Is internal parametricity useful in theorem proving?

Theorems for Free for Free: Parametricity, With and Without Types

We prove the Fundamental Property of this logical relation and that it is sound with respect to contextual equivalence. To demonstrate the utility of ...

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, ...

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 ...

Parametricity in an Impredicative Sort - DROPS

[P] hh and this proof may be used by the abstraction theorem to prove parametricity for terms involving the axiom. Or P is provably not parametric: there exists ...

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

This allows us to express, within the SaLT type system, where in the original CuMin program nondeterminism is or is not actually used (Section 6) ...

A Reasonably Exceptional Type Theory - PLEIAD

... parametricity translation for type theory. While in. System F, parametricity has to be stated and proved externally, the expressiveness of type theory makes ...

Internal Parametricity for Cubical Type Theory - Evan Cavallo

I Theorem 10. There is a term of type LEM → ⊥. Proof. We refute the weak excluded middle WLEM := (X:U) → (b ...

Internal and Observational Parametricity for Cubical Agda - Lirias

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

In Idris, can I prove free theorems, e.g. the only (total) function of type ...

This answer is useful. 16. Save this answer. Show activity on this post. You can't. Theorems like this ("free theorem") follow from the ...

Compiler are theorem prover - Reddit

Very helpful and very interesting — thank you. Can you say a little more about “parametricity theorems”? This is the first I've heard of ...

Free theorems and runtime type representations - Penn Engineering

Reynolds's abstraction theorem [21], often referred to as the parametricity theorem, can be used to derive ... The parametricity theorem can be proved with the ...

The Marriage of Univalence and Parametricity - Hal-Inria

In CIC and MLTT, univalence cannot be proven and is therefore defined as an axiom. Because the proof of Theorem 3.3 starts by using the ...

Cohesive Internal Parametricity for Cubical Type Theory - HoTT/UF

... parametricity in higher-dimensional theorem-proving. On the way, I'll highlight the similarities between internal parametricity and cubical type theory.

Relational parametricity. Why the paper "Theorems for Free" is hard ...

... theorems would be trivial equations of the form x = x. - TFF does not present the proof of the parametricity theorem in a way that ...

Parametricity-exploiting proofs in Agda - Stack Overflow

I tried using it to prove parametricity , but it's not at all obvious to me how the LFT-derived free theorem of my function type is useful here.

Higher Inductive Types and Parametricity in Cubical Type Theory

It is proposed that parametricity is a useful tool for this purpose, and a bare-bones schema for indexed higher inductive types, complete with a ...

[PDF] Parametric Cubical Type Theory | Semantic Scholar

A computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of ...

Categorical Models for Abadi-Plotkin's Logic for Parametricity

Thus parametricity is useful for proving ... This proves Theorem 6.2. 9. Parametric Internal Models ... show that the internal model in F is parametric with ...

Internal Parametricity, without an Interval - ACM Digital Library

Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the ...

Failure is Not an Option: An Exceptional Type Theory.

parametricity is used to derive theorems for free, while realizability constrains ... which relies on internal parametricity to recover ... made for proving inside ...