Prove2Me

The Lean 4 theorem `adjoint_comp_self_of_nested` in the `ChapterH8` chapter of the timepiece formalization

Proved
BookProof.ChapterH8.adjoint_comp_self_of_nested

by leonardopedro · Oct 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

timepiece

The Lean 4 theorem adjoint_comp_self_of_nested in the ChapterH8 chapter of the timepiece formalization.

Preamble
-- Generated from ChapterH8.lean — theorem BookProof.ChapterH8.adjoint_comp_self_of_nested
import Mathlib
import Definitions.Def_ChapterH8
open BookProof.ChapterH8

variable {K E : Type*} [Field K] [AddCommGroup E] [Module K E]
variable {E F G : Type*}
  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
variable {E F G : Type*}
  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]
variable {E F G : Type*}
  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]


noncomputable section


open BookProof.ChapterH4 BookProof.ChapterH5 BookProof.ChapterH6

open ContinuousLinearMap in
Formal statement
theorem BookProof.ChapterH8.adjoint_comp_self_of_nested (Vn : F →L[ℂ] E) (Vm : G →L[ℂ] E) (J : F →L[ℂ] G)
    (hJ : Vn = Vm.comp J)
    (hVm : (adjoint Vm).comp Vm = ContinuousLinearMap.id ℂ G)
    (hJJ : (adjoint J).comp J = ContinuousLinearMap.id ℂ F) :
    (adjoint Vn).comp Vn = ContinuousLinearMap.id ℂ F := by sorry
Source
https://github.com/leonardopedro/timepiece/blob/61595bc/BookProof/ChapterH8.lean

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