Refactored core

Moved all curnel code into curnel/, and split the two kernel modules
into separate files. Now the trusted core and the module/#lang stuff are
in separate files. The #lang is now reprovided by "cur.rkt", which
should also provide core agnostic sugar.

README.md

@@ -31,4 +31,5 @@ translation defined in [Proofs for Free](http://staff.city.ac.uk/~ross/papers/pr
 Open up anything in `stdlib/` to see some standard dependent-type
 formalisms.
 
-Open up `redex-curnel.rkt` to see the entire "trusted" core.
+Open up `curnel/redex-core.rkt` to see the entire "trusted" (after a
+large test suite) core.

cur.rkt

@@ -0,0 +1,4 @@
+#lang racket/base
+
+(require (rename-in "curnel/redex-lang.rkt" [provide real-provide]))
+(provide (rename-out [real-provide provide]) (all-from-out "curnel/redex-lang.rkt"))

curnel/redex-core.rkt

@@ -0,0 +1,923 @@
+#lang racket/base
+
+(require
+  racket/function
+  (only-in racket/set set=?)
+  redex/reduction-semantics)
+
+(provide
+  (all-defined-out))
+
+(set-cache-size! 10000)
+
+(module+ test
+  (require rackunit)
+  (define-syntax-rule (check-holds (e ...))
+    (check-true
+      (judgment-holds (e ...))))
+  (define-syntax-rule (check-not-holds (e ...))
+    (check-false
+      (judgment-holds (e ...)))))
+
+;; References:
+;; http://www3.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf
+;; https://www.cs.uoregon.edu/research/summerschool/summer11/lectures/oplss-herbelin1.pdf
+;; http://www.emn.fr/z-info/ntabareau/papers/universe_polymorphism.pdf
+;; http://people.cs.kuleuven.be/~jesper.cockx/Without-K/Pattern-matching-without-K.pdf
+;; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.74
+;; http://eb.host.cs.st-andrews.ac.uk/writings/thesis.pdf
+
+;; cicL is the core language of Cur. Very similar to TT (Idirs core)
+;; and Luo's UTT. Used to be similar to CIC, hence the name.
+;; Surface langauge should provide short-hand, such as -> for
+;; non-dependent function types, and type inference.
+(define-language cicL
+  (i ::= natural)
+  (U ::= (Unv i))
+  (x ::= variable-not-otherwise-mentioned)
+  (v ::= (Π (x : t) t) (λ (x : t) t) elim x U)
+  (t e ::= v (t t)))
+
+(define x? (redex-match? cicL x))
+(define t? (redex-match? cicL t))
+(define e? (redex-match? cicL e))
+(define v? (redex-match? cicL v))
+(define U? (redex-match? cicL U))
+
+(module+ test
+  (define-term Type (Unv 0))
+  (check-true (x? (term T)))
+  (check-true (x? (term A)))
+  (check-true (x? (term truth)))
+  (check-true (x? (term zero)))
+  (check-true (x? (term s)))
+  (check-true (t? (term zero)))
+  (check-true (t? (term s)))
+  (check-true (x? (term nat)))
+  (check-true (t? (term nat)))
+  (check-true (t? (term A)))
+  (check-true (t? (term S)))
+  (check-true (U? (term (Unv 0))))
+  (check-true (U? (term Type)))
+  (check-true (e? (term (λ (x_0 : (Unv 0)) x_0))))
+  (check-true (v? (term (λ (x_0 : (Unv 0)) x_0))))
+  (check-true (t? (term (λ (x_0 : (Unv 0)) x_0))))
+  (check-true (t? (term (λ (x_0 : (Unv 0)) x_0)))))
+
+;; 'A'
+;; Types of Universes
+(define-judgment-form cicL
+  #:mode (unv-ok I O)
+  #:contract (unv-ok U U)
+
+  [(where i_1 ,(add1 (term i_0)))
+   -----------------
+   (unv-ok (Unv i_0) (Unv i_1))])
+
+;; 'R'
+;; Kinding, I think
+(define-judgment-form cicL
+  #:mode (unv-kind I I O)
+  #:contract (unv-kind U U U)
+
+  [----------------
+    (unv-kind (Unv 0) (Unv 0) (Unv 0))]
+
+  [----------------
+    (unv-kind (Unv 0) (Unv i) (Unv i))]
+
+  [----------------
+    (unv-kind (Unv i) (Unv 0) (Unv 0))]
+
+  [(where i_3 ,(max (term i_1) (term i_2)))
+   ----------------
+   (unv-kind (Unv i_1) (Unv i_2) (Unv i_3))])
+
+(define-judgment-form cicL
+  #:mode (α-equivalent I I)
+  #:contract (α-equivalent t t)
+
+  [----------------- "α-x"
+                     (α-equivalent x x)]
+
+  [----------------- "α-U"
+                     (α-equivalent U U)]
+
+  [(α-equivalent t_1 (subst t_3 x_1 x_0))
+   (α-equivalent t_0 t_2)
+   ----------------- "α-binder"
+   (α-equivalent (any (x_0 : t_0) t_1)
+                 (any (x_1 : t_2) t_3))]
+
+  [(α-equivalent e_0 e_2)
+   (α-equivalent e_1 e_3)
+   ----------------- "α-app"
+   (α-equivalent (e_0 e_1) (e_2 e_3))])
+(module+ test
+  (check-holds (α-equivalent x x))
+  (check-not-holds (α-equivalent x y))
+  (check-holds (α-equivalent (λ (x : A) x) (λ (y : A) y))))
+
+;; NB: Substitution is hard
+;; NB: Copy and pasted from Redex examples
+(define-metafunction cicL
+  subst-vars : (x any) ... any -> any
+  [(subst-vars (x_1 any_1) x_1) any_1]
+  [(subst-vars (x_1 any_1) (any_2 ...))
+   ((subst-vars (x_1 any_1) any_2) ...)]
+  [(subst-vars (x_1 any_1) any_2) any_2]
+  [(subst-vars (x_1 any_1) (x_2 any_2) ... any_3)
+   (subst-vars (x_1 any_1) (subst-vars (x_2 any_2) ... any_3))]
+  [(subst-vars any) any])
+
+(define-metafunction cicL
+  subst : t x t -> t
+  [(subst U x t) U]
+  [(subst x x t) t]
+  [(subst x_0 x t) x_0]
+  [(subst (any (x : t_0) t_1) x t)
+   (any (x : (subst t_0 x t)) t_1)]
+  [(subst (any (x_0 : t_0) t_1) x t)
+   (any (x_new : (subst (subst-vars (x_0 x_new) t_0) x t))
+        (subst (subst-vars (x_0 x_new) t_1) x t))
+   (where x_new
+          ,(variable-not-in
+             (term (x_0 t_0 x t t_1))
+             (term x_0)))]
+  [(subst (e_0 e_1) x t) ((subst e_0 x t) (subst e_1 x t))]
+  [(subst elim x t) elim])
+(module+ test
+  (check-true (t? (term (Π (a : A) (Π (b : B) ((and A) B))))))
+  (check-holds
+    (α-equivalent
+      (Π (a : S) (Π (b : B) ((and S) B)))
+      (subst (Π (a : A) (Π (b : B) ((and A) B))) A S))))
+;; NB:
+;; TODO: Why do I not have tests for substitutions?!?!?!?!?!?!?!!?!?!?!?!?!?!!??!?!
+
+(define-metafunction cicL
+  subst-all : t (x ...) (e ...) -> t
+  [(subst-all t () ()) t]
+  [(subst-all t (x_0 x ...) (e_0 e ...))
+   (subst-all (subst t x_0 e_0) (x ...) (e ...))])
+
+;; TODO: I think a lot of things can be simplified if I rethink how
+;; TODO: model contexts, telescopes, and such.
+(define-extended-language cic-redL cicL
+  (E   ::= hole (v E) (E e)) ;; call-by-value
+  ;; Σ (signature). (inductive-name : type ((constructor : type) ...))
+  (Σ   ::= ∅ (Σ (x : t ((x : t) ...))))
+  (Ξ Φ ::= hole (Π (x : t) Ξ)) ;;(Telescope)
+  ;; NB: Does an apply context correspond to a substitution (γ)?
+  (Θ   ::= hole (Θ e)))        ;;(Apply context)
+(define Σ? (redex-match? cic-redL Σ))
+(define Ξ? (redex-match? cic-redL Ξ))
+(define Φ? (redex-match? cic-redL Φ))
+(define Θ? (redex-match? cic-redL Θ))
+(define E? (redex-match? cic-redL E))
+
+(module+ test
+  ;; TODO: Rename these signatures, and use them in all future tests.
+  (define Σ (term (∅ (nat : (Unv 0) ((zero : nat) (s : (Π (x : nat) nat)))))))
+  (define Σ0 (term ∅))
+  (define Σ3 (term (∅ (and : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Unv 0))) ()))))
+  (define Σ4 (term (∅ (and : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Unv 0)))
+                             ((conj : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Π (a : A) (Π (b : B) ((and A) B)))))))))))
+  (check-true (Σ? Σ0))
+  (check-true (Σ? Σ))
+  (check-true (Σ? Σ4))
+  (check-true (Σ? Σ3))
+  (check-true (Σ? Σ4))
+  (define sigma (term ((((((∅ (true : (Unv 0) ((T : true))))
+                           (false : (Unv 0) ()))
+                          (equal : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Unv 0)))
+                                 ()))
+                         (nat : (Unv 0) ()))
+                        (heap : (Unv 0) ()))
+                       (pre : (Π (temp808 : heap) (Unv 0)) ()))))
+  (check-true (Σ? (term (∅ (true : (Unv 0) ((T : true)))))))
+  (check-true (Σ? (term (∅ (false : (Unv 0) ())))))
