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Renamed languages to reflect facts: cic to tt

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William J. Bowman 2015-09-30 16:16:03 -04:00
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curnel/redex-core.rkt
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@ -27,20 +27,19 @@
;; 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
;; ttL is the core language of Cur. Very similar to TT (Idirs core) and Luo's UTT. Surface
;; langauge should provide short-hand, such as -> for non-dependent function types, and type
;; inference.
(define-language ttL
(i ::= natural)
(U ::= (Unv i))
(x ::= variable-not-otherwise-mentioned)
(t e ::= (Π (x : t) t) (λ (x : t) t) (elim t t) x U (t t)))
(define x? (redex-match? cicL x))
(define t? (redex-match? cicL t))
(define e? (redex-match? cicL e))
(define U? (redex-match? cicL U))
(define x? (redex-match? ttL x))
(define t? (redex-match? ttL t))
(define e? (redex-match? ttL e))
(define U? (redex-match? ttL U))
(module+ test
(define-term Type (Unv 0))
@ -63,7 +62,7 @@
;; 'A'
;; Types of Universes
(define-judgment-form cicL
(define-judgment-form ttL
#:mode (unv-ok I O)
#:contract (unv-ok U U)
@ -73,7 +72,7 @@
;; 'R'
;; Kinding, I think
(define-judgment-form cicL
(define-judgment-form ttL
#:mode (unv-kind I I O)
#:contract (unv-kind U U U)
@ -90,7 +89,7 @@
----------------
(unv-kind (Unv i_1) (Unv i_2) (Unv i_3))])
(define-judgment-form cicL
(define-judgment-form ttL
#:mode (α-equivalent I I)
#:contract (α-equivalent t t)
@ -115,13 +114,13 @@
(check-not-holds (α-equivalent x y))
(check-holds (α-equivalent (λ (x : A) x) (λ (y : A) y))))
(define-metafunction cicL
(define-metafunction ttL
fresh-x : any ... -> x
[(fresh-x any ...) ,(variable-not-in (term (any ...)) (term x))])
;; NB: Substitution is hard
;; NB: Copy and pasted from Redex examples
(define-metafunction cicL
(define-metafunction ttL
subst-vars : (x any) ... any -> any
[(subst-vars (x_1 any_1) x_1) any_1]
[(subst-vars (x_1 any_1) (any_2 ...))
@ -131,7 +130,7 @@
(subst-vars (x_1 any_1) (subst-vars (x_2 any_2) ... any_3))]
[(subst-vars any) any])
(define-metafunction cicL
(define-metafunction ttL
subst : t x t -> t
[(subst U x t) U]
[(subst x x t) t]
@ -157,7 +156,7 @@
;; NB:
;; TODO: Why do I not have tests for substitutions?!?!?!?!?!?!?!!?!?!?!?!?!?!!??!?!
(define-metafunction cicL
(define-metafunction ttL
subst-all : t (x ...) (e ...) -> t
[(subst-all t () ()) t]
[(subst-all t (x_0 x ...) (e_0 e ...))
@ -165,7 +164,7 @@
;; 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
(define-extended-language tt-redL ttL
;; NB: (in-hole Θv (elim x U)) is only a value when it's a partially applied elim.
;; TODO: Perhaps (elim x U) should step to an eta-expanded version of elim
(v ::= x U (Π (x : t) t) (λ (x : t) t) (elim x U) (in-hole Θv x) (in-hole Θv (elim x U)))
@ -173,20 +172,20 @@
(E ::= hole (E e) (v E)
(Π (x : v) E)
(Π (x : E) e))
;; TODO: Σ should probably be moved to cicL, since elim is there.
;; TODO: Σ should probably be moved to ttL, since elim is there.
;; 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)
(Θv ::= hole (Θv v)))
(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 Θv? (redex-match? cic-redL Θv))
(define E? (redex-match? cic-redL E))
(define v? (redex-match? cic-redL v))
(define Σ? (redex-match? tt-redL Σ))
(define Ξ? (redex-match? tt-redL Ξ))
(define Φ? (redex-match? tt-redL Φ))
(define Θ? (redex-match? tt-redL Θ))
(define Θv? (redex-match? tt-redL Θv))
(define E? (redex-match? tt-redL E))
(define v? (redex-match? tt-redL v))
(module+ test
(check-true (v? (term (λ (x_0 : (Unv 0)) x_0))))
@ -224,7 +223,7 @@
())))))
(check-true (Σ? sigma)))
(define-metafunction cic-redL
(define-metafunction tt-redL
append-Σ : Σ Σ -> Σ
[(append-Σ Σ ∅) Σ]
[(append-Σ Σ_2 (Σ_1 (x : t ((x_c : t_c) ...))))
