Merge branch 'redex-with-binding'
This commit is contained in:
William J. Bowman
commit
25e475e289
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@ -45,7 +45,10 @@
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(t e ::= (Π (x : t) t) (λ (x : t) t) (elim x U) x U (t t))
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;; Σ (signature). (inductive-name : type ((constructor : type) ...))
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;; NB: Σ is a map from a name x to a pair of it's type and a map of constructor names to their types
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(Σ ::= ∅ (Σ (x : t ((x : t) ...)))))
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(Σ ::= ∅ (Σ (x : t ((x : t) ...))))
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#:binding-forms
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(λ (x : t) e #:refers-to x)
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(Π (x : t_0) t_1 #:refers-to x))
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(define x? (redex-match? ttL x))
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(define t? (redex-match? ttL t))
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@ -134,73 +137,25 @@
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----------------
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(unv-kind (Unv i_1) (Unv i_2) (Unv i_3))])
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;;; ------------------------------------------------------------------------
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;;; Substitution and α-equivalence
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(define-judgment-form ttL
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#:mode (α-equivalent I I)
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#:contract (α-equivalent t t)
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[----------------- "α-x"
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(α-equivalent x x)]
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[----------------- "α-U"
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(α-equivalent U U)]
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[(α-equivalent t_1 (subst t_3 x_1 x_0))
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(α-equivalent t_0 t_2)
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----------------- "α-binder"
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(α-equivalent (any (x_0 : t_0) t_1)
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(any (x_1 : t_2) t_3))]
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[(α-equivalent e_0 e_2)
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(α-equivalent e_1 e_3)
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----------------- "α-app"
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(α-equivalent (e_0 e_1) (e_2 e_3))]
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[(α-equivalent e_0 e_2)
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(α-equivalent e_1 e_3)
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----------------- "α-elim"
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(α-equivalent (elim e_0 e_1) (elim e_2 e_3))])
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(define-metafunction ttL
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[(α-equivalent? t_0 t_1)
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,(alpha-equivalent? ttL (term t_0) (term t_1))])
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(module+ test
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;; NB: Hack to allow checking contexts or terms without explicitly defining on contexts
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(default-equiv (lambda (x y) (term (α-equivalent (in-hole ,x Type) (in-hole ,y Type)))))
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(check-equiv? (term x) (term x))
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(check-not-equiv? (term x) (term y))
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(check-equiv? (term (λ (x : A) x)) (term (λ (y : A) y))))
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(default-equiv (lambda (x y) (term (α-equivalent ,x ,y)))))
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;; NB: Copy and pasted from Redex examples
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(define-metafunction ttL
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subst-vars : (x any) ... any -> any
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[(subst-vars (x_1 any_1) x_1) any_1]
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[(subst-vars (x_1 any_1) (any_2 ...))
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((subst-vars (x_1 any_1) any_2) ...)]
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[(subst-vars (x_1 any_1) any_2) any_2]
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[(subst-vars (x_1 any_1) (x_2 any_2) ... any_3)
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(subst-vars (x_1 any_1) (subst-vars (x_2 any_2) ... any_3))]
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[(subst-vars any) any])
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;; NB TODO: Why do I not have tests for substitutions?!?!?!?!?!?!?!!?!?!?!?!?!?!!??!?!
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(define-metafunction ttL
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subst : t x t -> t
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[(subst U x t) U]
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[(subst x x t) t]
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[(subst x_0 x t) x_0]
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[(subst (any (x : t_0) t_1) x t)
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(any (x : (subst t_0 x t)) t_1)]
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[(subst (any (x_0 : t_0) t_1) x t)
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(any (x_new : (subst (subst-vars (x_0 x_new) t_0) x t))
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(subst (subst-vars (x_0 x_new) t_1) x t))
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(where x_new ,(variable-not-in (term (x_0 t_0 x t t_1)) (term x_0)))]
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[(subst (e_0 e_1) x t) ((subst e_0 x t) (subst e_1 x t))]
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[(subst (elim e_0 e_1) x t) (elim (subst e_0 x t) (subst e_1 x t))])
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[(subst t_0 x t_1)
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(substitute t_0 x t_1)])
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(module+ test
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(check-true (t? (term (Π (a : A) (Π (b : B) ((and A) B))))))
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(check-equiv?
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(term (Π (a : S) (Π (b : B) ((and S) B))))
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(term (subst (Π (a : A) (Π (b : B) ((and A) B))) A S))))
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(check-true
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(term
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(α-equivalent?