+  (check-true (Σ? (term (∅ (equal : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Unv 0)))
+                                  ())))))
+  (check-true (Σ? (term (∅ (nat : (Unv 0) ())))))
+  (check-true (Σ? (term (∅ (pre : (Π (temp808 : heap) (Unv 0)) ())))))
+
+  (check-true (Σ? (term ((∅ (true : (Unv 0) ((T : true)))) (false : (Unv 0) ())))))
+  (check-true (Σ? (term (((∅ (true : (Unv 0) ((T : true)))) (false : (Unv 0) ()))
+                                                            (equal : (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Unv 0)))
+                                                                   ())))))
+  (check-true (Σ? sigma)))
+
+(define-metafunction cic-redL
+  append-Σ : Σ Σ -> Σ
+  [(append-Σ Σ ∅) Σ]
+  [(append-Σ Σ_2 (Σ_1 (x : t ((x_c : t_c) ...))))
+   ((append-Σ Σ_2 Σ_1) (x : t ((x_c : t_c) ...)))])
+
+;; TODO: Test
+;; TODO: Isn't this just plug?
+(define-metafunction cic-redL
+  apply-telescope : t Ξ -> t
+  [(apply-telescope t hole) t]
+  [(apply-telescope t_0 (Π (x : t) Ξ)) (apply-telescope (t_0 x) Ξ)])
+
+(define-metafunction cic-redL
+  apply-telescopes : t (Ξ ...) -> t
+  [(apply-telescopes t ()) t]
+  [(apply-telescopes t (Ξ_0 Ξ_rest ...))
+   (apply-telescopes (apply-telescope t Ξ_0) (Ξ_rest ...))])
+
+;; NB: Depends on clause order
+(define-metafunction cic-redL
+  select-arg : x (x ...) (Θ e) -> e
+  [(select-arg x (x x_r ...) (in-hole Θ (hole e))) e]
+  [(select-arg x (x_1 x_r ...) (in-hole Θ (hole e)))
+   (select-arg x (x_r ...) Θ)])
+
+(define-metafunction cic-redL
+  method-lookup : Σ x x (Θ e) -> e
+  [(method-lookup Σ x_D x_ci Θ)
+   (select-arg x_ci (x_0 ...) Θ)
+   (where ((x_0 : t_0) ...) (constructors-for Σ x_D))])
+(module+ test
+  (check-equal?
+    (term (method-lookup ,Σ nat s
+                         ((hole (s zero))
+                          (λ (x : nat) (λ (ih-x : nat) (s (s zero)))))))
+    (term (λ (x : nat) (λ (ih-x : nat) (s (s zero)))))))
+
+;; Create the recursive applications of elim to the recursive
+;; arguments of an inductive constructor.
+;; TODO: Poorly documented, and poorly tested.
+(define-metafunction cic-redL
+  elim-recur : x e Θ Θ Θ (x ...) -> Θ
+  [(elim-recur x_D e_P Θ
+               (in-hole Θ_m (hole e_mi))
+               (in-hole Θ_i (hole (in-hole Θ_r x_ci)))
+               (x_c0 ... x_ci x_c1 ...))
+   ((elim-recur x_D e_P Θ Θ_m Θ_i (x_c0 ... x_ci x_c1 ...))
+    (in-hole Θ (((elim x_D) (in-hole Θ_r x_ci)) e_P)))]
+  [(elim-recur x_D e_P Θ Θ_i Θ_nr (x ...)) hole])
+(module+ test
+  (check-true
+    (redex-match? cic-redL (in-hole Θ_i (hole (in-hole Θ_r zero))) (term (hole zero))))
+  (check-equal?
+    (term (elim-recur nat (λ (x : nat) nat)
+                      ((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
+                      ((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
+                      (hole zero)
+                      (zero s)))
+    (term (hole (((((elim nat) zero) (λ (x : nat) nat)) (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x))))))))
+  (check-equal?
+    (term (elim-recur nat (λ (x : nat) nat)
+                      ((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
+                      ((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
+                      (hole (s zero))
+                      (zero s)))
+    (term (hole (((((elim nat) (s zero)) (λ (x : nat) nat)) (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))))))
+
+(define-judgment-form cic-redL
+  #:mode (length-match I I)
+  #:contract (length-match Θ (x ...))
+  [----------------------
+    (length-match hole ())]
+
+  [(length-match Θ (x_r ...))
+   ----------------------
+   (length-match (Θ e) (x x_r ...))])
+
+(define cic-->
+  (reduction-relation cic-redL
+    (--> (Σ (in-hole E ((any (x : t_0) t_1) t_2)))
+         (Σ (in-hole E (subst t_1 x t_2)))
+         -->β)
+    (--> (Σ (in-hole E (in-hole Θ_m (((elim x_D) (in-hole Θ_i x_ci)) e_P))))
+         (Σ (in-hole E (in-hole Θ_r (in-hole Θ_i e_mi))))
+         (where ((x_c : t_c) ... (x_ci : t_ci) (x_cr : t_cr) ...) (constructors-for Σ x_D))
+         ;; There should be a number of methods equal to the number of
+         ;; constructors; to ensure E doesn't capture methods
+         (judgment-holds (length-match Θ_m (x_c ... x_ci x_cr ...)))
+         (where e_mi (method-lookup Σ x_D x_ci Θ_m))
+         (where Θ_r (elim-recur x_D e_P Θ_m Θ_m Θ_i (x_c ... x_ci x_cr ...)))
+         -->elim)))
+
+(define-metafunction cic-redL
+  reduce : Σ e -> e
+  [(reduce Σ e) e_r
+                (where (_ e_r) ,(car (apply-reduction-relation* cic--> (term (Σ e)))))])
+(module+ test
+  (check-equal? (term (reduce ∅ (Unv 0))) (term (Unv 0)))
+  (check-equal? (term (reduce ∅ ((λ (x : t) x) (Unv 0)))) (term (Unv 0)))
+  (check-equal? (term (reduce ∅ ((Π (x : t) x) (Unv 0)))) (term (Unv 0)))
+  ;; NB: This test fails because not currently reducing under binders.
+  (check-equal? (term (reduce ∅ (Π (x : t) ((Π (x_0 : t) x_0) (Unv 0)))))
+                (term (Π (x : t) (Unv 0))))
+  (check-equal? (term (reduce ∅ (Π (x : t) ((Π (x_0 : t) (x_0 x)) x))))
+                (term (Π (x : t) ((Π (x_0 : t) (x_0 x)) x))))
+  (check-equal? (term (reduce ,Σ (((((elim nat) zero) (λ (x : nat) nat))
+                                                      (s zero))
+                                   (λ (x : nat) (λ (ih-x : nat)
+                                                       (s (s x)))))))
+                (term (s zero)))
+  (check-equal? (term (reduce ,Σ (((((elim nat) (s zero)) (λ (x : nat) nat))
+                                                          (s zero))
+                                   (λ (x : nat) (λ (ih-x : nat)
+                                                       (s (s x)))))))
+                (term (s (s zero))))
+  (check-equal? (term (reduce ,Σ (((((elim nat) (s (s (s zero)))) (λ (x : nat) nat))
+                                                                  (s zero))
+                                   (λ (x : nat) (λ (ih-x : nat) (s (s x)))))))
+                (term (s (s (s (s zero))))))
+
+  (check-equal?
+    (term (reduce ,Σ
+                  (((((elim nat) (s (s zero))) (λ (x : nat) nat))
+                    (s (s zero)))
+                   (λ (x : nat) (λ (ih-x : nat) (s ih-x))))))
+    (term (s (s (s (s zero)))))))
+
+(define-judgment-form cic-redL
+  #:mode (equivalent I I I)
+  #:contract (equivalent Σ t t)
+
+  [(where t_2 (reduce Σ t_0))
+   (where t_3 (reduce Σ t_1))
+   (α-equivalent t_2 t_3)
+   ----------------- "≡-αβ"
+   (equivalent Σ t_0 t_1)])
+
+(define-extended-language cic-typingL cic-redL
+  ;; NB: There may be a bijection between Γ and Ξ. That's
+  ;; NB: interesting.
+  (Γ   ::= ∅ (Γ x : t)))
+(define Γ? (redex-match? cic-typingL Γ))
+
+(define-metafunction cic-typingL
+  append-Γ : Γ Γ -> Γ
+  [(append-Γ Γ ∅) Γ]
+  [(append-Γ Γ_2 (Γ_1 x : t))
+   ((append-Γ Γ_2 Γ_1) x : t)])
+
+;; NB: Depends on clause order
+(define-metafunction cic-typingL
+  lookup-Γ : Γ x -> t or #f
+  [(lookup-Γ ∅ x) #f]
+  [(lookup-Γ (Γ x : t) x) t]
+  [(lookup-Γ (Γ x_0 : t_0) x_1) (lookup-Γ Γ x_1)])
+
+;; NB: Depends on clause order
+(define-metafunction cic-redL
+  lookup-Σ : Σ x -> t or #f
+  [(lookup-Σ ∅ x) #f]
+  [(lookup-Σ (Σ (x : t ((x_1 : t_1) ...))) x) t]
+  [(lookup-Σ (Σ (x_0 : t_0 ((x_1 : t_1) ... (x : t) (x_2 : t_2) ...))) x) t]
+  [(lookup-Σ (Σ (x_0 : t_0 ((x_1 : t_1) ...))) x) (lookup-Σ Σ x)])
+
+;; NB: Depends on clause order
+(define-metafunction cic-typingL
+  remove : Γ x -> Γ
+  [(remove ∅ x) ∅]
+  [(remove (Γ x : t) x) Γ]
+  [(remove (Γ x_0 : t_0) x_1) (remove Γ x_1)])
+
+;; TODO: Add positivity checking.
+(define-metafunction cicL
+  positive : t any -> #t or #f
