`add_apply`

proved

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module: IndisputableMonolith.Algebra.F2Power
domain: Algebra
line: 72 · github
papers citing: none yet

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IndisputableMonolith.Algebra.F2Power on GitHub at line 72.

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depends on

formal source

  69/-- Pointwise XOR. -/
  70instance : Add (F2Power D) := ⟨fun u v => fun i => xor (u i) (v i)⟩
  71
  72@[simp] theorem add_apply (u v : F2Power D) (i : Fin D) :
  73    (u + v) i = xor (u i) (v i) := rfl
  74
  75/-- Negation in characteristic 2 is the identity. -/
  76instance : Neg (F2Power D) := ⟨fun v => v⟩
  77
  78@[simp] theorem neg_eq_self (v : F2Power D) : -v = v := rfl
  79
  80/-- Subtraction reduces to addition in characteristic 2. -/
  81instance : Sub (F2Power D) := ⟨fun u v => u + v⟩
  82
  83@[simp] theorem sub_eq_add (u v : F2Power D) : u - v = u + v := rfl
  84
  85instance : AddCommGroup (F2Power D) where
  86  add := (· + ·)
  87  zero := 0
  88  neg := Neg.neg
  89  add_assoc u v w := by
  90    funext i
  91    show xor (xor (u i) (v i)) (w i) = xor (u i) (xor (v i) (w i))
  92    cases u i <;> cases v i <;> cases w i <;> rfl
  93  zero_add v := by
  94    funext i
  95    show xor false (v i) = v i
  96    cases v i <;> rfl
  97  add_zero v := by
  98    funext i
  99    show xor (v i) false = v i
 100    cases v i <;> rfl
 101  neg_add_cancel v := by
 102    funext i

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