refactor: def of symmetric trilin function
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jstoobysmith
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e36c61b331
24 changed files with 279 additions and 246 deletions
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@ -31,7 +31,7 @@ def B₀ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₀_cubic (S T : (SM 3).charges) : cubeTriLin (B₀, S, T) =
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lemma B₀_cubic (S T : (SM 3).charges) : cubeTriLin B₀ S T =
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6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₀, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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@ -44,7 +44,7 @@ def B₁ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₁_cubic (S T : (SM 3).charges) : cubeTriLin (B₁, S, T) =
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lemma B₁_cubic (S T : (SM 3).charges) : cubeTriLin B₁ S T =
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3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₁, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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@ -57,7 +57,7 @@ def B₂ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₂_cubic (S T : (SM 3).charges) : cubeTriLin (B₂, S, T) =
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lemma B₂_cubic (S T : (SM 3).charges) : cubeTriLin B₂ S T =
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3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₂, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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@ -70,7 +70,7 @@ def B₃ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₃_cubic (S T : (SM 3).charges) : cubeTriLin (B₃, S, T) =
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lemma B₃_cubic (S T : (SM 3).charges) : cubeTriLin B₃ S T =
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2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₃, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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@ -84,7 +84,7 @@ def B₄ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₄_cubic (S T : (SM 3).charges) : cubeTriLin (B₄, S, T) =
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lemma B₄_cubic (S T : (SM 3).charges) : cubeTriLin B₄ S T =
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(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₄, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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@ -98,7 +98,7 @@ def B₅ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₅_cubic (S T : (SM 3).charges) : cubeTriLin (B₅, S, T) =
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lemma B₅_cubic (S T : (SM 3).charges) : cubeTriLin B₅ S T =
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(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₅, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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@ -112,7 +112,7 @@ def B₆ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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| _, _ => 0
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)
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lemma B₆_cubic (S T : (SM 3).charges) : cubeTriLin (B₆, S, T) =
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lemma B₆_cubic (S T : (SM 3).charges) : cubeTriLin B₆ S T =
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3* (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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@ -130,8 +130,8 @@ def B : Fin 7 → (SM 3).charges := fun i =>
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| 6 => B₆
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lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 0, B i, S) = 0 := by
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change cubeTriLin (B₀, B i, S) = 0
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cubeTriLin (B 0) (B i) S = 0 := by
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change cubeTriLin B₀ (B i) S = 0
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rw [B₀_cubic]
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fin_cases i <;>
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simp at hi <;>
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@ -139,55 +139,55 @@ lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
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lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 1, B i, S) = 0 := by
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change cubeTriLin (B₁, B i, S) = 0
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cubeTriLin (B 1) (B i) S = 0 := by
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change cubeTriLin B₁ (B i) S = 0
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rw [B₁_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 2, B i, S) = 0 := by
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change cubeTriLin (B₂, B i, S) = 0
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cubeTriLin (B 2) (B i) S = 0 := by
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change cubeTriLin B₂ (B i) S = 0
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rw [B₂_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 3, B i, S) = 0 := by
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change cubeTriLin (B₃, B i, S) = 0
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cubeTriLin (B 3) (B i) S = 0 := by
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change cubeTriLin (B₃) (B i) S = 0
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rw [B₃_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 4, B i, S) = 0 := by
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change cubeTriLin (B₄, B i, S) = 0
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cubeTriLin (B 4) (B i) S = 0 := by
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change cubeTriLin (B₄) (B i) S = 0
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rw [B₄_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 5, B i, S) = 0 := by
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change cubeTriLin (B₅, B i, S) = 0
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cubeTriLin (B 5) (B i) S = 0 := by