+  [(positive any_1 any_2) #t])
+;; NB: These tests may or may not fail because positivity checking is not implemented.
+(module+ test
+  (check-true (term (positive nat nat)))
+  (check-true (term (positive (Π (x : (Unv 0)) (Π (y : (Unv 0)) (Unv 0))) #f)))
+  (check-true (term (positive (Π (x : nat) nat) nat)))
+  ;; (nat -> nat) -> nat
+  ;; Not sure if this is actually supposed to pass
+  (check-false (term (positive (Π (x : (Π (y : nat) nat)) nat) nat)))
+  ;; ((Unv 0) -> nat) -> nat
+  (check-true (term (positive (Π (x : (Π (y : (Unv 0)) nat)) nat) nat)))
+  ;; (((nat -> (Unv 0)) -> nat) -> nat)
+  (check-true (term (positive (Π (x : (Π (y : (Π (x : nat) (Unv 0))) nat)) nat) nat)))
+  ;; Not sure if this is actually supposed to pass
+  (check-false (term (positive (Π (x : (Π (y : (Π (x : nat) nat)) nat)) nat) nat)))
+
+  (check-true (term (positive (Unv 0) #f))))
+
+;; Holds when the signature Σ and typing context Γ are well-formed.
+(define-judgment-form cic-typingL
+  #:mode (wf I I)
+  #:contract (wf Σ Γ)
+
+  [----------------- "WF-Empty"
+   (wf ∅ ∅)]
+
+  [(type-infer Σ Γ t t_0)
+   (wf Σ Γ)
+   ----------------- "WF-Var"
+   (wf Σ (Γ x : t))]
+
+  [(wf Σ ∅)
+   (type-infer Σ ∅ t_D U_D)
+   (type-infer Σ (∅ x_D : t_D) t_c U_c) ...
+   ;; NB: Ugh this should be possible with pattern matching alone ....
+   (side-condition ,(map (curry equal? (term Ξ_D)) (term (Ξ_D* ...))))
+   (side-condition ,(map (curry equal? (term x_D)) (term (x_D* ...))))
+   (side-condition (positive t_D (t_c ...)))
+   ----------------- "WF-Inductive"
+   (wf (Σ (x_D : (name t_D (in-hole Ξ_D t))
+               ;; Checks that a constructor for x actually produces an x, i.e., that
+               ;; the constructor is well-formed.
+               ((x_c : (name t_c (in-hole Ξ_D* (in-hole Φ (in-hole Θ x_D*))))) ...))) ∅)])
+(module+ test
+  (check-true (judgment-holds (wf ,Σ0 ∅)))
+  (check-true (redex-match? cic-redL (in-hole Ξ (Unv 0)) (term (Unv 0))))
+  (check-true (redex-match? cic-redL (in-hole Ξ (in-hole Φ (in-hole Θ nat)))
+                            (term (Π (x : nat) nat))))
+  (define (bindings-equal? l1 l2)
+    (map set=? l1 l2))
+  (check-pred
+    (curry bindings-equal?
+           (list (list
+                   (make-bind 'Ξ (term (Π (x : nat) hole)))
+                   (make-bind 'Φ (term hole))
+                   (make-bind 'Θ (term hole)))
+                 (list
+                   (make-bind 'Ξ (term hole))
+                   (make-bind 'Φ (term (Π (x : nat) hole)))
+                   (make-bind 'Θ (term hole)))))
+    (map match-bindings (redex-match cic-redL (in-hole Ξ (in-hole Φ (in-hole Θ nat)))
+                                     (term (Π (x : nat) nat)))))
+  (check-pred
+    (curry bindings-equal?
+           (list
+             (list
+               (make-bind 'Φ (term (Π (x : nat) hole)))
+               (make-bind 'Θ (term hole)))))
+    (map match-bindings (redex-match cic-redL (in-hole hole (in-hole Φ (in-hole Θ nat)))
+                                     (term (Π (x : nat) nat)))))
+
+  (check-true
+    (redex-match? cic-redL
+                  (in-hole hole (in-hole hole (in-hole hole nat)))
+                  (term nat)))
+  (check-true
+    (redex-match? cic-redL
+                  (in-hole hole (in-hole (Π (x : nat) hole) (in-hole hole nat)))
+                  (term (Π (x : nat) nat))))
+  (check-holds (wf (∅ (nat : (Unv 0) ())) ∅))
+
+  (check-holds (wf ,Σ0 ∅))
+  (check-holds (type-infer ∅ ∅ (Unv 0) U))
+  (check-holds (type-infer ∅ (∅ nat : (Unv 0)) nat U))
+  (check-holds (type-infer ∅ (∅ nat : (Unv 0)) (Π (x : nat) nat) U))
+  (check-true (term (positive nat (nat (Π (x : nat) nat)))))
+  (check-holds
+    (wf (∅ (nat : (Unv 0) ((zero : nat)))) ∅))
+  (check-holds
+    (wf (∅ (nat : (Unv 0) ((s : (Π (x : nat) nat))))) ∅))
+  (check-holds (wf ,Σ ∅))
+
+  (check-holds (wf ,Σ3 ∅))
+  (check-holds (wf ,Σ4 ∅))
+  (check-holds (wf (∅ (truth : (Unv 0) ())) ∅))
+  (check-holds (wf ∅ (∅ x : (Unv 0))))
+  (check-holds (wf (∅ (nat : (Unv 0) ())) (∅ x : nat)))
+  (check-holds (wf (∅ (nat : (Unv 0) ())) (∅ x : (Π (x : nat) nat)))))
+
+;; Returns the inductive hypotheses required for eliminating the
+;; inductively defined type x_D with motive t_P, where the telescope
+;; Φ are the inductive arguments to a constructor for x_D
+(define-metafunction cic-redL
+  hypotheses-for : x t Φ -> Φ
+  [(hypotheses-for x_D t_P hole) hole]
+  [(hypotheses-for x_D t_P (Π (x : (in-hole Φ (in-hole Θ x_D))) Φ_1))
+   ;; TODO: Thread through Σ for reduce
+   (Π (x_h : (in-hole Φ (reduce ∅ ((in-hole Θ t_P) (apply-telescope x Φ)))))
+      (hypotheses-for x_D t_P Φ_1))
+   ;; NB: Lol hygiene
+   (where x_h ,(string->symbol (format "~a-~a" 'ih (term x))))])
+
+;; Returns the inductive arguments to a constructor for the
+;; inducitvely defined type x_D, where the telescope Φ are the
+;; non-parameter arguments to the constructor.
+(define-metafunction cic-redL
+  inductive-args : x Φ -> Φ
+  [(inductive-args x_D hole) hole]
+  [(inductive-args x_D (Π (x : (in-hole Φ (in-hole Θ x_D))) Φ_1))
+   (Π (x : (in-hole Φ (in-hole Θ x_D))) (inductive-args x_D Φ_1))]
+  ;; NB: Depends on clause order
+  [(inductive-args x_D (Π (x : t) Φ_1))
+   (inductive-args x_D Φ_1)])
+
+;; Returns the methods required for eliminating the inductively
+;; defined type x_D, whose parameters/indices are Ξ_pi and whose
+;; constructors are ((x_ci : t_ci) ...), with motive t_P.
+(define-metafunction cic-redL
+  methods-for : x Ξ t ((x : t) ...) -> Ξ
+  [(methods-for x_D Ξ_pi t_P ()) hole]
+  [(methods-for x_D Ξ_pi t_P ((x_ci : (in-hole Ξ_pi (in-hole Φ (in-hole Θ x_D))))
+                                    (x_c : t) ...))
+   (Π (x_mi : (in-hole Ξ_pi (in-hole Φ (in-hole Φ_h
+                                                ;; NB: Manually reducing types because no conversion
+                                                ;; NB: rule
+                                                ;; TODO: Thread through Σ for reduce
+                                                ;; TODO: Might be able to remove this now that I have
+                                                ;; TODO: equivalence in type-check
+                                                (reduce ∅ ((in-hole Θ t_P) (apply-telescopes x_ci (Ξ_pi Φ))))))))
+      (methods-for x_D Ξ_pi t_P ((x_c : t) ...)))
+   (where Φ_h (hypotheses-for x_D t_P (inductive-args x_D Φ)))
+   ;; NB: Lol hygiene
+   (where x_mi ,(string->symbol (format "~a-~a" 'm (term x_ci))))])
+(module+ test
+  (check-equal?
+    (term (methods-for nat hole P ((zero : nat) (s : (Π (x : nat) nat)))))
+    (term (Π (m-zero : (P zero))
+             (Π (m-s : (Π (x : nat) (Π (ih-x : (P x)) (P (s x)))))
+                hole))))
+  (check-equal?
+    (term (methods-for nat hole (λ (x : nat) nat) ((zero : nat) (s : (Π (x : nat) nat)))))
+    (term (Π (m-zero : nat) (Π (m-s : (Π (x : nat) (Π (ih-x : nat) nat)))
+                                   hole))))
+  (check-equal?
+    (term (methods-for and (Π (A : Type) (Π (B : Type) hole))
+                       (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B)) true)))
+                       ((conj : (Π (A : Type) (Π (B : Type) (Π (a : A) (Π (b : B)
+                                                                             ((and A) B)))))))))
+    (term (Π (m-conj : (Π (A : Type) (Π (B : Type) (Π (a : A) (Π (b : B) true)))))
+             hole)))
+  (check-true (x? (term false)))
+  (check-true (Ξ? (term hole)))
+  (check-true (t? (term (λ (y : false) (Π (x : Type) x)))))
+  (check-true (redex-match? cicL ((x : t) ...) (term ())))
+  (check-equal?
+    (term (methods-for false hole (λ (y : false) (Π (x : Type) x))
