remove dep example
This commit is contained in:
Stephen Chang
parent
7c841dc750
commit
441ec0e5d7
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@ -1,334 +0,0 @@
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#lang turnstile/lang
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; Π λ ≻ ⊢ ≫ → ∧ (bidir ⇒ ⇐)
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(provide (rename-out [#%type *])
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Π → ∀ = eq-refl eq-elim
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Nat Z S nat-ind
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λ #%app ann
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define define-type-alias)
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#;(begin-for-syntax
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(define old-ty= (current-type=?))
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(current-type=?
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(λ (t1 t2)
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(displayln (stx->datum t1))
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(displayln (stx->datum t2))
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(old-ty= t1 t2)))
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(current-typecheck-relation (current-type=?)))
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;(define-syntax-category : kind)
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(define-base-type Nat)
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(define-internal-type-constructor →)
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(define-internal-binding-type ∀)
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;; TODO: how to do Type : Type
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(define-typed-syntax (Π ([X:id : τ_in] ...) τ_out) ≫
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[[X ≫ X- : τ_in] ... ⊢ [τ_out ≫ τ_out- ⇒ _][τ_in ≫ τ_in- ⇒ _] ...]
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-------
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[⊢ (∀- (X- ...) (→- τ_in- ... τ_out-)) ⇒ #,(expand/df #'#%type)])
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;; abbrevs for Π
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(define-simple-macro (→ τ_in ... τ_out)
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#:with (X ...) (generate-temporaries #'(τ_in ...))
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(Π ([X : τ_in] ...) τ_out))
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(define-simple-macro (∀ (X ...) τ)
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(Π ([X : #%type] ...) τ))
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;; ~Π pattern expander
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(begin-for-syntax
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(define-syntax ~Π
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(pattern-expander
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(syntax-parser
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[(_ ([x:id : τ_in] ... (~and (~literal ...) ooo)) τ_out)
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#'(~∀ (x ... ooo) (~→ τ_in ... ooo τ_out))]
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[(_ ([x:id : τ_in] ...) τ_out)
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#'(~∀ (x ...) (~→ τ_in ... τ_out))]))))
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;; equality -------------------------------------------------------------------
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(define-internal-type-constructor =)
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(define-typed-syntax (= t1 t2) ≫
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[⊢ t1 ≫ t1- ⇒ _] [⊢ t2 ≫ t2- ⇒ _]
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;; #:do [(printf "t1: ~a\n" (stx->datum #'t1-))
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;; (printf "t2: ~a\n" (stx->datum #'t2-))]
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; [t1- τ= t2-]
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---------------------
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[⊢ (=- t1- t2-) ⇒ #,(expand/df #'#%type)])
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(define-typed-syntax (eq-refl e) ≫
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[⊢ e ≫ e- ⇒ _]
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----------
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[⊢ (#%app- void-) ⇒ (= e- e-)])
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(define-typed-syntax (eq-elim t P pt w peq) ≫
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[⊢ t ≫ t- ⇒ ty]
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; [⊢ P ≫ P- ⇒ _]
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; [⊢ pt ≫ pt- ⇒ _]
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; [⊢ w ≫ w- ⇒ _]
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; [⊢ peq ≫ peq- ⇒ _]
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[⊢ P ≫ P- ⇐ (→ ty #%type)]
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[⊢ pt ≫ pt- ⇐ (P- t-)]
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[⊢ w ≫ w- ⇐ ty]
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[⊢ peq ≫ peq- ⇐ (= t- w-)]
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--------------
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[⊢ (#%app- void-) ⇒ (P- w-)])
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;; lambda and #%app -----------------------------------------------------------
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;; TODO: add case with expected type + annotations
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;; - check that annotations match expected types
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(define-typed-syntax λ
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[(_ ([x:id : τ_in] ...) e) ≫
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[[x ≫ x- : τ_in] ... ⊢ [e ≫ e- ⇒ τ_out][τ_in ≫ τ_in- ⇒ _] ...]
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-------
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[⊢ (λ- (x- ...) e-) ⇒ (Π ([x- : τ_in-] ...) τ_out)]]
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[(_ (y:id ...) e) ⇐ (~Π ([x:id : τ_in] ...) τ_out) ≫
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[[x ≫ x- : τ_in] ... ⊢ #,(substs #'(x ...) #'(y ...) #'e) ≫ e- ⇐ τ_out]
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---------
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[⊢ (λ- (x- ...) e-)]])
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;; classes for matching number literals
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(begin-for-syntax
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(define-syntax-class nat
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(pattern (~or n:exact-nonnegative-integer (_ n:exact-nonnegative-integer))
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#:attr val
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#'n))
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(define-syntax-class nats
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(pattern (n:nat ...) #:attr vals #'(n.val ...)))
