Universes for discrimination proofs

I can't decide if this is trivial or not. What does this construction mean/say?

module Universe where

data N0 : Set where

data Bool : Set where
 True : Bool
 False : Bool

data Direction : Set where
 Left : Direction
 Right : Direction

data Eq (T : Set) : T -> T -> Set where
 refl : (t : T) -> Eq T t t

-- Eq Set Bool Direction -> N0
--  is not provable
--  (techincally I would have to define Eq in Set1 but we can pretend universe polymorphism)

mutual
 data U : Set where
  n0 : U
  bool : U
  direction : U
  eq : (A : U) -> T A -> T A -> U
  pi : (A : U) -> (B : T A -> U) -> U
 
 T : U -> Set
 T n0 = N0
 T bool = Bool
 T direction = Direction
 T (eq A x y) = Eq (T A) x y
 T (pi A B) = (a : T A) -> T (B a)

mutual
 data U1 : Set where
  n0' : U1
  bool' : U1
  direction' : U1
  eq' : (A : U1) -> T1 A -> T1 A -> U1
  pi' : (A : U1) -> (B : T1 A -> U1) -> U1
  u' : U1
  lift : U -> U1
 
 T1 : U1 -> Set
 T1 n0' = N0
 T1 bool' = Bool
 T1 direction' = Direction
 T1 (eq' A x y) = Eq (T1 A) x y
 T1 (pi' A B) = (a : T1 A) -> T1 (B a)
 T1 u' = U
 T1 (lift A) = T A

T' : U -> Set
T' n0 = T n0
T' bool = T bool
T' direction = T n0
T' (eq A x y) = T n0
T' (pi A B) = T n0

rew : forall {A B : U} -> Eq U A B -> T' A -> T' B
rew (refl _) prf = prf

discr : Eq U bool direction -> N0
discr prf = rew prf True

corr : T1 (pi' (eq' u' bool direction) \_ -> n0')
corr = discr

Agda Supports Eta - Cont is a monad

Since Agda 2 now supports eta in the definitional equality, here is a celebratory proof that Cont is a monad!

module eta where

data _==_ {T : Set} : T -> T -> Set where refl : (x : T) -> x == x

cont : Set -> Set -> Set
cont a r = (a -> r) -> r

return : forall {a r} -> a -> cont a r
return a f = f a

_>>=_ : forall {a b r} -> cont a r -> (a -> cont b r) -> cont b r
m >>= k = \c -> m (\a -> k a c)

cont-left-identity : forall {a b r} -> (o : a) -> (f : a -> cont b r) ->
 (return o >>= f) == f o
cont-left-identity o f = refl (f o)

cont-right-identity : forall {a r} -> (n : cont a r) ->
 (n >>= return) == n
cont-right-identity n = refl n

cont-associativity : forall {a b c r} -> (n : cont a r) -> (f : a -> cont b r) -> (g : b -> cont c r) ->
 ((n >>= f) >>= g) == (n >>= (\x -> (f x) >>= g))
cont-associativity n f g = refl ((n >>= f) >>= g)

Inventing a concrete syntax

ind F : ★.

ind T : ★;
I : T.

ind Bool : ★;
true : Bool;
false : Bool.

ind N : ★;
O : N;
S : N → N.

ind Ord : ★
OO : Ord;
OS : Ord → Ord;
OL : (N → Ord) → Ord.

ind Fin : N → ★;
fz : (n:N)Fin (S n);
fs : (n:N)Fin n → Fin (S n).

ind Eq (A:★)(x:A):A → ★;
refl : Eq A x x.

ind JMEq (A:★)(x:A):(B:★)B → ★;
JMrefl : JMEq A x A x.

ind So : Bool → ★;
oh : So true.

ind Ex (T:★)(P:T → ★):★;
witness : (t : T)Prf(P t) → Ex T P.

ind Acc (A:★)(R:A → A → ★):A → ★;
below : ((y:A)R y x → Acc A R y) → Acc A R x.

ind W (A:★)(B:A → ★):★;
sup : (a:A)(f:B a → W A B)W A B.

