refactor: Move constructors
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6 changed files with 421 additions and 169 deletions
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@ -6,7 +6,6 @@ Authors: Joseph Tooby-Smith
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import Mathlib.Logic.Function.CompTypeclasses
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.Normed.Field.Basic
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import Mathlib.LinearAlgebra.Matrix.Trace
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/-!
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# Real Lorentz Tensors
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@ -423,141 +422,6 @@ def contr {X : Type} (T : Marked d X 2)
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/-!
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# Tensors from reals, vectors and matrices
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Note that that these definitions are not equivariant with respect to an
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action of the Lorentz group. They are provided for constructive purposes.
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-/
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/-- A 0-tensor from a real number. -/
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def ofReal (d : ℕ) (r : ℝ) : RealLorentzTensor d Empty where
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color := fun _ => Colors.up
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coord := fun _ => r
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/-- A marked 1-tensor with a single up index constructed from a vector.
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Note: This is not the same as rising indices on `ofVecDown`. -/
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def ofVecUp {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.up
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A marked 1-tensor with a single down index constructed from a vector.
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Note: This is not the same as lowering indices on `ofVecUp`. -/
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def ofVecDown {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.down
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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/-- A tensor with two up indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatUpUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A tensor with two down indices constructed from a matrix.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index up and the second index down.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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@[simps!]
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def ofMatUpDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.up
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- A marked 2-tensor with the first index down and the second index up.
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Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.down
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
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lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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simp only [Fin.isValue, contr, IndexValue, Equiv.cast_apply, trace, diag_apply, ofReal,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
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apply Finset.sum_congr rfl
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intro x _
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rfl
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/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
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lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-- Multiplying `ofVecUp` with `ofVecDown` gives the dot product. -/
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@[simp]
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lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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congrSet (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
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(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
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@[simp]
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lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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congrSet (Equiv.equivEmpty (Empty ⊕ Empty))
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(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext' ?_ ?_
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, congrSet_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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/-!
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## Rising and lowering indices
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Rising or lowering an index corresponds to changing the color of that index.
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@ -568,14 +432,6 @@ Rising or lowering an index corresponds to changing the color of that index.
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/-!
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## Action of the Lorentz group
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-/
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/-! TODO: Define the action of the Lorentz group on the sets of Tensors. -/
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/-!
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## Graphical species and Lorentz tensors
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-/
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