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(Π (a : S) (Π (b : B) ((and S) B)))
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(subst (Π (a : A) (Π (b : B) ((and A) B))) A S)))))
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(define-metafunction ttL
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subst-all : t (x ...) (e ...) -> t
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@ -724,9 +679,17 @@
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[(where t_2 (reduce Σ t_0))
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(where t_3 (reduce Σ t_1))
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(α-equivalent t_2 t_3)
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(side-condition (α-equivalent? t_2 t_3))
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----------------- "≡-αβ"
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(equivalent Σ t_0 t_1)])
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(module+ test
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(define-syntax-rule (check-equivalent e1 e2)
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(check-holds (equivalent ∅ e1 e2)))
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(check-equivalent
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(λ (x : Type) x) (λ (y : Type) y))
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(check-equivalent
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(Π (x : Type) x) (Π (y : Type) y))
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)
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;;; ------------------------------------------------------------------------
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;;; Type checking and synthesis
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@ -953,8 +916,8 @@
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(check-holds (unv-kind (Unv 0) (Unv 0) (Unv 0)))
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(check-holds (type-infer ∅ (∅ x_0 : (Unv 0)) (Π (x_2 : x_0) (Unv 0)) t))
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(check-holds (type-infer ∅ ∅ (λ (x : (Unv 0)) x) (Π (x : (Unv 0)) (Unv 0))))
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(check-holds (type-infer ∅ ∅ (λ (y : (Unv 0)) (λ (x : y) x))
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(check-holds (type-check ∅ ∅ (λ (x : (Unv 0)) x) (Π (x : (Unv 0)) (Unv 0))))
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(check-holds (type-check ∅ ∅ (λ (y : (Unv 0)) (λ (x : y) x))
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(Π (y : (Unv 0)) (Π (x : y) y))))
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(check-equal? (list (term (Unv 1)))
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@ -1046,25 +1009,25 @@
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((((elim nat (Unv 0)) nat) (s zero)) zero)
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nat))
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(define lam (term (λ (nat : (Unv 0)) nat)))
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(check-equal?
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(list (term (Π (nat : (Unv 0)) (Unv 0))))
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(judgment-holds (type-infer ,Σ0 ∅ ,lam t) t))
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(check-equal?
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(list (term (Π (nat : (Unv 0)) (Unv 0))))
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(judgment-holds (type-infer ,Σ ∅ ,lam t) t))
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(check-equal?
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(list (term (Π (x : (Π (y : (Unv 0)) y)) nat)))
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(judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (x : (Π (y : (Unv 0)) y)) (x nat))
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t) t))
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(check-equal?
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(list (term (Π (y : (Unv 0)) (Unv 0))))
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(judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (y : (Unv 0)) y) t) t))
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(check-equal?
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(list (term (Unv 0)))
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(judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅
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(check-equivalent
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(Π (nat : (Unv 0)) (Unv 0))
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,(car (judgment-holds (type-infer ,Σ0 ∅ ,lam t) t)))
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(check-equivalent
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(Π (nat : (Unv 0)) (Unv 0))
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,(car (judgment-holds (type-infer ,Σ ∅ ,lam t) t)))
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(check-equivalent
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(Π (x : (Π (y : (Unv 0)) y)) nat)
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,(car (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (x : (Π (y : (Unv 0)) y)) (x nat))
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t) t)))
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(check-equivalent
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(Π (y : (Unv 0)) (Unv 0))
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,(car (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅ (λ (y : (Unv 0)) y) t) t)))
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(check-equivalent
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(Unv 0)
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,(car (judgment-holds (type-infer (∅ (nat : (Unv 0) ())) ∅
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((λ (x : (Π (y : (Unv 0)) (Unv 0))) (x nat))
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(λ (y : (Unv 0)) y))
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t) t))
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t) t)))
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(check-equal?
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(list (term (Unv 0)) (term (Unv 1)))
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(judgment-holds
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2
info.rkt
2
info.rkt
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@ -1,6 +1,6 @@
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#lang info
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(define collection "cur")
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(define deps '("base" "rackunit-lib" ("redex-lib" #:version "1.6")))
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(define deps '("base" "rackunit-lib" ("redex-lib" #:version "1.8" #;redex-1-paul-public)))
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(define build-deps '("scribble-lib" "racket-doc" "sandbox-lib"))
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(define scribblings '(("scribblings/cur.scrbl" (multi-page))))
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(define pkg-desc "Dependent types with parenthesis and meta-programming.")
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