refactor: Change case of type and props
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jstoobysmith
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58 changed files with 695 additions and 696 deletions
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@ -24,102 +24,102 @@ open BigOperators
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namespace PlaneSeven
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₀ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₀ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 0, 0 => 1
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| 0, 1 => - 1
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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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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₁ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₁ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 1, 0 => 1
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| 1, 1 => - 1
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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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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₂ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₂ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 2, 0 => 1
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| 2, 1 => - 1
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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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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₃ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₃ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 3, 0 => 1
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| 3, 1 => - 1
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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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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₄ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₄ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 4, 0 => 1
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| 4, 1 => - 1
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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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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₅ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₅ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 5, 0 => 1
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| 5, 1 => - 1
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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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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₆ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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def B₆ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 1, 2 => 1
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| 2, 2 => -1
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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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/-- The charge assignments forming a basis of the plane. -/
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@[simp]
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def B : Fin 7 → (SM 3).charges := fun i =>
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def B : Fin 7 → (SM 3).Charges := fun i =>
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match i with
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| 0 => B₀
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| 1 => B₁
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@ -129,7 +129,7 @@ def B : Fin 7 → (SM 3).charges := fun i =>
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| 5 => B₅
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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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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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rw [B₀_cubic]
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@ -138,7 +138,7 @@ lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ 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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rw [B₁_cubic]
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@ -146,7 +146,7 @@ lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
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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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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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rw [B₂_cubic]
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@ -154,7 +154,7 @@ lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
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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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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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rw [B₃_cubic]
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@ -162,7 +162,7 @@ lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
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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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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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rw [B₄_cubic]
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@ -170,7 +170,7 @@ lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
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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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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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rw [B₅_cubic]
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@ -178,7 +178,7 @@ lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
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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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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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rw [B₆_cubic]
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@ -186,7 +186,7 @@ lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
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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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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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fin_cases i
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exact B₀_Bi_cubic h S
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@ -279,7 +279,7 @@ theorem B_in_accCube (f : Fin 7 → ℚ) : accCube (∑ i, f i • B i) = 0 := b
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rw [Bi_Bj_Bk_cubic]
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simp
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lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).isSolution (∑ i, f i • B i) := by
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lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).IsSolution (∑ i, f i • B i) := by
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let X := chargeToAF (∑ i, f i • B i) (by
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rw [map_sum]
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apply Fintype.sum_eq_zero
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@ -337,8 +337,8 @@ theorem basis_linear_independent : LinearIndependent ℚ B := by
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end PlaneSeven
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theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).charges),
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LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).isSolution (∑ i, f i • B i) := by
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theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).Charges),
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LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).IsSolution (∑ i, f i • B i) := by
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use PlaneSeven.B
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apply And.intro
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exact PlaneSeven.basis_linear_independent
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