Rice-like Theorem for Injective Types

• The paper demonstrates that, constructively, any algebraically injective type supports only trivial decidable properties unless weak excluded middle is assumed.
• It employs a strategy of decomposing types via embeddings and exploiting Ω-paths to show that non-trivial properties necessitate WEM.
• The findings extend Rice’s theorem from recursion theory, revealing stringent constraints on definable properties within Univalent Foundations.

A Rice-like theorem for injective types asserts a strong limitation on the existence of non-trivial decidable properties for algebraically injective types in Univalent Foundations. It shows that, constructively, any injective type admits only trivial decidable properties unless weak excluded middle (WEM) holds. This result identifies a deep analogy with Rice's theorem in classical recursion theory and highlights significant constraints in the theory of injective types, especially in constructive and homotopical settings (Jong et al., 18 Jan 2026).

1. Preliminaries: Injective Types and Decidable Properties

Let $\UU, \VV, \WW$ denote universes in Univalent Foundations. For a type $D:\WW$, $D$ is $(\UU, \VV)$–injective if, for every embedding $j: X \hookrightarrow Y$ with $X:\UU$, $Y:\VV$, and every map $f:X \to D$, there exists an extension $f/j: Y \to D$ such that $f/j \circ j = f$. This generalizes classical injectivity by internalizing extension along embeddings.

A property of a type $D$ is a map $P: D \to \Prop \subset \UU$. It is decidable if $\forall d:D. P(d) \lor \neg P(d)$ and non-trivial if both fibers $\{d \mid P(d)\}$ and $\{d \mid \neg P(d)\}$ are inhabited—i.e., $P$ partitions $D$ into two non-empty pieces.

2. Formal Statement of the Rice-Like Theorem

The central Rice-like theorem is formulated as follows. If $D:\WW$ is $(\UU, \VV)$–injective and one has a non-trivial decidable property $P:D\to\Prop \subset \UU$, then weak excluded middle holds in $\UU$:

$\forall Q:\Prop. \neg Q \lor \neg\neg Q.$

Equivalently, if $f: D \to \Two$ is extensional and both fibers $\operatorname{fiber}_f(0)$ and $\operatorname{fiber}_f(1)$ are inhabited, then the existence of such an $f$ yields WEM. Therefore, under the failure of WEM in $\UU$, no injective $D$ admits a non-trivial decidable property (Jong et al., 18 Jan 2026).

3. Proof Strategy and Key Lemmas

The proof consists of three main ingredients:

• Decomposition: Any non-trivial decidable property $f:D\to\Two$ corresponds to a decomposition of $D$ into two inhabited fibers. Formally, the type of decompositions is

$\operatorname{decomp}\,D = \Sigma (f: D \to \Two).\, \operatorname{fiber}_f(0) \times \operatorname{fiber}_f(1).$

• Injectives Admit $\Omega$-Paths: For the type of truth-values $\Omega_\UU : \UU^+$, there is an embedding $\Two \hookrightarrow \Omega_\UU$. Injectivity implies that for any $x_0, x_1: D$, there exists $g:\Omega_\UU \to D$ with $g(\bot) = x_0,\, g(\top) = x_1$. Thus, injectives admit paths indexed over the type of propositions.
• Decomposition of $\Omega$ Implies WEM: If $\phi : \Omega \to \Two$ admits both fibers inhabited, then for any $Q:\Prop$,

$$R = (\phi(\bot) \times Q) + (\phi(\top) \times \neg Q)$$

ensures that $\phi(R) = \bot$ implies $\neg Q$ and $\phi(R)=\top$ implies $\neg\neg Q$. Therefore, $\neg Q \lor \neg\neg Q$.

Combining these steps, any decomposition $f:D\to\Two$ yields, via an $\Omega$-path $g$ and composition $\phi = f\circ g$, a decomposition of $\Omega$ itself, and thus WEM.

4. Consequences for Constructive Mathematics

An immediate corollary is that several important injective types cannot admit non-trivial decidable properties unless WEM holds. These include:

• The universe $\UU$ itself
• The subuniverse $\Omega_\UU$ of propositions
• The type of ordinals in $\UU$
• The type of iterative (multi)sets
• Carriers of sup-lattices or pointed directed complete posets (dcpos)
• Types of (small) $\infty$-magmas, monoids, and groups

Thus, in the absence of WEM, all decidable properties on such injective types are necessarily trivial (i.e., constant functions to $0$ or $1$). This has led to the formulation that injective types are indecomposable constructively (Jong et al., 18 Jan 2026).

5. Relation to Excluded Middle and Classical Logic

In classical logic, where excluded middle holds, every pointed type is injective, and all propositions are decidable, so injective types may admit many decompositions. This sharply contrasts with the constructive (univalent) setting, where injectivity leads to indecomposability unless classical reasoning is introduced. The Rice-like theorem thus identifies injectivity as conferring a form of "computation-theoretic universality" in the constructive setting: no algorithm can distinguish non-trivial properties without importing classical logic.

6. Broader Implications and Related Examples

Further analysis establishes that all injective types have this indecomposability property constructively. There are also several structural counterexamples: types with an apartness relation supporting two points apart cannot be injective unless WEM holds; the type of inhabited types is injective if and only if all propositions are projective; specific counterexamples include the type of booleans, simple types, Dedekind reals, or conatural numbers, whose injectivity would entail WEM itself.

A plausible implication is the strong structural restriction injectivity imposes on the landscape of definable and decidable properties in constructive mathematics, paralleling the phenomenon of "no non-trivial computable invariants" known from recursion theory and classical Rice's theorem (Jong et al., 18 Jan 2026).

7. Summary Table: Logical Status of Decidable Properties on Injective Types

Injective Type Example	Decidable Non-trivial Properties	Logical Principle Required
$\UU$, $\Omega_\UU$, ordinals	None, unless WEM	Weak excluded middle
Iterative (multi)sets	None, unless WEM	Weak excluded middle
Pointed dcpos, sup-lattices	None, unless WEM	Weak excluded middle
(Small) $\infty$-magmas, groups	None, unless WEM	Weak excluded middle

If WEM fails, only trivial decidable properties exist for these injective types.

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The Rice-like theorem for injective types, as established by (Jong et al., 18 Jan 2026), uncovers a deep obstruction to constructive decomposability for injective types in Univalent Foundations, connecting logic, type-theoretic structure, and the theory of decidable properties.