Lysxia

(* formalization attached to my answer to https://cstheory.stackexchange.com/questions/53855/can-we-use-relational-parametricity-to-simplify-the-type-forall-a-a-to-a *)
From Coq Require Import Lia.

Definition U : Type := forall A : Type, ((A -> A) -> A) -> A.

(* Try enumerating solutions by hand. After the initial [intros],
   we can either complete the term with [exact xi],
   or continue deeper with [apply f; intros xi]. *)
Goal U.
Proof.

Lysxia

#!/usr/bin/env cabal
{- cabal:
build-depends: base, effectful-core
-}

-- Usage: cabal run Coroutine.hs

{-# LANGUAGE
  DataKinds,
  FlexibleContexts,

Lysxia

(* https://proofassistants.stackexchange.com/questions/3841/assistance-using-destruct-on-an-equality-proof-for-functors/3843 *)

(* Proof of associativity of functor composition without UIP *)

(* 1. Intuition: treat equality as a proper algebraic structure:
      think of it as a category (aka. the discrete category), not an equivalence relation. *)
(* 1a. An equality between morphisms, f = g, becomes the existence of a morphism between f and g. *)
(* 1b. In the definition of a category, homsets are thus categories.
       The resulting structure is known as a 2-category. https://ncatlab.org/nlab/show/2-category
       You can kepp applying the same generalization to the morphism category, resulting

Lysxia

{-# LANGUAGE BangPatterns #-}

import Criterion
import Criterion.Main
import Data.List (inits, scanl')
import Data.Primitive.Array
import GHC.Exts (RealWorld)
import System.IO.Unsafe

-- base implementation, using queues

Lysxia

(* A trick to print the simplified type of something *)

Definition type_of {A : Type} (a : A) : Type := A.
Arguments type_of _ /.
Notation simpl_type_of a := ltac:(let A := eval simpl in (type_of a) in exact A).

(* Example *)

Parameter p : forall x, 3 + x = x.

Lysxia

{- counter.hs: count comments and lines of code in .lagda.md files.

Usage:

  cat src/*.lagda.md|runghc counter.hs

Counting method:
Triple backticks either start or end a code block.
Triple backtick lines are not counted.
Code lines starting with `--` are counted as comments

Lysxia

-- Q: How to fix the last two commented lines?
--
-- Summary: Two intrinsically typed calculi A and B, and a translation from A to B.
-- The translation consists of a translation on types, followed by a translation on terms
-- indexed by the translation on types.
-- Agda won't let me case-split in the translation of (term) variables.
--
-- Signs of trouble:
-- - green slime in _A.∋_ and _B.∋_
-- - with clauses (rather than only pattern-matching on arguments)

Lysxia

{-# LANGUAGE RankNTypes, GADTs #-}
module O where

data Coyoneda g a where
  Coyoneda :: forall g a x. g x -> (x -> a) -> Coyoneda g a

newtype Obj f g = Obj { unObj :: forall a. f a -> Coyoneda g (a, Obj f g) }

compose :: Obj f g -> Obj g h -> Obj f h
compose (Obj a) (Obj b) = Obj (\f ->

Lysxia

From Coq Require Import List.
Import ListNotations.

Inductive bintree : Type :=
  | Leaf : bintree
  | Node : bintree -> nat -> bintree -> bintree.

Inductive tree :=
  | TNode : nat -> list tree -> tree.

Lysxia

data Freer (F : Set → Set) (A : Set) : Set₁ where
  pure : A → Freer F A
  bind : (B : Set) → F B → (B → Freer F A) → Freer F A

data Freest {A : Set} (J : A → Set) (F : A → Set) (a : A) : Set where
  pure : J a → Freest J F a
  bind : (b : A) → F b → (J b → Freest J F a) → Freest J F a

data R {A : Set} (J : A → Set) (F : A → Set) : Set → Set where
  mkR : (a : A) → F a → R J F (J a)