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change cubeTriLin (B₅) (B i) S = 0
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rw [B₅_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 6, B i, S) = 0 := by
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change cubeTriLin (B₆, B i, S) = 0
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cubeTriLin (B 6) (B i) S = 0 := by
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change cubeTriLin (B₆) (B i) S = 0
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rw [B₆_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma Bi_Bj_ne_cubic {i j : Fin 7} (h : i ≠ j) (S : (SM 3).charges) :
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cubeTriLin (B i, B j, S) = 0 := by
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cubeTriLin (B i) (B j) S = 0 := by
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fin_cases i
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exact B₀_Bi_cubic h S
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exact B₁_Bi_cubic h S
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@ -198,57 +198,57 @@ lemma Bi_Bj_ne_cubic {i j : Fin 7} (h : i ≠ j) (S : (SM 3).charges) :
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exact B₆_Bi_cubic h S
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lemma B₀_B₀_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 0, B 0, B i) = 0 := by
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change cubeTriLin (B₀, B₀, B i) = 0
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cubeTriLin (B 0) (B 0) (B i) = 0 := by
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change cubeTriLin (B₀) (B₀) (B i) = 0
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rw [B₀_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₁_B₁_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 1, B 1, B i) = 0 := by
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change cubeTriLin (B₁, B₁, B i) = 0
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cubeTriLin (B 1) (B 1) (B i) = 0 := by
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change cubeTriLin (B₁) (B₁) (B i) = 0
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rw [B₁_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₂_B₂_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 2, B 2, B i) = 0 := by
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change cubeTriLin (B₂, B₂, B i) = 0
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cubeTriLin (B 2) (B 2) (B i) = 0 := by
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change cubeTriLin (B₂) (B₂) (B i) = 0
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rw [B₂_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₃_B₃_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 3, B 3, B i) = 0 := by
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change cubeTriLin (B₃, B₃, B i) = 0
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cubeTriLin (B 3) (B 3) (B i) = 0 := by
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change cubeTriLin (B₃) (B₃) (B i) = 0
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rw [B₃_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₄_B₄_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 4, B 4, B i) = 0 := by
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change cubeTriLin (B₄, B₄, B i) = 0
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cubeTriLin (B 4) (B 4) (B i) = 0 := by
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change cubeTriLin (B₄) (B₄) (B i) = 0
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rw [B₄_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₅_B₅_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 5, B 5, B i) = 0 := by
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change cubeTriLin (B₅, B₅, B i) = 0
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cubeTriLin (B 5) (B 5) (B i) = 0 := by
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change cubeTriLin (B₅) (B₅) (B i) = 0
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rw [B₅_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₆_B₆_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 6, B 6, B i) = 0 := by
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change cubeTriLin (B₆, B₆, B i) = 0
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cubeTriLin (B 6) (B 6) (B i) = 0 := by
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change cubeTriLin (B₆) (B₆) (B i) = 0
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rw [B₆_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma Bi_Bi_Bj_cubic (i j : Fin 7) :
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cubeTriLin (B i, B i, B j) = 0 := by
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cubeTriLin (B i) (B i) (B j) = 0 := by
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fin_cases i
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exact B₀_B₀_Bi_cubic
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exact B₁_B₁_Bi_cubic
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@ -259,7 +259,7 @@ lemma Bi_Bi_Bj_cubic (i j : Fin 7) :
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exact B₆_B₆_Bi_cubic
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lemma Bi_Bj_Bk_cubic (i j k : Fin 7) :
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cubeTriLin (B i, B j, B k) = 0 := by
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cubeTriLin (B i) (B j) (B k) = 0 := by
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by_cases hij : i ≠ j
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exact Bi_Bj_ne_cubic hij (B k)
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simp at hij
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@ -267,7 +267,7 @@ lemma Bi_Bj_Bk_cubic (i j k : Fin 7) :
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exact Bi_Bi_Bj_cubic j k
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theorem B_in_accCube (f : Fin 7 → ℚ) : accCube (∑ i, f i • B i) = 0 := by
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change cubeTriLin _ = 0
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change cubeTriLin _ _ _ = 0
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rw [cubeTriLin.map_sum₁₂₃]
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apply Fintype.sum_eq_zero
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intro i
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