Try to add an inductive definition of a multivector

Author: eric-wieser

src/geometric_algebra/nursery/chisolm.lean

@@ -34,10 +34,133 @@ namespace geometric_algebra
 
 section
 
+
 parameters
-{G₀ : Type*} [field G₀]
-{G₁ : Type*} [add_comm_group G₁] [vector_space G₀ G₁]
-{G : Type*} [ring G] [algebra G₀ G] [geometric_algebra G₀ G₁ G]
+{G₀ : Type u} [field G₀]
+{G₁ : Type u} [add_comm_group G₁] [vector_space G₀ G₁]
+{G : Type u} [ring G] [algebra G₀ G] [geometric_algebra G₀ G₁ G]
+
+
+-- define a blade representation
+-- TODO: how to describe the fact that the vector argument to `graded` must be orthogonal?
+inductive blade : nat → Type u
+| scalar : G₀ → blade 0
+| vector : G₁ → blade 1
+| graded {n : ℕ} : G₁ → blade (n + 1) → blade (n + 2)
+namespace blade
+  instance g0_coe : has_coe G₀ (blade 0) := { coe := blade.scalar }
+  instance g1_coe : has_coe G₁ (blade 1) := { coe := blade.vector }
+
+  -- define zero and one on the blades
+  def zero : Π (n : ℕ), blade n
+  | 0 := blade.scalar (0 : G₀)
+  | 1 := blade.vector (0 : G₁)
+  | (nat.succ $ nat.succ n) := blade.graded (0 : G₁) (zero $ nat.succ n)
+  instance has_zero {n : ℕ} : has_zero (blade n) := { zero := zero n }
+  instance r0_has_one : has_one (blade 0) := { one := (1 : G₀) }
+
+  -- define add on the blades
+  def r0_add (a : blade 0) (b : blade 0) : blade 0 := begin
+    cases a with a',
+    cases b with b', 
+    exact blade.scalar (a' + b')
+  end
+  instance r0_has_add : has_add (blade 0) := { add := r0_add }
+  def r1_add (a : blade 1) (b : blade 1) : blade 1 := begin
+    cases a,
+    cases b,
+    exact blade.vector (a_1 + b_1)
+  end
+  instance r1_has_add : has_add (blade 1) := { add := r1_add }
+end blade
+
+-- define a sum-of-blades representation
+inductive hom_mv : nat → Type u
+| scalar : blade 0 -> hom_mv 0
+| vector : blade 1 -> hom_mv 1
+| graded {n : ℕ} : list (blade $ nat.succ $ nat.succ n) → (hom_mv $ nat.succ $ nat.succ n)
+
+namespace hom_mv
+  def coe : Π {n : ℕ}, (blade n) → hom_mv n
+  | 0 := hom_mv.scalar
+  | 1 := hom_mv.vector
+  | (r + 2) := λ b, hom_mv.graded [b]
+  instance has_blade_coe {r : ℕ} : has_coe (blade r) (hom_mv r) := { coe := coe }
+  instance has_g0_coe : has_coe G₀ (hom_mv 0) := { coe := λ s, coe s }
+  instance has_g1_coe : has_coe G₁ (hom_mv 1) := { coe := λ s, coe s }
+
+  -- define zero and one on the hom_mvs
+  instance has_zero {n : ℕ} : has_zero (hom_mv n) := { zero := (0 : blade n) }
+  instance r0_has_one : has_one (hom_mv 0) := { one := (1 : blade 0) }
+
+  def add : Π {n : ℕ}, hom_mv n → hom_mv n → hom_mv n
+  | 0 := λ a b, begin
+    cases a with a',
+    cases b with b',
+    exact hom_mv.scalar (a' + b'),
+  end
+  | 1 := λ a b, begin
+    cases a,
+    cases b,
+    exact hom_mv.vector (a_1 + b_1)
+  end
+  | (n + 2) := λ a b,begin
+    cases a,
+    cases b,
+    exact hom_mv.graded (a_a ++ b_a),
+  end
+  instance has_add {n : ℕ} : has_add (hom_mv n) := { add := add }
+end hom_mv
+
+inductive multivector : nat → Type u
+| scalar : hom_mv 0 → multivector 0
+| augment {n : ℕ} : multivector n → hom_mv (nat.succ n) → multivector (nat.succ n)
+
+
+namespace mv
+  -- define zero and one on the multivectors
+  def mv_zero : Π (n : ℕ), multivector n
+  | 0 := multivector.scalar (0 : G₀)
+  | 1 := multivector.augment (mv_zero 0) 0
+  | (nat.succ $ nat.succ n) := multivector.augment (mv_zero $ nat.succ n) (hom_mv.graded [])
+  def mv_one : Π (n : ℕ), multivector n
+  | 0 := multivector.scalar (1 : G₀)
+  | 1 := multivector.augment (mv_one 0) 0
+  | (nat.succ $ nat.succ n) := multivector.augment (mv_one $ nat.succ n) (hom_mv.graded [])
+  instance mv_has_zero {n : ℕ} : has_zero (multivector n) := { zero := mv_zero n }
+  instance mv_has_one {n : ℕ} : has_one (multivector n) := { one := mv_one n }
+
+  -- blades are coercible to multivectors
+  def hom_mv_coe : Π {n : ℕ}, (hom_mv n) -> (multivector n)
+  | nat.zero := λ b, multivector.scalar b
+  | (nat.succ n) := λ b, multivector.augment (mv_zero n) b
+  instance has_hom_mv_coe  {n : ℕ} : has_coe (hom_mv n) (multivector n) := { coe := hom_mv_coe }
+  instance has_g0_coe : has_coe G₀ (multivector 0) := { coe := λ s, hom_mv_coe $ hom_mv.scalar $ blade.scalar s }
+  instance has_g1_coe : has_coe G₁ (multivector 1) := { coe := λ v, hom_mv_coe $ hom_mv.vector $ blade.vector v }
+
+  -- multivectors are up-coercible
+  def augment_coe {n : ℕ} (mv : multivector n) : multivector (nat.succ n) := multivector.augment mv 0
+  instance has_augment_coe {n : ℕ} : has_coe (multivector n) (multivector (nat.succ n)) := { coe := augment_coe }
+
+  def mv_add : Π {n : ℕ}, multivector n → multivector n → multivector n
+  | 0 := λ a b, begin
+    cases a,
+    cases b,
+    exact multivector.scalar (a_1 + b_1)
+  end
+  | (nat.succ n) := λ a b, begin
+    cases a with z z a' ar,
+    cases b with z z b' br,
+    exact multivector.augment (mv_add a' b') (ar + br)
+  end
+  instance has_add {n: ℕ} : has_add (multivector n) := { add := mv_add }
+end mv
+
+parameter v : G₁
+
+--  Addition!
+#check ((2 + v) : multivector 1)
+#check ((2 + v) : multivector 2)
 
 def fᵥ : G₁ →+ G := f₁ G₀