+                       ()))
+    (term hole)))
+
+;; Returns the inductively defined type that x constructs
+;; NB: Depends on clause order
+(define-metafunction cic-redL
+  constructor-of : Σ x -> x
+  [(constructor-of (Σ (x : t ((x_0 : t_0) ... (x_c : t_c) (x_1 : t_1) ...)))
+                   x_c) x]
+  [(constructor-of (Σ (x_1 : t_1 ((x_c : t) ...))) x)
+   (constructor-of Σ x)])
+(module+ test
+  (check-equal?
+    (term (constructor-of ,Σ zero))
+    (term nat))
+  (check-equal?
+    (term (constructor-of ,Σ s))
+    (term nat)))
+
+;; Returns the constructors for the inductively defined type x_D in
+;; the signature Σ
+(define-metafunction cic-redL
+  constructors-for : Σ x -> ((x : t) ...) or #f
+  ;; NB: Depends on clause order
+  [(constructors-for ∅ x_D) #f]
+  [(constructors-for (Σ (x_D : t_D ((x : t) ...))) x_D)
+   ((x : t) ...)]
+  [(constructors-for (Σ (x_1 : t_1 ((x : t) ...))) x_D)
+   (constructors-for Σ x_D)])
+(module+ test
+  (check-equal?
+    (term (constructors-for ,Σ nat))
+    (term ((zero : nat) (s : (Π (x : nat) nat)))))
+  (check-equal?
+    (term (constructors-for ,sigma false))
+    (term ())))
+
+;; Holds when an apply context Θ provides arguments that match the
+;; telescope Ξ
+(define-judgment-form cic-typingL
+  #:mode (telescope-types I I I I)
+  #:contract (telescope-types Σ Γ Θ Ξ)
+
+  [----------------- "TT-Hole"
+   (telescope-types Σ Γ hole hole)]
+
+  [(type-check Σ Γ e t)
+   (telescope-types Σ Γ Θ Ξ)
+   ----------------- "TT-Match"
+   (telescope-types Σ Γ (in-hole Θ (hole e)) (Π (x : t) Ξ))])
+(module+ test
+  (check-holds
+    (telescope-types ,Σ ∅ (hole zero) (Π (x : nat) hole)))
+  (check-true
+    (redex-match? cic-redL (in-hole Θ (hole e))
+                  (term ((hole zero) (λ (x : nat) x)))))
+  (check-holds
+    (telescope-types ,Σ ∅ (hole zero)
+                     (methods-for nat hole
+                                  (λ (x : nat) nat)
+                                  ((zero : nat)))))
+  (check-holds
+    (type-check ,Σ ∅ (λ (x : nat) (λ (ih-x : nat) x))
+                (Π (x : nat) (Π (ih-x : nat) nat))))
+  (check-holds
+    (telescope-types ,Σ ∅
+                     ((hole zero)
+                      (λ (x : nat) (λ (ih-x : nat) x)))
+                     (methods-for nat hole
+                                  (λ (x : nat) nat)
+                                  (constructors-for ,Σ nat))))
+  (check-holds
+    (telescope-types (,Σ4 (true : (Unv 0) ((tt : true))))
+                     ∅ (hole (λ (A : (Unv 0)) (λ (B : (Unv 0))
+                                                    (λ (a : A) (λ (b : B) tt)))))
+                     (methods-for and (Π (A : Type) (Π (B : Type) hole))
+                                  (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B)) true)))
+                                  (constructors-for ,Σ4 and))))
+  (check-holds
+    (telescope-types ,sigma (∅ x : false)
+                     hole
+                     (methods-for false hole (λ (y : false) (Π (x : Type) x))
+                                  ()))))
+
+;; TODO: Bi-directional and inference?
+;; TODO: http://www.cs.ox.ac.uk/ralf.hinze/WG2.8/31/slides/stephanie.pdf
+
+;; Holds when e has type t under signature Σ and typing context Γ
+(define-judgment-form cic-typingL
+  #:mode (type-infer I I I O)
+  #:contract (type-infer Σ Γ e t)
+
+  [(unv-ok U_0 U_1)
+   (wf Σ Γ)
+   ----------------- "DTR-Axiom"
+   (type-infer Σ Γ U_0 U_1)]
+
+  [(where t (lookup-Σ Σ x))
+   ----------------- "DTR-Inductive"
+   (type-infer Σ Γ x t)]
+
+  [(where t (lookup-Γ Γ x))
+   ----------------- "DTR-Start"
+   (type-infer Σ Γ x t)]
+
+  [(type-infer Σ Γ t_0 U_1)
+   (type-infer Σ (Γ x : t_0) t U_2)
+   (unv-kind U_1 U_2 U)
+   ----------------- "DTR-Product"
+   (type-infer Σ Γ (Π (x : t_0) t) U)]
+
+  [(type-infer Σ Γ e_0 (Π (x_0 : t_0) t_1))
+   (type-infer Σ Γ e_1 t_2)
+   (equivalent Σ t_0 t_2)
+   ----------------- "DTR-Application"
+   (type-infer Σ Γ (e_0 e_1) (subst t_1 x_0 e_1))]
+
+  [(type-infer Σ (Γ x : t_0) e t_1)
+   (type-infer Σ Γ (Π (x : t_0) t_1) U)
+   ----------------- "DTR-Abstraction"
+   (type-infer Σ Γ (λ (x : t_0) e) (Π (x : t_0) t_1))]
+
+  [(type-infer Σ Γ x_D (in-hole Ξ U_D))
+   (type-infer Σ Γ e_D (in-hole Θ_ai x_D))
+   (type-infer Σ Γ e_P (in-hole Ξ_1 (Π (x : (in-hole Θ_Ξ x_D)) U_P)))
+   (equivalent Σ (in-hole Ξ (Unv 0)) (in-hole Ξ_1 (Unv 0)))
+   ;; methods
+   (telescope-types Σ Γ Θ_m (methods-for x_D Ξ e_P (constructors-for Σ x_D)))
+   ----------------- "DTR-Elim_D"
+   (type-infer Σ Γ (in-hole Θ_m (((elim x_D) e_D) e_P)) (reduce Σ ((in-hole Θ_ai e_P) e_D)))])
+
+(define-judgment-form cic-typingL
+  #:mode (type-check I I I I)
+  #:contract (type-check Σ Γ e t)
+
+  [(type-infer Σ Γ e t_0)
+   (equivalent Σ t t_0)
+   ----------------- "DTR-Check"
+   (type-check Σ Γ e t)])
+(module+ test
+  (check-holds (type-infer ∅ ∅ (Unv 0) (Unv 1)))
+  (check-holds (type-infer ∅ (∅ x : (Unv 0)) (Unv 0) (Unv 1)))
+  (check-holds (type-infer ∅ (∅ x : (Unv 0)) x (Unv 0)))
+  (check-holds (type-infer ∅ ((∅ x_0 : (Unv 0)) x_1 : (Unv 0))
+                           (Π (x_3 : x_0) x_1) (Unv 0)))
+  (check-holds (type-infer ∅ (∅ x_0 : (Unv 0)) x_0 U_1))
+  (check-holds (type-infer ∅ ((∅ x_0 : (Unv 0)) x_2 : x_0) (Unv 0) U_2))
+  (check-holds (unv-kind (Unv 0) (Unv 0) (Unv 0)))
+  (check-holds (type-infer ∅ (∅ x_0 : (Unv 0)) (Π (x_2 : x_0) (Unv 0)) t))
+
+  (check-holds (type-infer ∅ ∅ (λ (x : (Unv 0)) x) (Π (x : (Unv 0)) (Unv 0))))
+  (check-holds (type-infer ∅ ∅ (λ (y : (Unv 0)) (λ (x : y) x))
+                           (Π (y : (Unv 0)) (Π (x : y) y))))
+
+  (check-equal? (list (term (Unv 1)))
+                (judgment-holds
+                  (type-infer ∅ ((∅ x1 : (Unv 0)) x2 : (Unv 0)) (Π (t6 : x1) (Π (t2 : x2) (Unv 0)))
+                              U)
+                  U))
+  ;; ---- Elim
+  ;; TODO: Clean up/Reorganize these tests
+  (check-true
+    (redex-match? cic-typingL
+                  (in-hole Θ_m (((elim x_D) e_D) e_P))
+                  (term ((((elim truth) T) (Π (x : truth) (Unv 1))) (Unv 0)))))
+  (define Σtruth (term (∅ (truth : (Unv 0) ((T : truth))))))
+  (check-holds (type-infer ,Σtruth ∅ truth (in-hole Ξ U)))
+  (check-holds (type-infer ,Σtruth ∅ T (in-hole Θ_ai truth)))
+  (check-holds (type-infer ,Σtruth ∅ (λ (x : truth) (Unv 1))
+                           (in-hole Ξ (Π (x : (in-hole Θ truth)) U))))
+  (check-equal?
+    (term (methods-for truth hole (λ (x : truth) (Unv 1)) ((T : truth))))
+    (term (Π (m-T : (Unv 1)) hole)))
+  (check-holds (telescope-types ,Σtruth ∅ (hole (Unv 0))
+                                (methods-for truth hole
+                                             (λ (x : truth) (Unv 1))
+                                             ((T : truth)))))
+  (check-holds (type-check (∅ (truth : (Unv 0) ((T : truth))))
+                           ∅
+                           ((((elim truth) T) (λ (x : truth) (Unv 1))) (Unv 0))
+                           (Unv 1)))
+
+  (check-not-holds (type-check (∅ (truth : (Unv 0) ((T : truth))))
+                               ∅
+                               (((((elim truth) T) (Unv 1)) Type) Type)
+                               (Unv 1)))
+  (check-holds
+    (type-infer ∅ ∅ (Π (x2 : (Unv 0)) (Unv 0)) U))
+  (check-holds
+    (type-infer ∅ (∅ x1 : (Unv 0)) (λ (x2 : (Unv 0)) (Π (t6 : x1) (Π (t2 : x2) (Unv 0))))
+                t))
+  (define-syntax-rule (nat-test syn ...)
+                      (check-holds (type-infer ,Σ syn ...)))
+  (nat-test ∅ (Π (x : nat) nat) (Unv 0))
+  (nat-test ∅ (λ (x : nat) x) (Π (x : nat) nat))
+
+  (check-holds