@ -232,25 +231,25 @@
;; TODO: Test
;; TODO: Maybe this should be called "apply-to-telescope"
(define-metafunction cic-redL
(define-metafunction tt-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
(define-metafunction tt-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
(define-metafunction tt-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
(define-metafunction tt-redL
method-lookup : Σ x x (Θ e) -> e
[(method-lookup Σ x_D x_ci Θ)
(select-arg x_ci (x_0 ...) Θ)
@ -266,7 +265,7 @@
;; arguments of an inductive constructor.
;; TODO: Poorly documented, and poorly tested.
;; Probably the source of issue #20
(define-metafunction cic-redL
(define-metafunction tt-redL
elim-recur : x U e Θ Θ Θ (x ...) -> Θ
[(elim-recur x_D U e_P Θ
(in-hole Θ_m (hole e_mi))
@ -277,7 +276,7 @@
[(elim-recur x_D U 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))))
(redex-match? tt-redL (in-hole Θ_i (hole (in-hole Θ_r zero))) (term (hole zero))))
(check-equal?
(term (elim-recur nat Type (λ (x : nat) nat)
((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
@ -299,7 +298,7 @@
(λ (x : nat) (λ (ih-x : nat) (s (s x)))))
(s zero))))))
(define-judgment-form cic-redL
(define-judgment-form tt-redL
#:mode (length-match I I)
#:contract (length-match Θ (x ...))
[----------------------
@ -309,7 +308,7 @@
----------------------
(length-match (Θ e) (x x_r ...))])
(define-judgment-form cic-redL
(define-judgment-form tt-redL
#:mode (telescope-match I I)
#:contract (telescope-match Θ Ξ)
@ -321,8 +320,8 @@
(telescope-match (Θ e) (Π (x : t) Ξ))])
(define cic-->
(reduction-relation cic-redL
(define tt-->
(reduction-relation tt-redL
(--> (Σ (in-hole E ((any (x : t_0) t_1) t_2)))
(Σ (in-hole E (subst t_1 x t_2)))
-->β)
@ -362,17 +361,17 @@
(where Θ_r (elim-recur x_D U v_P (in-hole Θv_p Θv_m) Θv_m Θv_i (x_c* ...)))
-->elim)))
(define-metafunction cic-redL
(define-metafunction tt-redL
step : Σ e -> e
[(step Σ e)
e_r
(where (_ e_r) ,(car (apply-reduction-relation cic--> (term (Σ e)))))])
(where (_ e_r) ,(car (apply-reduction-relation tt--> (term (Σ e)))))])
(define-metafunction cic-redL
(define-metafunction tt-redL
reduce : Σ e -> e
[(reduce Σ e)
e_r
(where (_ e_r) ,(let ([r (apply-reduction-relation* cic--> (term (Σ e))
(where (_ e_r) ,(let ([r (apply-reduction-relation* tt--> (term (Σ e))
#:cache-all? #t)])
(unless (null? (cdr r))
(error "Church-rosser broken" r))
@ -442,7 +441,7 @@
(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
zero))))))
(define-judgment-form cic-redL
(define-judgment-form tt-redL
#:mode (equivalent I I I)
#:contract (equivalent Σ t t)
@ -452,27 +451,27 @@
----------------- "≡-αβ"
(equivalent Σ t_0 t_1)])
(define-extended-language cic-typingL cic-redL
(define-extended-language tt-typingL tt-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 Γ? (redex-match? tt-typingL Γ))
(define-metafunction cic-typingL
(define-metafunction tt-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
(define-metafunction tt-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
(define-metafunction tt-redL
lookup-Σ : Σ x -> t or #f
[(lookup-Σ ∅ x) #f]
[(lookup-Σ (Σ (x : t ((x_1 : t_1) ...))) x) t]
@ -480,14 +479,14 @@
[(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
(define-metafunction tt-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
(define-metafunction ttL
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.
@ -508,7 +507,7 @@
(check-true (term (positive (Unv 0) #f))))
;; Holds when the signature Σ and typing context Γ are well-formed.
(define-judgment-form cic-typingL
(define-judgment-form tt-typingL
#:mode (wf I I)
#:contract (wf Σ Γ)
@ -534,8 +533,8 @@
((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)))
(check-true (redex-match? tt-redL (in-hole Ξ (Unv 0)) (term (Unv 0))))
(check-true (redex-match? tt-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))
@ -549,7 +548,7 @@
(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)))
(map match-bindings (redex-match tt-redL (in-hole Ξ (in-hole Φ (in-hole Θ nat)))
(term (Π (x : nat) nat)))))
(check-pred
(curry bindings-equal?
