`multiplicative`

proved

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module: IndisputableMonolith.Algebra.CostAlgebra
domain: Algebra
line: 774 · github
papers citing: none yet

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IndisputableMonolith.Algebra.CostAlgebra on GitHub at line 774.

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 771instance : CoeFun JAut (fun _ => PosReal → PosReal) := ⟨fun f => f.1⟩
 772
 773/-- Multiplicativity of a `J`-automorphism. -/
 774theorem multiplicative (f : JAut) (x y : PosReal) :
 775    f (posMul x y) = posMul (f x) (f y) :=
 776  f.2.1 x y
 777
 778/-- Cost preservation of a `J`-automorphism. -/
 779theorem preserves_cost (f : JAut) (x : PosReal) :
 780    J (f x).1 = J x.1 :=
 781  f.2.2 x
 782
 783@[ext] theorem ext {f g : JAut} (h : ∀ x : PosReal, f x = g x) : f = g := by
 784  apply Subtype.ext
 785  funext x
 786  exact h x
 787
 788/-- The identity automorphism. -/
 789def id : JAut :=
 790  ⟨fun x => x, by
 791    constructor
 792    · intro _ _
 793      rfl
 794    · intro _
 795      rfl⟩
 796
 797/-- The reciprocal automorphism. -/
 798noncomputable def reciprocal : JAut :=
 799  ⟨posInv, by
 800    constructor
 801    · intro x y
 802      apply Subtype.ext
 803      simp [posMul, posInv, mul_comm]
 804    · intro x

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