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; extract list of quoted numbers
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(define stx->nat (syntax-parser [n:nat (stx-e #'n.val)]))
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(define (stx->nats stx) (stx-map stx->nat stx))
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(define (stx+ ns) (apply + (stx->nats ns)))
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(define (delta op-stx args)
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(syntax-parse op-stx
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[(~literal +-) (stx+ args)]
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[(~literal zero?-) (apply zero? (stx->nats args))])))
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(define-typed-syntax #%app
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[(_ e_fn e_arg ...) ≫ ; apply lambda
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; #:do[(printf "applying (1) ~a\n" (stx->datum #'e_fn))]
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[⊢ e_fn ≫ (~and e_fn- (_ (x:id ...) e ~!)) ⇒ (~Π ([X : τ_inX] ...) τ_outX)]
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#:do[(printf "e_fn-: ~a\n" (stx->datum #'e_fn-))
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(printf "args: ~a\n" (stx->datum #'(e_arg ...)))]
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#:fail-unless (stx-length=? #'[τ_inX ...] #'[e_arg ...])
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(num-args-fail-msg #'e_fn #'[τ_inX ...] #'[e_arg ...])
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[⊢ e_arg ≫ e_argX- ⇒ ty-argX] ... ; typechecking args must be fold; do in 2 steps
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#:do[(define (ev e)
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(syntax-parse e
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; [_ #:do[(printf "eval: ~a\n" (stx->datum e))] #:when #f #'(void)]
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[(~or _:id
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; ((~literal #%plain-lambda) . _)
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(~= _ _)
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~Nat
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((~literal quote) _))
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e]
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;; handle nums
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[((~literal #%plain-app)
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(~and op (~or (~literal +-) (~literal zero?-)))
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. args:nats)
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#`#,(delta #'op #'args.vals)]
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[((~literal #%plain-app) (~and f ((~literal #%plain-lambda) . b)) . rst)
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(expand/df #`(#%app f . #,(stx-map ev #'rst)))]
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[(x ...)
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;; #:do[(printf "t before: ~a\n" (typeof e))
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;; (printf "t after: ~a\n" (typeof #`#,(stx-map ev #'(x ...))))]
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(syntax-property #`#,(stx-map ev #'(x ...)) ': (typeof e))]
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[_ e] ; other literals
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#;[es (stx-map L #'es)]))]
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#:with (ty-arg ...)
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(stx-map
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(λ (t) (ev (substs #'(e_argX- ...) #'(X ...) t)))
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#'(ty-argX ...))
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#:with (e_arg- ...) (stx-map (λ (e t) (assign-type e t)) #'(e_argX- ...) #'(ty-arg ...))
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#:with (τ_in ... τ_out)
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(stx-map
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(λ (t) (ev (substs #'(e_arg- ...) #'(X ...) t)))
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#'(τ_inX ... τ_outX))
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; #:do[(printf "vars: ~a\n" #'(X ...))]
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; #:when (stx-andmap (λ (t1 t2)(displayln (stx->datum t1)) (displayln (stx->datum t2)) (displayln (typecheck? t1 t2)) #;(typecheck? t1 t2)) #'(ty-arg ...) #'(τ_in ...))
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;; #:do[(stx-map
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;; (λ (tx t) (printf "ty_in inst: \n~a\n~a\n" (stx->datum tx) (stx->datum t)))
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;; #'(τ_inX ...) #'(τ_in ...))]
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; [⊢ e_arg- ≫ _ ⇐ τ_in] ...
