y hy → p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hxhp2:p y hy⊢ p (x + y) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * ax:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:(superCommuteF a) x = a * x + x * ahp2:(superCommuteF a) y = a * y + y * a⊢ a * x + x * a + (a * y + y * a) = a * x + a * y + (x * a + y * a) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * ac:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hp1:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionicp:(a2 : 𝓕.FieldOpFreeAlgebra) → a2 ∈ statisticSubmodule fermionic → Prop := fun a2 hx => (superCommuteF a) a2 = a * a2 + a2 * a⊢ b ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebraha:a ∈ statisticSubmodule fermionichb:b ∈ statisticSubmodule fermionic⊢ a * b + b * a = b * a + a * b All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebrab:𝓕.FieldOpFreeAlgebra⊢ ↑(bosonicProjF a) * ↑(bosonicProjF b) - ↑(bosonicProjF b) * ↑(bosonicProjF a) + (↑(fermionicProjF a) * ↑(bosonicProjF b) - ↑(bosonicProjF b) * ↑(fermionicProjF a)) + (↑(bosonicProjF a) * ↑(fermionicProjF b) - ↑(fermionicProjF b) * ↑(bosonicProjF a) + (↑(fermionicProjF a) * ↑(fermionicProjF b) + ↑(fermionicProjF b) * ↑(fermionicProjF a))) = ↑(bosonicProjF a) * ↑(bosonicProjF b) - ↑(bosonicProjF b) * ↑(bosonicProjF a) + ↑(bosonicProjF a) * ↑(fermionicProjF b) - ↑(fermionicProjF b) * ↑(bosonicProjF a) + ↑(fermionicProjF a) * ↑(bosonicProjF b) - ↑(bosonicProjF b) * ↑(fermionicProjF a) + ↑(fermionicProjF a) * ↑(fermionicProjF b) + ↑(fermionicProjF b) * ↑(fermionicProjF a) All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs = bosonich2:¬ofList 𝓕.crAnStatistics φs' = bosonic⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') ∈ statisticSubmodule (fermionic + fermionic) exact superCommuteF_grade (ofCrAnListF_mem_statisticSubmodule_of _ _ (𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs = bosonich2:¬ofList 𝓕.crAnStatistics φs' = bosonic⊢ ofList 𝓕.crAnStatistics φs = fermionic All goals completed! 🐙)) (ofCrAnListF_mem_statisticSubmodule_of _ _ (𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph1:¬ofList 𝓕.crAnStatistics φs = bosonich2:¬ofList 𝓕.crAnStatistics φs' = bosonic⊢ ofList 𝓕.crAnStatistics φs' = fermionic All goals completed! 🐙))𝓕:FieldSpecificationφ:𝓕.CrAnFieldOpφ':𝓕.CrAnFieldOp⊢ (superCommuteF (ofCrAnListF [φ])) (ofCrAnListF [φ']) ∈ statisticSubmodule bosonic ∨ (superCommuteF (ofCrAnListF [φ])) (ofCrAnListF [φ']) ∈ statisticSubmodule fermionic All goals completed! 🐙𝓕:FieldSpecificationφ1:𝓕.CrAnFieldOpφ2:𝓕.CrAnFieldOpφ3:𝓕.CrAnFieldOphs:(superCommuteF (ofCrAnOpF φ2)) (ofCrAnOpF φ3) ∈ statisticSubmodule fermionich1:ofCrAnOpF φ1 ∈ statisticSubmodule fermionic⊢ (superCommuteF (ofCrAnOpF φ1)) ((superCommuteF (ofCrAnOpF φ2)) (ofCrAnOpF φ3)) ∈ statisticSubmodule (fermionic + fermionic) All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs)⊢ ∑ x_1, (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs) + ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) = ∑ x_1, ofCrAnListF (List.take (↑x_1) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑x_1]) + (superCommuteF y) (ofCrAnOpF φs[↑x_1])) * ofCrAnListF (List.drop (↑x_1 + 1) φs) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs)⊢ (fun x_1 => ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs) + ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) = fun x_1 => ofCrAnListF (List.take (↑x_1) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑x_1]) + (superCommuteF y) (ofCrAnOpF φs[↑x_1])) * ofCrAnListF (List.drop (↑x_1 + 1) φs) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs)n:Fin φs✝.length⊢ ofCrAnListF (List.take (↑n) φs) * (superCommuteF x) (ofCrAnOpF φs[↑n]) * ofCrAnListF (List.drop (↑n + 1) φs) + ofCrAnListF (List.take (↑n) φs) * (superCommuteF y) (ofCrAnOpF φs[↑n]) * ofCrAnListF (List.drop (↑n + 1) φs) = ofCrAnListF (List.take (↑n) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑n]) + (superCommuteF y) (ofCrAnOpF φs[↑n])) * ofCrAnListF (List.drop (↑n + 1) φs) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic}hpx:p x hx⊢ p (c • x) ⋯ All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule bosonicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule bosonic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = bosonic} All goals completed! 