是否存在可表示同形状归纳类型等价的类型论及相关证明机制？

Great question! This is exactly the sort of problem that homotopy type theory (HoTT) and its cubical variants excel at solving, and observational type theory (OTT) also has built-in mechanisms for this. Let's break down how each approach works:

Yes, This Is Possible in HoTT/Cubical Type Theory (via Univalence) #

The key here is the univalence axiom, which states that two types are equal if and only if they are equivalent (i.e., there exists a pair of mutually inverse functions between them). For your list1 and list2 types, this is straightforward to apply:

1. Construct an equivalence between list1 A and list2 A:
   We can define bidirectional functions that map each constructor of list1 to the corresponding constructor of list2 (and vice versa), then prove these functions are inverses. For example (in Cubical Agda, a cubical TT implementation with built-in univalence):

   open import Cubical.Core.Everything
   open import Cubical.Foundations.Everything
   
   data list1 (A : Type) : Type where
     nil1 : list1 A
     cons1 : A → list1 A → list1 A
   
   data list2 (A : Type) : Type where
     nil2 : list2 A
     cons2 : A → list2 A → list2 A
   
   -- Forward map: list1 → list2
   list1→list2 : ∀ {A} → list1 A → list2 A
   list1→list2 nil1 = nil2
   list1→list2 (cons1 x l) = cons2 x (list1→list2 l)
   
   -- Reverse map: list2 → list1
   list2→list1 : ∀ {A} → list2 A → list1 A
   list2→list1 nil2 = nil1
   list2→list1 (cons2 x l) = cons1 x (list2→list1 l)
   
   -- Prove these form a valid equivalence
   list1≃list2 : ∀ {A} → list1 A ≃ list2 A
   list1≃list2 = isoToEquiv (iso list1→list2 list2→list1
     (λ l → inductive-refl)  -- Inductively prove list2→list1∘list1→list2 = id
     (λ l → inductive-refl)) -- Inductively prove list1→list2∘list2→list1 = id
   

2. Use univalence to turn the equivalence into a type equality:
   The ua (univalence axiom) function converts our equivalence list1≃list2 into a path list1 A ≡ list2 A—a formal proof that the two types are equal.

3. Transport predicates and functions across the equality:
   With a type equality p : list1 A ≡ list2 A, the transport function lets us "carry" any predicate P : list1 A → Type over to list2 A, producing P' : list2 A → Type. Similarly, any function f : list1 A → X can be transported to f' : list2 A → X. This exactly matches your requirement: the structure of proofs/functions for list1 is preserved, with only constructor names/types replaced.

Observational Type Theory (OTT) Also Handles This Natively #

OTT takes a different approach: equality is defined by observational equivalence. For inductive types, two types are observationally equal if they have identical constructor structures (same number of constructors, each constructor has parameters of observationally equal types).

In OTT, list1 and list2 would be judged observationally equal by definition, no extra axioms needed. This means you can directly use a predicate or function defined for list1 A with list2 A, and vice versa—no explicit transport step is required. This aligns perfectly with your intuition that same-shaped inductive types should be interchangeable.

Defining Generic "List-Like" Functions #

To define a function like reverse : list-like A → list-like A that works for both list1 and list2, you have a few options:

• In HoTT/cubical TT: Define reverse on a standard list A type, then transport it to list1 A and list2 A using their equivalences to list A. Alternatively, you can define a higher inductive type that abstracts the "list signature" (nil + cons), then show list1 and list2 are instances of this signature.
• In OTT: Since list1 and list2 are observationally equal, you can define reverse for one and use it directly with the other, or define it generically over any type with the list-shaped constructor structure.

内容的提问来源于stack exchange，提问作者LogicChains