@ -557,15 +556,15 @@
(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)))
(map match-bindings (redex-match tt-redL (in-hole hole (in-hole Φ (in-hole Θ nat)))
(term (Π (x : nat) nat)))))
(check-true
(redex-match? cic-redL
(redex-match? tt-redL
(in-hole hole (in-hole hole (in-hole hole nat)))
(term nat)))
(check-true
(redex-match? cic-redL
(redex-match? tt-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) ())) ∅))
@ -591,7 +590,7 @@
;; 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
(define-metafunction tt-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))
@ -604,7 +603,7 @@
;; 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
(define-metafunction tt-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))
@ -615,7 +614,7 @@
;; Returns the methods required for eliminating the inductively
;; defined type x_D, whose constructors are ((x_ci : t_ci) ...), with motive t_P.
(define-metafunction cic-redL
(define-metafunction tt-redL
methods-for : x t ((x : t) ...) -> Ξ
[(methods-for x_D t_P ()) hole]
[(methods-for x_D t_P ((x_ci : (in-hole Φ (in-hole Θ x_D)))
@ -651,7 +650,7 @@
(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-true (redex-match? ttL ((x : t) ...) (term ())))
(check-equal?
(term (methods-for false (λ (y : false) (Π (x : Type) x))
()))
@ -659,7 +658,7 @@
;; Returns the inductively defined type that x constructs
;; NB: Depends on clause order
(define-metafunction cic-redL
(define-metafunction tt-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]
@ -674,7 +673,7 @@
(term nat)))
;; TODO: Define generic traversals of Σ and Γ ?
(define-metafunction cic-redL
(define-metafunction tt-redL
parameters-of : Σ x -> Ξ
[(parameters-of (Σ (x_D : (in-hole Ξ U) ((x : t) ...))) x_D)
Ξ]
@ -690,7 +689,7 @@
;; Returns the constructors for the inductively defined type x_D in
;; the signature Σ
(define-metafunction cic-redL
(define-metafunction tt-redL
constructors-for : Σ x -> ((x : t) ...) or #f
;; NB: Depends on clause order
[(constructors-for ∅ x_D) #f]
@ -708,7 +707,7 @@
;; Holds when an apply context Θ provides arguments that match the
;; telescope Ξ
(define-judgment-form cic-typingL
(define-judgment-form tt-typingL
#:mode (telescope-types I I I I)
#:contract (telescope-types Σ Γ Θ Ξ)
@ -723,7 +722,7 @@
(check-holds
(telescope-types ,Σ ∅ (hole zero) (Π (x : nat) hole)))
(check-true
(redex-match? cic-redL (in-hole Θ (hole e))
(redex-match? tt-redL (in-hole Θ (hole e))
(term ((hole zero) (λ (x : nat) x)))))
(check-holds
(telescope-types ,Σ ∅ (hole zero)
@ -757,7 +756,7 @@
;; 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
(define-judgment-form tt-typingL
#:mode (type-infer I I I O)
#:contract (type-infer Σ Γ e t)
@ -826,7 +825,7 @@
;; applied to the paramters and discriminant
(apply-telescope x_P Ξ_P*D)))))])
(define-judgment-form cic-typingL
(define-judgment-form tt-typingL
#:mode (type-check I I I I)
#:contract (type-check Σ Γ e t)
@ -857,7 +856,7 @@
;; ---- Elim
;; TODO: Clean up/Reorganize these tests
(check-true
(redex-match? cic-typingL
(redex-match? tt-typingL
(in-hole Θ_m (((elim x_D U) e_D) e_P))
(term ((((elim truth Type) T) (Π (x : truth) (Unv 1))) (Unv 0)))))
(define Σtruth (term (∅ (truth : (Unv 0) ((T : truth))))))
@ -1014,7 +1013,7 @@
(type-infer ,Σ4 (((∅ P : Type) Q : Type) ab : ((and P) Q))
ab (in-hole Θ and)))
(check-true
(redex-match? cic-redL
(redex-match? tt-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
@ -1085,7 +1084,7 @@
(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
(redex-match? tt-typingL
((in-hole Θ_m ((elim x_D U) e_P)) e_D)
(term (((elim false (Unv 1)) (λ (y : false) (Π (x : Type) x)))
x))))
@ -1153,12 +1152,12 @@
(type-check ,Σ= ∅ ,refl-elim nat))
(check-true
(redex-match?
cic-redL
tt-redL
(Σ (in-hole E (in-hole Θ ((elim x_D U) e_P))))
(term (,Σ= ,refl-elim))))
(check-true
(redex-match?
cic-redL
tt-redL
(in-hole (Θ_p (in-hole Θ_i x_ci)) Θ_m)
(term (((((hole
(λ (A1 : (Unv 0)) (λ (x1 : A1) zero))) bool) true) true) ((refl bool) true)))))