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#:do[(printf "res e =\n~a\n" (stx->datum (substs #'(e_arg- ...) #'(x ...) #'e)))
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(printf "res t = ~a\n" (stx->datum (substs #'(e_arg- ...) #'(X ...) #'τ_out)))]
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#:with res-e (let L ([e (substs #'(e_arg- ...) #'(x ...) #'e)]) ; eval
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(syntax-parse e
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[(~or _:id
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((~literal #%plain-lambda) . _)
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(~Π ([_ : _] ...) _)
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(~= _ _)
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~Nat)
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e]
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;; handle nums
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[((~literal #%plain-app)
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(~and op (~or (~literal +-) (~literal zero?-)))
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. args:nats)
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#`#,(delta #'op #'args.vals)]
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[((~literal #%plain-app) . rst)
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(expand/df #`(#%app . #,(stx-map L #'rst)))]
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[_ e] ; other literals
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#;[es (stx-map L #'es)]))
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;; #:with res-ty (syntax-parse (substs #'(e_arg- ...) #'(X ...) #'τ_out)
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;; [((~literal #%plain-app) . rst) (expand/df #'(#%app . rst))]
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;; [other-ty #'other-ty])
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--------
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[⊢ res-e #;#,(substs #'(e_arg- ...) #'(x ...) #'e) ⇒ τ_out
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#;#,(substs #'(e_arg- ...) #'(X ...) #'τ_out)]]
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[(_ e_fn e_arg ... ~!) ≫ ; apply var
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; #:do[(printf "applying (2) ~a\n" (stx->datum #'e_fn))]
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[⊢ e_fn ≫ e_fn- ⇒ ty-fn]
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; #:do[(printf "e_fn- ty: ~a\n" (stx->datum #'ty-fn))]
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[⊢ e_fn ≫ _ ⇒ (~Π ([X : τ_inX] ...) τ_outX)]
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; #:do[(printf "e_fn- no: ~a\n" (stx->datum #'e_fn-))
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; (printf "args: ~a\n" (stx->datum #'(e_arg ...)))]
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;; #:with e_fn- (syntax-parse #'e_fn*
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;; [((~literal #%plain-app) . rst) (expand/df #'(#%app . rst))]
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;; [other #'other])
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#:fail-unless (stx-length=? #'[τ_inX ...] #'[e_arg ...])
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(num-args-fail-msg #'e_fn #'[τ_inX ...] #'[e_arg ...])
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[⊢ e_arg ≫ e_argX- ⇒ ty-argX] ... ; typechecking args must be fold; do in 2 steps
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#:do[(define (ev e)
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(syntax-parse e
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; [_ #:do[(printf "eval: ~a\n" (stx->datum e))] #:when #f #'(void)]
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[(~or _:id
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; ((~literal #%plain-lambda) . _)
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(~= _ _)
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~Nat
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((~literal quote) _))
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e]
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;; handle nums
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[((~literal #%plain-app)
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(~and op (~or (~literal +-) (~literal zero?-)))
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. args:nats)
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#`#,(delta #'op #'args.vals)]
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[((~literal #%plain-app) (~and f ((~literal #%plain-lambda) . b)) . rst)
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(expand/df #`(#%app f . #,(stx-map ev #'rst)))]
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[(x ...)
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;; #:do[(printf "t before: ~a\n" (typeof e))
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;; (printf "t after: ~a\n" (typeof #`#,(stx-map ev #'(x ...))))]
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(syntax-property #`#,(stx-map ev #'(x ...)) ': (typeof e))]
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[_ e] ; other literals
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#;[es (stx-map L #'es)]))]
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#:with (ty-arg ...)
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(stx-map
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(λ (t) (ev (substs #'(e_argX- ...) #'(X ...) t)))
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#'(ty-argX ...))
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#:with (e_arg- ...) (stx-map (λ (e t) (assign-type e t)) #'(e_argX- ...) #'(ty-arg ...))
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#:with (τ_in ... τ_out)
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(stx-map
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(λ (t) (ev (substs #'(e_arg- ...) #'(X ...) t)))
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#'(τ_inX ... τ_outX))
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;; #:do[(printf "vars: ~a\n" #'(X ...))]
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; #:when (stx-andmap (λ (e t1 t2)(displayln (stx->datum e))(displayln (stx->datum t1)) (displayln (stx->datum t2)) (displayln (typecheck? t1 t2)) #;(typecheck? t1 t2)) #'(e_arg ...)#'(ty-arg ...) #'(τ_in ...))
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;; #:do[(stx-map
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;; (λ (tx t) (printf "ty_in inst: \n~a\n~a\n" (stx->datum tx) (stx->datum t)))
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;; #'(τ_inX ...) #'(τ_in ...))]
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; [⊢ e_arg ≫ _ ⇐ τ_in] ...
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; #:do[(printf "res e2 =\n~a\n" (stx->datum #'(#%app- e_fn- e_arg- ...)))
