{-
A polynomial model of a Martin-Löf type theory. Based on:
- https://www.repository.cam.ac.uk/handle/1810/254394
- https://gist.github.com/bobatkey/0d1f04057939905d35699f1b1c323736
The model supports the negation of function extensionality, together with the
eta rule for functions and a wealth of type formers. So this is a nice
formalization of the unprovability of function extensionality in a reasonably
feature-rich type theory.
As a bonus, there's also a game-semantic generalization of the polynomial model.
The formalization for this is not complete, I include operations but skip the equations.
This project was checked with Agda 2.6.2 with standard library 1.6.
Metatheory:
- We use funext and UIP.
- We use Agda's native Level to model universe levels. This is convenient because we get
the built-in level solving for free. We could also use a deeper inductive-recursive embedding,
but I don't think there's much benefit to that here.
-}
module README where
-- A bundle of imports and lemmas. Lemmas are mostly about shuffling transports
-- and properties of the binary sum type, which is prominently used in the model.
open import Lib
open import CwF -- category-with-families
open import Pi
open import Sigma
open import Univ -- countable hierarchy of Coquand-style universes
open import Nat
open import Identity
open import Unit
open import Empty
open import Bool
open import NoFunExt -- refutation of function extensionality
open import Games -- game semantics