evalInduction — Mathlib

Lean4
/-- (co-)Induction principle for `eval`. If a property `C` holds of any point `a` evaluating to `b`
which is either terminal (meaning `a = b`) or where the next point also satisfies `C`, then it
holds of any point where `eval f a` evaluates to `b`. This formalizes the notion that if
`eval f a` evaluates to `b` then it reaches terminal state `b` in finitely many steps. -/
@[elab_as_elim]
def evalInduction {σ} {f : σ → Option σ} {b : σ} {C : σ → Sort*} {a : σ} (h : b ∈ eval f a)
    (H : ∀ a, b ∈ eval f a → (∀ a', f a = some a' → C a') → C a) : C a :=
  PFun.fixInduction h fun a' ha' h' ↦ H _ ha' fun b' e ↦ h' _ <| Part.mem_some_iff.2 <| by rw [e]; rfl
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