+    (type-infer ,Σ ∅ nat (in-hole Ξ U)))
+  (check-holds
+    (type-infer ,Σ ∅ zero (in-hole Θ_ai nat)))
+  (check-holds
+    (type-infer ,Σ ∅ (λ (x : nat) nat)
+                (in-hole Ξ (Π (x : (in-hole Θ nat)) U))))
+  (nat-test ∅ (((((elim nat) zero) (λ (x : nat) nat)) zero)
+                                   (λ (x : nat) (λ (ih-x : nat) x)))
+            nat)
+  (nat-test ∅ nat (Unv 0))
+  (nat-test ∅ zero nat)
+  (nat-test ∅ s (Π (x : nat) nat))
+  (nat-test ∅ (s zero) nat)
+  (nat-test ∅ (((((elim nat) zero) (λ (x : nat) nat))
+                             (s zero))
+                                   (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
+            nat)
+  (nat-test (∅ n : nat)
+            (((((elim nat) n) (λ (x : nat) nat)) zero) (λ (x : nat) (λ (ih-x : nat) x)))
+            nat)
+  (check-holds
+    (type-check (,Σ (bool : (Unv 0) ((btrue : bool) (bfalse : bool))))
+                (∅ n2 : nat)
+                (((((elim nat) n2) (λ (x : nat) bool)) btrue) (λ (x : nat) (λ (ih-x : bool) bfalse)))
+                bool))
+  (check-not-holds
+    (type-check ,Σ ∅
+                ((((elim nat) zero) nat) (s zero))
+                nat))
+  (define lam (term (λ (nat : (Unv 0)) nat)))
+  (check-equal?
+    (list (term (Π (nat : (Unv 0)) (Unv 0))))
+    (judgment-holds (type-infer ,Σ0 ∅ ,lam t) t))
+  (check-equal?
+    (list (term (Π (nat : (Unv 0)) (Unv 0))))
+    (judgment-holds (type-infer ,Σ ∅ ,lam t) t))
+  (check-equal?
+    (list (term (Π (x : (Π (y : (Unv 0)) y)) nat)))
+    (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (x : (Π (y : (Unv 0)) y)) (x nat))
+                                t) t))
+  (check-equal?
+    (list (term (Π (y : (Unv 0)) (Unv 0))))
+    (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (y : (Unv 0)) y) t) t))
+  (check-equal?
+    (list (term (Unv 0)))
+    (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅
+                                ((λ (x : (Π (y : (Unv 0)) (Unv 0))) (x nat))
+                                 (λ (y : (Unv 0)) y))
+                                t) t))
+  (check-equal?
+    (list (term (Unv 0)) (term (Unv 1)))
+    (judgment-holds
+      (type-infer ,Σ4 ∅ (Π (S : (Unv 0)) (Π (B : (Unv 0)) (Π (a : S) (Π (b : B) ((and S) B)))))
+                  U) U))
+  (check-holds
+    (type-check ,Σ4 (∅ S : (Unv 0)) conj (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Π (a : A) (Π (b : B) ((and A) B)))))))
+  (check-holds
+    (type-check ,Σ4 (∅ S : (Unv 0))
+                conj (Π (P : (Unv 0)) (Π (Q : (Unv 0)) (Π (x : P) (Π (y : Q) ((and P) Q)))))))
+  (check-holds
+    (type-check ,Σ4 (∅ S : (Unv 0)) S (Unv 0)))
+  (check-holds
+    (type-check ,Σ4 (∅ S : (Unv 0)) (conj S)
+                (Π (B : (Unv 0)) (Π (a : S) (Π (b : B) ((and S) B))))))
+  (check-holds
+    (type-check ,Σ4 (∅ S : (Unv 0)) (conj S)
+                (Π (B : (Unv 0)) (Π (a : S) (Π (b : B) ((and S) B))))))
+  (check-holds
+    (type-check ,Σ4 ∅ (λ (S : (Unv 0)) (conj S))
+                (Π (S : (Unv 0)) (Π (B : (Unv 0)) (Π (a : S) (Π (b : B) ((and S) B)))))))
+  (check-holds
+    (type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                ((((conj true) true) tt) tt)
+                ((and true) true)))
+  (check-holds
+    (type-infer ,Σ4 ∅ and (in-hole Ξ U_D)))
+  (check-holds
+    (type-infer (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                ((((conj true) true) tt) tt)
+                (in-hole Θ and)))
+  (check-holds
+    (type-infer (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B)) true)))
+                (in-hole Ξ (Π (x : (in-hole Θ_Ξ and)) U_P))))
+  (check-holds
+    (type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                ((((elim and) ((((conj true) true) tt) tt))
+                  (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
+                                                 true))))
+                 (λ (A : (Unv 0))
+                    (λ (B : (Unv 0))
+                       (λ (a : A)
+                          (λ (b : B) tt)))))
+                true))
+  (check-true (Γ? (term (((∅ P : (Unv 0)) Q : (Unv 0)) ab : ((and P) Q)))))
+  (check-holds
+    (type-infer ,Σ4 ∅ and (in-hole Ξ U)))
+  (check-holds
+    (type-infer ,Σ4 (((∅ P : Type) Q : Type) ab : ((and P) Q))
+                ab (in-hole Θ and)))
+  (check-true
+    (redex-match? cic-redL
+                  (in-hole Ξ (Π (x : (in-hole Θ and)) U))
+                  (term (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Π (x : ((and A) B)) (Unv 0)))))))
+  (check-holds
+    (type-infer ,Σ4 (((∅ P : Type) Q : Type) ab : ((and P) Q))
+                (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
+                                               ((and B) A))))
+                (in-hole Ξ (Π (x : (in-hole Θ and)) U))))
+  (check-holds
+    (equivalent ,Σ4
+                (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Π (x : ((and A) B)) (Unv 0))))
+                (Π (P : (Unv 0)) (Π (Q : (Unv 0)) (Π (x : ((and P) Q)) (Unv 0))))))
+  (check-holds
+    (type-infer ,Σ4 ∅
+                (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
+                                               ((and B) A))))
+                (in-hole Ξ (Π (x : (in-hole Θ_Ξ and)) U_P))))
+  (check-holds
+    (type-check ,Σ4
+                (((∅ P : (Unv 0)) Q : (Unv 0)) ab : ((and P) Q))
+                ((((elim and) ab)
+                  (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
+                                                 ((and B) A)))))
+                 (λ (A : (Unv 0))
+                    (λ (B : (Unv 0))
+                       (λ (a : A)
+                          (λ (b : B) ((((conj B) A) b) a))))))
+                ((and Q) P)))
+  (check-holds
+    (type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B)) ((and B) A))))
+                (Π (A : (Unv 0)) (Π (B : (Unv 0)) (Π (x : ((and A) B)) (Unv 0))))))
+  (check-holds
+    (type-infer (,Σ4 (true : (Unv 0) ((tt : true))))
+                ((∅ A : Type) B : Type)
+                (conj B)
+                t))
+  (check-holds
+    (type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
+                ((((elim and) ((((conj true) true) tt) tt))
+                  (λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
+                                                 ((and B) A)))))
+                 (λ (A : (Unv 0))
+                    (λ (B : (Unv 0))
+                       (λ (a : A)
+                          (λ (b : B) ((((conj B) A) b) a))))))
+                ((and true) true)))
+  (define gamma (term (∅ temp863 : pre)))
+  (check-holds (wf ,sigma ∅))
+  (check-holds (wf ,sigma ,gamma))
+  (check-holds
+    (type-infer ,sigma ,gamma (Unv 0) t))
+  (check-holds
+    (type-infer ,sigma ,gamma pre t))
+  (check-holds
+    (type-check ,sigma (,gamma tmp863 : pre) (Unv 0) (Unv 1)))
+  (check-holds
+    (type-infer ,sigma ,gamma pre t))
+  (check-holds
+    (type-check ,sigma (,gamma tmp863 : pre) (Unv 0) (Unv 1)))
+  (check-holds
+    (type-infer ,sigma (,gamma x : false) false (in-hole Ξ U_D)))
+  (check-holds
+    (type-infer ,sigma (,gamma x : false) x (in-hole Θ false)))
+  (check-holds
+    (type-infer ,sigma (,gamma x : false) (λ (y : false) (Π (x : Type) x))
+                (in-hole Ξ (Π (x : (in-hole Θ false)) U))))
+  (check-true
+    (redex-match? cic-typingL (in-hole Θ_m (((elim x_D) e_D) e_P))
+                  (term (((elim false) x) (λ (y : false) (Π (x : Type) x))))))
+  (check-holds
+    (type-check ,sigma (,gamma x : false)
+                (((elim false) x) (λ (y : false) (Π (x : Type) x)))
+                (Π (x : (Unv 0)) x))))