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; (printf "res t2 = ~a\n" (stx->datum (substs #'(e_arg- ...) #'(X ...) #'τ_out)))]
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;; #:with res-e (syntax-parse #'e_fn-
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;; [((~literal #%plain-lambda) . _) (expand/df #'(#%app e_fn- e_arg- ...))]
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;; [other #'(#%app- e_fn- e_arg- ...)])
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--------
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[⊢ (#%app- e_fn- e_arg- ...) ⇒ τ_out
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#;#,(expand/df (substs #'(e_arg- ...) #'(X ...) #'τ_out))]])
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(define-typed-syntax (ann e (~datum :) τ) ≫
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[⊢ e ≫ e- ⇐ τ]
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--------
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[⊢ e- ⇒ τ])
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(define-typed-syntax (if e1 e2 e3) ≫
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[⊢ e1 ≫ e1- ⇒ _]
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[⊢ e2 ≫ e2- ⇒ ty]
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[⊢ e3 ≫ e3- ⇒ _]
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#:do[(displayln #'(e1 e2 e3))]
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--------------
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[⊢ (#%app- void-) ⇒ ty])
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;; top-level ------------------------------------------------------------------
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(define-syntax define-type-alias
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(syntax-parser
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[(_ alias:id τ);τ:any-type)
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#'(define-syntax- alias
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(make-variable-like-transformer #'τ))]
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#;[(_ (f:id x:id ...) ty)
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#'(define-syntax- (f stx)
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(syntax-parse stx
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[(_ x ...)
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#:with τ:any-type #'ty
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#'τ.norm]))]))
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(define-typed-syntax define
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#;[(_ x:id (~datum :) τ:type e:expr) ≫
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;[⊢ e ≫ e- ⇐ τ.norm]
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#:with y (generate-temporary #'x)
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--------
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[≻ (begin-
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(define-syntax x (make-rename-transformer (⊢ y : τ.norm)))
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(define- y (ann e : τ.norm)))]]
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[(_ x:id e) ≫
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;This won't work with mutually recursive definitions
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[⊢ e ≫ e- ⇒ _]
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#:with y (generate-temporary #'x)
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#:with y+props (transfer-props #'e- #'y #:except '(origin))
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--------
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[≻ (begin-
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(define-syntax x (make-rename-transformer #'y+props))
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(define- y e-))]]
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#;[(_ (f [x (~datum :) ty] ... (~or (~datum →) (~datum ->)) ty_out) e ...+) ≫
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#:with f- (add-orig (generate-temporary #'f) #'f)
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--------
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[≻ (begin-
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(define-syntax- f
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(make-rename-transformer (⊢ f- : (→ ty ... ty_out))))
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(define- f-
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(stlc+lit:λ ([x : ty] ...)
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(stlc+lit:ann (begin e ...) : ty_out))))]])
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;; peano nums -----------------------------------------------------------------
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(define-typed-syntax Z
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[_:id ≫ --- [⊢ 0 ⇒ Nat]])
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(define-typed-syntax (S n) ≫
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[⊢ n ≫ n- ⇐ Nat]
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-----------
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[⊢ (#%app- +- n- 1) ⇒ Nat])
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#;(define-typed-syntax (sub1 n) ≫
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[⊢ n ≫ n- ⇐ Nat]
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#:do[(displayln #'n-)]
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-----------
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[⊢ (#%app- -- n- 1) ⇒ Nat])