CPS transformation using delimited continuations

;; Idea: Use reified continuations to implement a CPS converter
;;
;; in the expression: (+ (* x x) (* y y))
;; the continuation of (* x x) is (lambda (x-squared) (+ x-squared (* y y)))
;;
;; similiarly,
;;
;; in the expression: (list '+ (list '* 'x 'x) (list '* 'y 'y))
;; the continuation of (list '* 'x 'x) is (lambda (x-squared) (list '+ x-squared (list '* 'y 'y)))

(require scheme/control) ;; shift/reset delimited continuations

;; > (reset (list '+ (shift k (list '* 'x 'x))     (list '* 'y 'y)))
;; (* x x)
;; > (reset (list '+ (shift k (k '?))              (list '* 'y 'y)))
;; (+ ? (* y y))
;; > (reset (list '+ (shift k (k (list '* 'x 'x))) (list '* 'y 'y)))
;; (+ (* x x) (* y y))
;; > (reset (list '+ (shift k `(let ((x-squared ,(list '* 'x 'x))) ,(k 'x-squared))) (list '* 'y 'y)))
;; (let ((x-squared (* x x))) (+ x-squared (* y y)))


;; CPS applications should throw the result value into a continuation
;; so (f x y z) turns into (f x y z (lambda (result) (continuation result)))

(define (apply# f . args) (shift k (let ((g (gensym "g"))) (reset `(,f ,@args (lambda (,g) ,(k g)))))))

;; Some examples of apply# in action:
;;
;; > `(k ,(apply# 'f--> 'x 'y 'z))
;; (f--> x y z (lambda (g349) (k g349)))
;; > (apply# 'f--> (apply# '+--> 2 3) 'y 'z)
;; (+--> 2 3 (lambda (g345) (f--> g345 y z (lambda (g346) g346))))
;; > (apply# 'f--> (apply# '+--> 2 3) (apply# '*--> 'x 'y) 'z)
;; (+--> 2 3 (lambda (g415) (*--> x y (lambda (g416) (f--> g415 g416 z (lambda (g417) g417))))))

;; What about syntax like IF? clearly apply# would be wrong (due to evaluation order) so define new syntax!
(define-syntax if#
  (syntax-rules ()
    ((if# <cond> <then> <else>) (shift k `(if ,<cond> ,(reset (k <then>)) ,(reset (k <else>)))))))

;; Examples:
;;
;; > (if# (apply# 'zero?--> 'n) ''yes ''no)
;; (zero?--> n (lambda (g374) (if g374 'yes 'no)))
;; > `(display ,(if# (apply# 'zero?--> 'n) ''yes ''no))
;; (zero?--> n (lambda (g850) (if g850 (display 'yes) (display 'no))))

(define-syntax define#
  (syntax-rules ()
    ((define# (name/args ...) body) `(define (name/args ... k-->) ,(reset `(k--> ,body))))))

;; That's enough now to CPS convert entire procedures:

;; > (define# (fact-iter--> n acc)
;;     (if# (apply# 'zero?--> 'n)
;;          'acc
;;          (apply# 'fact-iter--> (apply# '---> 'n 1) (apply# '*--> 'acc 'n))))
(define (fact-iter--> n acc k-->)
  (zero?--> n (lambda (g875)
   (if g875
       (k--> acc)
       (---> n 1 (lambda (g876)
        (*--> acc n (lambda (g877)
         (fact-iter--> g876 g877 (lambda (g878) (k--> g878)))))))))))

;; Test it! (this CPS format is a subset of Scheme)
(define (zero?--> n k) (k (zero? n)))
(define (*--> x y k) (k (* x y)))
(define (---> x y k) (k (- x y)))
;; > (fact-iter--> 7 1 display)
;; 5040


;; Now a function to CPS convert based on all this is trivial, it's just a fold
;; that replaces if with if#, define with define# and applications with apply#!

(define (if#-thunked cond then-thunk else-thunk) (if# cond (then-thunk) (else-thunk)))
(define (define#-thunked name/args body-thunk)
  (let ((k--> (gensym "k-->"))) `(define (,@name/args ,k-->) ,(reset `(,k--> ,(body-thunk))))))
(define (cps term)
  (if (pair? term)
      (case (car term)
        ((quote) `',term)
        ((if) (if#-thunked (cps (cadr term)) (lambda () (cps (caddr term))) (lambda () (cps (cadddr term)))))
        ((define) (define#-thunked (cadr term) (lambda () (cps (caddr term)))))
        (else (apply apply# (map cps term))))
      term))

;; > (cps '(define (fact-iter-2 n acc) (if (zero? n) acc (fact-iter-2 (- n 1) (* n acc)))))
(define (fact-iter-2--> n acc k-->)
  (zero?--> n (lambda (g443)
   (if g443
       (k--> acc)
       (---> n 1 (lambda (g444)
        (*--> n acc (lambda (g445)
         (fact-iter-2--> g444 g445 (lambda (g446)
          (k--> g446)))))))))))
;; > (fact-iter-2--> 8 1 display)
;; 40320