curnel/redex-lang.rkt

@@ -0,0 +1,437 @@
+#lang racket/base
+;; This module just provide module language sugar over the redex model.
+
+;; TODO: Strip to racket/base as much as possible.
+;; TODO: Remove trace,pretty, debugging stuff
+(require
+  racket/pretty
+  "redex-core.rkt"
+  redex/reduction-semantics
+  racket/provide-syntax
+  (for-syntax
+    (except-in racket import)
+    syntax/parse
+    racket/pretty
+    racket/syntax
+    (except-in racket/provide-transform export)
+    racket/require-transform
+    (except-in "redex-core.rkt" remove)
+    redex/reduction-semantics))
+(provide
+  ;; Basic syntax
+  for-syntax
+  only-in
+  all-defined-out
+  rename-in
+  prefix-in
+  submod
+  #%module-begin
+  begin
+  (rename-out
+    [dep-module+ module+]
+    [dep-provide provide]
+    [dep-require require]
+
+    [dep-lambda lambda]
+    [dep-lambda λ]
+    [dep-app #%app]
+
+    [dep-forall forall]
+    [dep-forall ∀]
+
+    [dep-inductive data]
+
+    [dep-elim elim]
+
+    [dep-var #%top]
+
+    ;      [dep-datum #%datum]
+    [dep-define define])
+  Type
+  ;; DYI syntax extension
+  define-syntax
+  begin-for-syntax
+  define-for-syntax
+  (for-syntax (all-from-out syntax/parse))
+  syntax-case
+  syntax-rules
+  define-syntax-rule
+  (for-syntax (all-from-out racket))
+  ;; reflection
+  (for-syntax
+    cur-expand
+    type-infer/syn
+    type-check/syn?
+    normalize/syn
+    cur-equal?)
+  run)
+
+(begin-for-syntax
+  ;; TODO: Gamma and Sigma seem to get reset inside a module+
+  (define gamma
+    (make-parameter (term ∅)
+                    (lambda (x)
+                      (unless (Γ? x)
+                        (error 'core-error "We built a bad gamma ~s" x))
+                      x)))
+
+  (define sigma
+    (make-parameter (term ∅)
+                    (lambda (x)
+                      (unless (Σ? x)
+                        (error 'core-error "We built a bad sigma ~s" x))
+                      x)))
+
+  (define (extend-Γ/term env x t)
+    (term (,(env) ,x : ,t)))
+
+  (define (extend-Γ/term! env x t) (env (extend-Γ/term env x t)))
+
+  (define (extend-Γ/syn env x t)
+    (extend-Γ/term env (syntax->datum x) (cur->datum t)))
+
+  (define (extend-Γ/syn! env x t) (env (extend-Γ/syn env x t)))
+
+  (define (extend-Σ/term env x t c*)
+    (term (,(env) (,x : ,t (,@c*)))))
+
+  (define (extend-Σ/term! env x t c*)
+    (env (extend-Σ/term env x t c*)))
+
+  (define (extend-Σ/syn env x t c*)
+    (extend-Σ/term env (syntax->datum x) (cur->datum t)
+                   (for/list ([c (syntax->list c*)])
+                     (syntax-case c ()
+                       [(c : ct)
+                        (parameterize ([gamma (extend-Γ/syn gamma x t)])
+                          (term (,(syntax->datum #'c) : ,(cur->datum #'ct))))]))))
+
+  (define (extend-Σ/syn! env x t c*)
+    (env (extend-Σ/syn env x t c*)))
+
+  (define bind-subst (make-parameter (list null null)))
+
+  (define (add-binding/term! x t)
+    (let ([vars (first (bind-subst))]
+          [exprs (second (bind-subst))])
+      (bind-subst (list (cons x vars) (cons t exprs)))))
+
+  ;; TODO: Still absurdly slow. Probably doing n^2 checks of sigma and
+  ;; gamma. And lookup on sigma, gamma are linear, so probably n^2 lookup time.
+  (define (type-infer/term t)
+    (let ([t (judgment-holds (type-infer ,(sigma) ,(gamma) ,t t_0) t_0)])
+      (and (pair? t) (car t))))
+
+  (define (type-check/term? e t)
+    (and (judgment-holds (type-check ,(sigma) ,(gamma) ,e ,t)) #t))
+
+  (define (syntax->curnel-syntax syn) (denote syn (cur->datum syn)))
+
+  (define (denote syn t)
+    (quasisyntax/loc
+      syn
+      (term (reduce #,(sigma) (subst-all #,(datum->syntax syn t) #,(first (bind-subst)) #,(second (bind-subst)))))))
+
+  ;; TODO: Blanket disarming is probably a bad idea.
+  (define orig-insp (variable-reference->module-declaration-inspector
+                      (#%variable-reference)))
+  (define (disarm syn) (syntax-disarm syn orig-insp))
+
+  ;; Locally expand everything down to core forms.
+  (define (core-expand syn)
+    (disarm
+      (local-expand
+        syn
+        'expression
+        (append (syntax-e #'(term reduce subst-all dep-var #%app λ Π elim Unv #%datum))))))
+
+  ;; Only type-check at the top-level, to prevent exponential
+  ;; type-checking. Redex is expensive enough.
+  ;; TODO: This results in less good error messages. Add an
+  ;; algorithm to find the smallest ill-typed term.
+  (define inner-expand? (make-parameter #f))
+
+  ;; Expand a piece of syntax into a curnel redex term
+  (define (cur->datum syn)
+    ;; Main loop; avoid type
+    (define reified-term
+      (parameterize ([inner-expand? #t])
+        (let cur->datum ([syn syn])
+          (syntax-parse (core-expand syn)
+            #:literals (term reduce #%app subst-all)
+            #:datum-literals (elim Π λ : Unv)
+            [x:id (syntax->datum #'x)]
+            [(subst-all e _ _) (syntax->datum #'e)]
+            [(reduce Σ e) (cur->datum #'e)]
+            [(term e) (cur->datum #'e)]
+            [(Unv i) (term (Unv ,(syntax->datum #'i)))]
+            ;; TODO: should really check that b is one of the binders
+            ;; Maybe make a syntax class for the binders, core forms,
+            ;; etc.
+            [(b:id (x:id : t) e)
+             (let* ([x (syntax->datum #'x)]
+                    [t (cur->datum #'t)]
+                    [e (parameterize ([gamma (extend-Γ/term gamma x t)])
+                         (cur->datum #'e))])
+               (term (,(syntax->datum #'b) (,x : ,t) ,e)))]
+            [(elim t e P m ...)
+             (let* ([t (cur->datum #'t)]
+                    [e (cur->datum #'e)]
+                    [P (cur->datum #'P)]
+                    [e (term (((elim ,t) ,e) ,P))])
+               (for/fold ([e e])
+                         ([m (syntax->list #'(m ...))])
+                 (term (,e ,(cur->datum m)))))]
+            [(#%app e1 e2)
+             (term (,(cur->datum #'e1) ,(cur->datum #'e2)))]))))
+    (unless (and inner-expand? (type-infer/term reified-term))
+      ;; TODO: is this really a syntax error?
+      (raise-syntax-error 'cur "term is ill-typed:"
+                          (begin (printf "Sigma: ~s~nGamma: ~s~n" (sigma) (gamma))
+                                 reified-term)
+                          syn))
+    reified-term)
+
+  ;; Reflection tools
+  (define (normalize/syn syn)
+    (denote
+      syn
+      (term (reduce ,(sigma) (subst-all ,(cur->datum syn) ,(first (bind-subst)) ,(second (bind-subst)))))))
+
+  (define (run-cur->datum syn)
+    (cur->datum (normalize/syn syn)))
+
+  ;; Are these two terms equivalent in type-systems internal equational reasoning?
+  (define (cur-equal? e1 e2)
+    (and (judgment-holds (equivalent ,(sigma) ,(run-cur->datum e1) ,(run-cur->datum e2)) #t)))
+
+  ;; TODO: OOps, type-infer doesn't return a cur term but a redex term
+  ;; wrapped in syntax bla. This is bad.
+  (define (type-infer/syn syn)
+    (let ([t (type-infer/term (run-cur->datum syn))])
+      (and t (datum->syntax syn t))))
+
+  (define (type-check/syn? syn type)
+    (type-check/term? (run-cur->datum syn) (run-cur->datum type)))
+
+  ;; Takes a Cur term syn and an arbitrary number of identifiers ls. The cur term is
+  ;; expanded until expansion reaches a Curnel form, or one of the
+  ;; identifiers in ls.
+  (define (cur-expand syn . ls)
+    (disarm
+      (local-expand
+        syn
+        'expression
+        (append (syntax-e #'(Type dep-inductive dep-lambda dep-app dep-elim dep-forall dep-var))
+                ls)))))
+
+;; TODO: OOps, run doesn't return a cur term but a redex term
+;; wrapped in syntax bla. This is bad.
+(define-syntax (run syn)
+  (syntax-case syn ()
+    [(_ expr) (normalize/syn #'expr)]))
+
+;; -----------------------------------------------------------------
