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theorem proved term proof

composeGenerators_preserves

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formal statement (Lean)

  50theorem composeGenerators_preserves
  51    {α G : Type*} {adm : Admissible α} {act : GenAction α G}
  52    (h_pres : Preserves adm act) (gs : List G) (x : α) (hadm : adm x) :
  53    adm (composeGenerators act gs x) := by

proof body

Term-mode proof.

  54  induction gs generalizing x with
  55  | nil => exact hadm
  56  | cons g gs ih =>
  57      unfold composeGenerators
  58      simp only [List.foldl_cons]
  59      have h1 : adm (act g x) := h_pres g x hadm
  60      exact ih (act g x) h1
  61
  62/-- A transformation `f : α → α` is *reachable* from generators `(act, gs)` if
  63    there is a generator-list whose composition agrees with `f` on every
  64    admissible input. -/

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