feat(LinearAlgebra/CliffordAlgebra): construction from a basis #10660
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eric-wieser
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| lemma single_mul_single_helper (i : ι) : | ||
| (mul_helper (Model.Index.single i) (Model.Index.single i)).2 = 0 := by |
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| (mul_helper (Model.Index.single i) (Model.Index.single i)).2 = 0 := by | |
| (mul_helper (Model.Index.single i) (Model.Index.single i)).2 = 0 := by |
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I suspect I will need some of these for #10660
| Index.reverseRecOn nil concat (.concat l i h) = (concat l i h <| Index.reverseRecOn nil concat l) := by | ||
| simp [Index.reverseRecOn, Index.concat] | ||
| rw [List.reverseRecOn] | ||
| simp |
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| simp | |
| simp? |
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| open List in | ||
| def Index.mulOfLt (i : ι) (l : Model.Index ι) (h : ∀ j ∈ l.1, i < j) : | ||
| (Finsupp.supported _ R {i' | i'.1.erase i <+ l}).comap |
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| (Finsupp.supported _ R {i' | i'.1.erase i <+ l}).comap | |
| (Finsupp.supported _ R {i' | i'.1.erase i <+ l}).comap |
| rw [LinearMap.map_mul_iff] | ||
| ext xi yi | ||
| dsimp [LinearMap.toSpanSingleton] | ||
| simp |
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| simp | |
| simp? |
| toFun f := liftToFun B f.val f.property | ||
| invFun F := ⟨F ∘ₗ ofFreeVS B, by | ||
| intro m | ||
| simp |
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| simp | |
| simp? |
kbuzzard
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I suspect I will need some of these for #10660
…to eric-wieser/clifford.basis
| | cons x xs => | ||
| simp only [List.cons_append] | ||
| congr | ||
| dsimp only [←List.cons_append] |
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| dsimp only [←List.cons_append] | |
| dsimp only [← List.cons_append] |
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| /-- (xX)Y = x(XY)-/ | ||
| lemma Index.cons_mul (l₁ l₂ : Model.Index ι) (a : ι) (h) : | ||
| (l₁.cons a h).mul B l₂ = |
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| (l₁.cons a h).mul B l₂ = | |
| (l₁.cons a h).mul B l₂ = |
| | nil => simp | ||
| | cons is i h ih => | ||
| rw [Index.cons_mul] | ||
| simp |
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| simp | |
| simp? |
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| /-- (Xx)Y = X(xY)-/ | ||
| lemma Index.concat_mul (l₁ l₂ : Model.Index ι) (a : ι) (h) : | ||
| (l₁.concat a h).mul B l₂ = |
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| (l₁.concat a h).mul B l₂ = | |
| (l₁.concat a h).mul B l₂ = |
| (l₁.concat a h).mul B l₂ = | ||
| (Model.ofFinsupp.symm (Index.singleMul B a l₂ : Model ι B)).sum fun ind val => val • l₁.mul B ind := by | ||
| rw [mul] | ||
| simp |
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| simp | |
| simp? |
| lemma Index.single_mul_single_same (i : ι) : | ||
| Model.Index.mul B (.single i) (.single i) = B i i • (1 : Model ι B) := by | ||
| rw [Model.Index.mul, single_eq_cons, recOn_cons, recOn_nil, ofFinsupp_symm_single, | ||
| Finsupp.sum_single_index, one_smul, ←single_eq_cons, singleMul_single_same] |
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| Finsupp.sum_single_index, one_smul, ←single_eq_cons, singleMul_single_same] | |
| Finsupp.sum_single_index, one_smul, ← single_eq_cons, singleMul_single_same] |
| erw [Model.Index.singleMul, Index.recOn] | ||
| rw [ltByCases, dif_pos hij, dif_neg hij.not_lt] | ||
| dsimp | ||
| rw [←Model.Index.singleMul] |
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| rw [←Model.Index.singleMul] | |
| rw [← Model.Index.singleMul] |
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dagurtomas
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I suspect I will need some of these for #10660
utensil
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Mar 26, 2024
I suspect I will need some of these for #10660
Louddy
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I suspect I will need some of these for #10660
callesonne
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I suspect I will need some of these for #10660
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This is adapted from https://github.com/eric-wieser/lftcm2023-clifford_algebra, which only worked for the special case of
Q = 0.