Is internal parametricity useful in theorem proving?

Internal parametricity refers to the ability to claim that ∀u:(∀t.t→t).u=λx.x and similarly for other pi types abstracting over types (this is a ...

Parametricity and data kinds - Proof Assistants Stack Exchange

... internal parametricity was an open question, I think). The only ... Is internal parametricity useful in theorem proving? 40 · What is ...

Internal and Observational Parametricity for Cubical Agda

Our parametricity theorem lets us obtain one-liner proofs of standard internal free theorems. We observe that the SRP is harder to prove than ...

Higher Inductive Types and Internal Parametricity for Cubical Type ...

... theory will be useful to establish the basic ... We have brought internal parametricity to cubical type theory, showing that the latter.

Internalizing Parametricity - Chalmers Publication Library

A calculus with an internal parametricity theorem . . . . . 31. 3.1 ... 𝐴 is parametric, and that ⟦𝐴⟧ proves its parametricity theorem. Hence each ...

Internal Parametricity for Cubical Type Theory - arXiv

Nevertheless, it has been common to treat proofs of equality as irrelevant: we prove equalities to check code correctness or to prove a theorem, ...

Internal and Observational Parametricity for Cubical Agda (POPL ...

... parametric proof assistant. We provide the first mechanized proofs of crucial theorems for internal parametricity, like the relativity theorem. We identify ...

Extending Cubical Agda with Internal Parametricity

Useful information can systematically be extracted by only looking at the type of a parametric function. These facts commonly known as “free theorems” [18] ...

[POPL'24] Internal parametricity, without an interval - YouTube

Internal Parametricity, without an Interval (Video, POPL 2024) Thorsten Altenkirch, Yorgo Chamoun, Ambrus Kaposi, and Michael Shulman ...

An encoding of parametricity in Agda - [email protected]

types. The type of the relativized function is precisely an instance of. Reynolds theorem for it, and its implementation is a proof. Post by Taral.

Internal Parametricity, without an Interval | Request PDF

Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, ...

Parametricity: Money for Nothing and Theorems for Free

The inability to come up with a counterexample doesn't prove a theorem. And yet there is one area of programming where such arguments work, and ...

Parametricity via Cohesion - arXiv

To prove the desired parametricity theorem ... internal parametricity for Cubical Type Theory to derive coherence theorems for the smash product.

Internal and Observational Parametricity for Cubical Agda - Lirias

Factoring the proof of a free theorem into an SRP proof obligation plus a call to the param theorem is already helpful. Such proofs are conceptually easier ( ...

Evan Cavallo, Internal parametricity and cubical type theory - YouTube

... theory, the type theory we designed that combines the two, and potential applications to higher-dimensional theorem proving. This is joint ...

Proof-relevant parametricity? - Fredrik Nordvall Forsberg

This is the Abstraction Theorem “one level up” for the proofs [t]1, which we will put to use in the next section. 6 Theorems about Proofs for Free. In Phil ...

Higher Inductive Types and Internal Parametricity for Cubical Type ...

... theory, internal parametricity, cohesive type theory. Page 3. iii. Abstract ... useful to establish the basic vocabulary and organizational ...

Parametricity and Proving Free Theorems for Functional-Logic ... - LIX

{bτ2c}}, and ˆg proves the inner level of set brackets to be cosmetic: ... Hence, Theorem 8.3 can be used with the term from that typing judgment as e0 ...

Free theorems and runtime type representations - Penn Engineering

The parametricity theorem can be proved with the following sequence of lemmas. Lemma 1 (Interpretation of types is a value relation). If Γ ` τ and. Γ ` δ then ...

Internalizing Parametricity - Chalmers Research

However these results are meta theorems, and although the theorems can be stated as internal propositions, they cannot be proved internally. In ...