🐙𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x) φs)) • (ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs))⊢ ∑ x_1, ((exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) + (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑x_1]) + (superCommuteF y) (ofCrAnOpF φs[↑x_1])) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x) φs)) • (ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs))⊢ (fun x_1 => (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) + (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))) = fun x_1 => (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑x_1]) + (superCommuteF y) (ofCrAnOpF φs[↑x_1])) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)x:𝓕.FieldOpFreeAlgebray:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hy:y ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))hpy:(superCommuteF y) (ofCrAnListF φs) = ∑ x, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x) φs)) • (ofCrAnListF (List.take (↑x) φs) * (superCommuteF y) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs))n:Fin φs✝.length⊢ (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • (ofCrAnListF (List.take (↑n) φs) * (superCommuteF x) (ofCrAnOpF φs[↑n]) * ofCrAnListF (List.drop (↑n + 1) φs)) + (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • (ofCrAnListF (List.take (↑n) φs) * (superCommuteF y) (ofCrAnOpF φs[↑n]) * ofCrAnListF (List.drop (↑n + 1) φs)) = (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • (ofCrAnListF (List.take (↑n) φs) * ((superCommuteF x) (ofCrAnOpF φs[↑n]) + (superCommuteF y) (ofCrAnOpF φs[↑n])) * ofCrAnListF (List.drop (↑n + 1) φs)) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)⊢ ∀ (a : ℂ) (x : 𝓕.FieldOpFreeAlgebra) (hx : x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}), p x hx → p (a • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:p x hx⊢ p (c • x) ⋯ 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))⊢ ∑ x_1, c • (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • c • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)c:ℂx:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))⊢ (fun x_1 => c • (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))) = fun x_1 => (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • c • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs)) 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)c:ℂx✝:𝓕.FieldOpFreeAlgebrahx:x ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic}hpx:(superCommuteF x) (ofCrAnListF φs) = ∑ x_1, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x_1) φs)) • (ofCrAnListF (List.take (↑x_1) φs) * (superCommuteF x) (ofCrAnOpF φs[↑x_1]) * ofCrAnListF (List.drop (↑x_1 + 1) φs))x:Fin φs✝.length⊢ c • (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x) φs)) • (ofCrAnListF (List.take (↑x) φs) * (superCommuteF x✝) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs)) = (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑x) φs)) • c • (ofCrAnListF (List.take (↑x) φs) * (superCommuteF x✝) (ofCrAnOpF φs[↑x]) * ofCrAnListF (List.drop (↑x + 1) φs)) All goals completed! 🐙 𝓕:FieldSpecificationa:𝓕.FieldOpFreeAlgebraφs:List 𝓕.CrAnFieldOpha:a ∈ statisticSubmodule fermionicp:(a : 𝓕.FieldOpFreeAlgebra) → a ∈ statisticSubmodule fermionic → Prop := fun a ha => (superCommuteF a) (ofCrAnListF φs) = ∑ n, (exchangeSign fermionic) (ofList 𝓕.crAnStatistics (List.take (↑n) φs)) • ofCrAnListF (List.take (↑n) φs) * (superCommuteF a) (ofCrAnOpF (φs.get n)) * ofCrAnListF (List.drop (↑n + 1) φs)⊢ a ∈ Submodule.span ℂ {a | ∃ φs, a = ofCrAnListF φs ∧ ofList 𝓕.crAnStatistics φs = fermionic} All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph:(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') ∈ statisticSubmodule fermionich0:¬(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') = 0hn:ofList 𝓕.crAnStatistics φs = ofList 𝓕.crAnStatistics φs'hc:¬ofList 𝓕.crAnStatistics φs = bosonic⊢ (superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') ∈ statisticSubmodule (fermionic + fermionic) exact superCommuteF_grade (ofCrAnListF_mem_statisticSubmodule_of _ _ (𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph:(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') ∈ statisticSubmodule fermionich0:¬(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') = 0hn:ofList 𝓕.crAnStatistics φs = ofList 𝓕.crAnStatistics φs'hc:¬ofList 𝓕.crAnStatistics φs = bosonic⊢ ofList 𝓕.crAnStatistics φs = fermionic All goals completed! 🐙)) (ofCrAnListF_mem_statisticSubmodule_of _ _ (𝓕:FieldSpecificationφs:List 𝓕.CrAnFieldOpφs':List 𝓕.CrAnFieldOph:(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') ∈ statisticSubmodule fermionich0:¬(superCommuteF (ofCrAnListF φs)) (ofCrAnListF φs') = 0hn:ofList 𝓕.crAnStatistics φs = ofList 𝓕.crAnStatistics φs'hc:¬ofList 𝓕.crAnStatistics φs = bosonic⊢ ofList 𝓕.crAnStatistics φs' = fermionic All goals completed! 🐙))