+;; Require/provide macros
+
+#| TODO NB XXX
+ | This is code some of the most hacky awful code I've ever written. But it works. ish
+ |#
+(begin-for-syntax
+  (define envs (list #'(void)))
+
+  (define (cur-identifier-bound? id)
+    (let ([x (syntax->datum id)])
+      (and (x? x)
+        (or (term (lookup-Γ ,(gamma) ,x))
+          (term (lookup-Σ ,(sigma) ,x))))))
+
+  (define (filter-cur-exports syn modes)
+    (partition (compose cur-identifier-bound? export-local-id)
+               (apply append (map (lambda (e) (expand-export e modes))
+                                  (syntax->list syn))))))
+(define-syntax extend-env-and-provide
+  (make-provide-transformer
+    (lambda (syn modes)
+      (syntax-case syn ()
+        [(_ e ...)
+         (let-values ([(cur ~cur) (filter-cur-exports #'(e ...) modes)])
+           #| TODO: Ignoring the built envs for now
+           (set! envs (for/list ([e cur])
+           (let* ([x (syntax->datum (export-local-id e))]
+                  [t (type-infer/term x)]
+                  [env (if (term (lookup ,(gamma) ,x)) #'gamma #'sigma)])
+             #`(extend-env/term! #,env #,(export-out-sym e) #,t))))
+           |#
+        ~cur)]))))
+
+(define-syntax (export-envs syn)
+  (syntax-case syn ()
+    [(_ gamma-out sigma-out bind-out)
+     #`(begin-for-syntax
+         (define gamma-out (term #,(gamma)))
+         (define sigma-out (term #,(sigma)))
+         (define bind-out '#,(bind-subst)))]))
+
+;; TODO: This can only handle a single provide form, otherwise generates multiple *-out
+(define-syntax (dep-provide syn)
+  (syntax-case syn ()
+    [(_ e ...)
+     (begin
+       #| TODO NB:
+        | Ignoring the built envs above, for now.  The local-lift export seems to get executed before
+        | the filtered environment is built.
+        |#
+       ;; TODO: rename out will need to rename variables in gamma and ; sigma.
+       (syntax-local-lift-module-end-declaration
+         #`(export-envs gamma-out sigma-out bind-out))
+       #`(provide (extend-env-and-provide e ...)
+                  (for-syntax gamma-out sigma-out bind-out)))]))
+(begin-for-syntax
+  (define out-gammas #`())
+  (define out-sigmas #`())
+  (define out-binds #`())
+  (define gn 0)
+  (define sn 0)
+  (define bn 0)
+  (define (filter-cur-imports syn)
+    (for/fold ([imports '()]
+               [sources '()])
+              ([req-spec (syntax->list syn)])
+      (let-values ([(more-imports more-sources) (expand-import req-spec)])
+        (values (for/fold ([imports imports])
+                          ([imp more-imports])
+                  (cond
+                    [(equal? (import-src-sym imp) 'gamma-out)
+                     (let ([new-id (format-id (import-orig-stx imp)
+                                              "gamma-out~a" gn)])
+                       ;; TODO: Fewer set!s
+                       ;; TODO: Do not DIY gensym
+                       (set! gn (add1 gn))
+                       (set! out-gammas
+                             #`(#,@out-gammas (gamma (term (append-Γ
+                                                             ,(gamma)
+                                                             ,#,new-id)))))
+                       (cons (struct-copy import imp [local-id new-id])
+                             imports))]
+                    ;; TODO: Many shared code between these two clauses
+                    [(equal? (import-src-sym imp) 'sigma-out)
+                     (let ([new-id (format-id (import-orig-stx imp)
+                                              "sigma-out~a" sn)])
+                       ;; TODO: Fewer set!s
+                       ;; TODO: Do not DIY gensym
+                       (set! sn (add1 sn))
+                       (set! out-sigmas
+                             #`(#,@out-sigmas (sigma (term (append-Σ
+                                                             ,(sigma)
+                                                             ,#,new-id)))))
+                       (cons (struct-copy import imp [local-id new-id])
+                             imports))]
+                    ;; TODO: Many shared code between these two clauses
+                    [(equal? (import-src-sym imp) 'bind-out)
+                     (let ([new-id (format-id (import-orig-stx imp)
+                                              "bind-out~a" bn)])
+                       ;; TODO: Fewer set!s
+                       ;; TODO: Do not DIY gensym
+                       (set! bn (add1 bn))
+                       (set! out-binds
+                             #`(#,@out-binds (bind-subst (list (append
+                                                                 (first #,new-id)
+                                                                 (first (bind-subst)))
+                                                               (append
+                                                                 (second #,new-id)
+                                                                 (second (bind-subst)))))))
+                       (cons (struct-copy import imp [local-id new-id])
+                             imports))]
+                    [else (cons imp imports)]))
+                (append sources more-sources))))))
+
+(define-syntax extend-env-and-require
+  (make-require-transformer
+    (lambda (syn)
+      (syntax-case syn ()
+        [(_ e ...) (filter-cur-imports #'(e ...))]))))
+
+;; TODO: rename in will need to rename variables in gamma and
+;; sigma.
+(define-syntax (import-envs syn)
+  (syntax-case syn ()
+    [(_) #`(begin-for-syntax #,@out-gammas #,@out-sigmas
+                             #,@out-binds)]))
+
+(define-syntax (dep-require syn)
+  (syntax-case syn ()
+    [(_ e ...)
+     #`(begin
+         (require (extend-env-and-require e ...))
+         (import-envs))]))
+
+(define-syntax (dep-module+ syn)
+  (syntax-case syn ()
+    [(_ name body ...)
+     #`(module+ name
+         (begin-for-syntax
+           (gamma (term #,(gamma)))
+           (sigma (term #,(sigma)))
+           (bind-subst '#,(bind-subst)))
+         body ...)]))
+
+;; -----------------------------------------------------------------
+;; Core wrapper macros
+;;
+;; TODO: Can these be simplified further?
+#;(define-syntax (dep-datum syn) (denote #'syn))
+(define-syntax (dep-lambda syn)
+  (syntax-case syn (:)
+    [(_ (x : t) e)
+     (syntax->curnel-syntax
+       (quasisyntax/loc syn (λ (x : t) e)))]))
+
+(define-syntax (dep-app syn)
+  (syntax-case syn ()
+    [(_ e1 e2)
+     (syntax->curnel-syntax
+       (quasisyntax/loc syn (#%app e1 e2)))]))
+
+(define-syntax (dep-forall syn)
+  (syntax-case syn (:)
+    [(_ (x : t) e)
+     (syntax->curnel-syntax
+       (quasisyntax/loc syn (Π (x : t) e)))]))
+
+(define-syntax (Type syn)
+  (syntax-case syn ()
+    [(_ i)
+     (syntax->curnel-syntax
+       (quasisyntax/loc syn (Unv i)))]
+    [_ (quasisyntax/loc syn (Type 0))]))
+
+(define-syntax (dep-inductive syn)
+  (syntax-case syn (:)
+    [(_ i : ti (x1 : t1) ...)
+     (begin
+       (extend-Σ/syn! sigma #'i #'ti #'((x1 : t1) ...))
+       #'(void))]))
+
+(define-syntax (dep-elim syn)
+  (syntax-case syn (:)
+    [(_ D e P method ...)
+     (syntax->curnel-syntax
+       (quasisyntax/loc syn (elim D e P method ...)))]))
+
+;; TODO: Not sure if this is the correct behavior for #%top
+(define-syntax (dep-var syn)
+  (syntax-case syn ()
+    [(_ . id) #`(term (reduce #,(sigma) id))]))
+
+;; TODO: Syntax-parse
+(define-syntax (dep-define syn)
+  (syntax-case syn (:)
+    [(_ (name (x : t)) e)
+     #'(dep-define name (dep-lambda (x : t) e))]
+    [(_ id e)
+     (let ([e (cur->datum #'e)]
+           [id (syntax->datum #'id)])
+       (extend-Γ/term! gamma id (type-infer/term e))
+       (add-binding/term! id e)
+       #'(void))]))

