`comp`

definition

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module: IndisputableMonolith.Algebra.CostAlgebra
domain: Algebra
line: 808 · github
papers citing: none yet

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IndisputableMonolith.Algebra.CostAlgebra on GitHub at line 808.

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formal source

 805      simpa [posInv] using (J_reciprocal x.1 x.2).symm⟩
 806
 807/-- Composition of `J`-automorphisms. -/
 808def comp (g f : JAut) : JAut :=
 809  ⟨fun x => g (f x), by
 810    constructor
 811    · intro x y
 812      change g (f (posMul x y)) = posMul (g (f x)) (g (f y))
 813      rw [f.multiplicative x y, g.multiplicative (f x) (f y)]
 814    · intro x
 815      rw [g.preserves_cost (f x), f.preserves_cost x]⟩
 816
 817@[simp] theorem comp_apply (g f : JAut) (x : PosReal) : comp g f x = g (f x) := rfl
 818
 819@[simp] theorem comp_id (f : JAut) : comp id f = f := by
 820  apply JAut.ext
 821  intro x
 822  rfl
 823
 824@[simp] theorem id_comp (f : JAut) : comp f id = f := by
 825  apply JAut.ext
 826  intro x
 827  rfl
 828
 829@[simp] theorem reciprocal_comp_reciprocal : comp reciprocal reciprocal = id := by
 830  apply JAut.ext
 831  intro x
 832  simp [comp, reciprocal, id]
 833
 834/-- Pointwise, a `J`-automorphism can only choose the identity branch or the
 835    reciprocal branch. -/
 836theorem eq_self_or_inv (f : JAut) (x : PosReal) :
 837    f x = x ∨ f x = posInv x := by
 838  have hJ : J (f x).1 = J x.1 := f.preserves_cost x

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