example.rkt

@@ -1,11 +1,13 @@
-#lang s-exp "redex-curnel.rkt"
+#lang s-exp "cur.rkt"
 
 ;; Use racket libraries over your dependently typed code!?!?
 ;; TODO: actually, I'm not sure this should work quite as well as it
 ;; seems to with check-equal?
 (require rackunit)
-(require (only-in "redex-curnel.rkt" [#%app real-app]
-                  [define real-define]))
+(require
+  (only-in "cur.rkt"
+    [#%app real-app]
+    [define real-define]))
 
 (define-syntax (#%app syn)
   (syntax-case syn ()

oll.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "redex-curnel.rkt"
+#lang s-exp "cur.rkt"
 ;; OLL: The OTT-Like Library
 ;; TODO: Add latex extraction
 ;; TODO: Automagically create a parser from bnf grammar

proofs-for-free.rkt

@@ -1,6 +1,9 @@
-#lang s-exp "redex-curnel.rkt"
-(require "stdlib/sugar.rkt" "stdlib/prop.rkt" racket/trace 
-         (for-syntax racket/syntax))
+#lang s-exp "cur.rkt"
+(require
+  "stdlib/sugar.rkt"
+  "stdlib/prop.rkt"
+  racket/trace
+  (for-syntax racket/syntax))
 
 ;; ---------
 (begin-for-syntax

stdlib/bool.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../redex-curnel.rkt"
+#lang s-exp "../cur.rkt"
 (provide bool btrue bfalse if bnot)
 
 (data bool : Type

stdlib/maybe.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../redex-curnel.rkt"
+#lang s-exp "../cur.rkt"
 (require "sugar.rkt")
 (provide maybe none some)
 

stdlib/nat.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../redex-curnel.rkt"
+#lang s-exp "../cur.rkt"
 (require "sugar.rkt" "bool.rkt")
 ;; TODO: override (all-defined-out) to enable exporting all these
 ;; properly.

stdlib/prop.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../redex-curnel.rkt"
+#lang s-exp "../cur.rkt"
 (require "sugar.rkt")
 ;; TODO: Handle multiple provide forms properly
 ;; TODO: Handle (all-defined-out) properly

stdlib/sugar.rkt

@@ -1,19 +1,22 @@
-#lang s-exp "../redex-curnel.rkt"
-(provide ->
-         ->*
-         forall*
-         lambda*
-         #%app
-         define
-         case*
-         define-type
-         define-theorem
-         qed
-         real-app
-         define-rec)
+#lang s-exp "../cur.rkt"
+(provide
+  ->
+  ->*
+  forall*
+  lambda*
+  #%app
+  define
+  case*
+  define-type
+  define-theorem
+  qed
+  real-app
+  define-rec)
 
-(require (only-in "../redex-curnel.rkt" [#%app real-app]
-                  [define real-define]))
+(require
+  (only-in "../cur.rkt"
+    [#%app real-app]
+    [define real-define]))
 
 
 (define-syntax (-> syn)

stdlib/tactics/base.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../../redex-curnel.rkt"
+#lang s-exp "../../cur.rkt"
 (require
   (for-syntax racket/syntax))
 (provide

stdlib/tactics/sartactics.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../../redex-curnel.rkt"
+#lang s-exp "../../cur.rkt"
 (require
   "base.rkt"
   (prefix-in basic: "standard.rkt")

stdlib/tactics/standard.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../../redex-curnel.rkt"
+#lang s-exp "../../cur.rkt"
 (require
   "base.rkt"
   (for-syntax racket/syntax))

stdlib/typeclass.rkt

@@ -1,4 +1,4 @@
-#lang s-exp "../redex-curnel.rkt"
+#lang s-exp "../cur.rkt"
 (require
   "nat.rkt"
   "bool.rkt"

stlc.rkt

@@ -1,6 +1,11 @@
-#lang s-exp "redex-curnel.rkt"
-(require "stdlib/nat.rkt" "stdlib/sugar.rkt" "oll.rkt"
-         "stdlib/maybe.rkt" "stdlib/bool.rkt" "stdlib/prop.rkt")
+#lang s-exp "cur.rkt"
+(require
+  "stdlib/nat.rkt"
+  "stdlib/sugar.rkt"
+  "oll.rkt"
+  "stdlib/maybe.rkt"
+  "stdlib/bool.rkt"
+  "stdlib/prop.rkt")
 
 (define-language stlc
   #:vars (x)