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;; generalized recursor over natural nums
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;; (cases dispatched in #%app)
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(define- (nat-ind- P z s n) (#%app- void))
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(define-syntax nat-ind
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(make-variable-like-transformer
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(assign-type
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#'nat-ind-
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#'(Π ([P : (→ Nat #%type)]
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[z : (P Z)]
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[s : (Π ([k : Nat]) (→ (P k) (P (S k))))]
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[n : Nat])
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(P n)))))
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#;(define-type-alias nat-ind
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(λ ([P : (→ Nat #%type)]
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[z : (P Z)]
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[s : (Π ([k : Nat]) (→ (P k) (P (S k))))]
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[n : Nat])
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#'(#%app- nat-ind- P z s n)))
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#;(define-typed-syntax (nat-ind P z s n) ≫
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[⊢ P ≫ P- ⇐ (→ Nat #%type)]
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; [⊢ b ≫ b- ⇒ _] ; zero
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; [⊢ c ≫ c- ⇒ _] ; succ
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; [⊢ d ≫ d- ⇒ _]
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[⊢ z ≫ z- ⇐ (P- Z)] ; zero
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[⊢ s ≫ s- ⇐ (Π ([k : Nat]) (→ (P- k) (P- (S k))))] ; succ
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[⊢ n ≫ n- ⇐ Nat]
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#:with res (if (typecheck? #'n- (expand/df #'Z))
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#'z-
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#'(s- (nat-ind P- z- s- (sub1 n-))))
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----------------
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[⊢ res ⇒ (P- n-)])
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; [≻ (P- d-)])
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@ -1,53 +0,0 @@
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#lang s-exp "../dep.rkt"
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(require "rackunit-typechecking.rkt")
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; Π → λ ∀ ≻ ⊢ ≫ ⇒
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;; examples from Prabhakar's Proust paper
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;; Peano nums -----------------------------------------------------------------
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(define-type-alias nat-rec
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(λ ([C : *])
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(λ ([zc : C][sc : (→ C C)])
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(λ ([n : Nat])
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(nat-ind (λ ([x : Nat]) C)
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zc
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(λ ([x : Nat]) sc)
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n)))))
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;(check-type nat-rec : (∀ (C) (→ C (→ C C) (→ Nat C))))
|
||||
|
||||
(define-type-alias plus
|
||||
(λ ([n : Nat])
|
||||
(((nat-rec (→ Nat Nat))
|
||||
(λ ([m : Nat]) m)
|
||||
(λ ([pm : (→ Nat Nat)])
|
||||
(λ ([x : Nat])
|
||||
(S (pm x)))))
|
||||
n)))
|
||||
|
||||
(check-type plus : (→ Nat (→ Nat Nat)))
|
||||
|
||||
;(check-type ((plus Z) Z) : Nat -> 0)
|
||||
;(check-type ((plus (S Z)) (S Z)) : Nat -> 2)
|
||||
;(check-type ((plus (S Z)) Z) : Nat -> 1)
|
||||
|
||||
;; TODO: implement nat-ind reductions
|
||||
;; plus zero left
|
||||
;; (check-type
|
||||
;; (λ ([n : Nat]) (eq-refl n))
|
||||
;; : (Π ([n : Nat]) (= ((plus Z) n) n)))
|
||||
|
||||
;; (check-type
|
||||
;; (λ ([n : Nat])
|
||||
;; (nat-ind (λ ([m : Nat]) (= ((plus m) Z) m))
|
||||
;; (eq-refl Z)
|
||||
;; (λ ([k : Nat])
|
||||
;; (λ ([p : (= ((plus k) Z) k)])
|
||||
;; (eq-elim ((plus k) Z)
|
||||
;; (λ ([W : Nat]) (= (S ((plus k) Z)) (S W)))
|
||||
;; (eq-refl (S ((plus k) Z)))
|
||||
;; k
|
||||
;; p)))
|
||||
;; n))
|
||||
;; : (Π ([n : Nat]) (= ((plus n) Z) n)))
|
||||
|
|
@ -1,147 +0,0 @@
|
|||
#lang s-exp "../dep.rkt"
|
||||
(require "rackunit-typechecking.rkt")
|
||||
|
||||
; Π → λ ∀ ≻ ⊢ ≫ ⇒
|
||||
|
||||
;; examples from Prabhakar's Proust paper
|
||||
|
||||
(check-type (λ ([x : *]) x) : (Π ([x : *]) *))
|
||||
(typecheck-fail ((λ ([x : *]) x) (λ ([x : *]) x))
|
||||
#:verb-msg "expected *, given (Π ([x : *]) *)")
|
||||
|
||||
;; transitivity of implication
|
||||
(check-type (λ ([A : *][B : *][C : *])
|
||||
(λ ([f : (→ B C)])
|
||||
(λ ([g : (→ A B)])
|
||||
(λ ([x : A])
|
||||
(f (g x))))))
|
||||
: (∀ (A B C) (→ (→ B C) (→ (→ A B) (→ A C)))))
|
||||
; unnested
|
||||
(check-type (λ ([A : *][B : *][C : *])
|
||||
(λ ([f : (→ B C)][g : (→ A B)])
|
||||
(λ ([x : A])
|
||||
(f (g x)))))
|
||||
: (∀ (A B C) (→ (→ B C) (→ A B) (→ A C))))
|
||||
;; no annotations
|
||||
(check-type (λ (A B C)
|
||||
(λ (f) (λ (g) (λ (x)
|
||||
(f (g x))))))
|
||||
: (∀ (A B C) (→ (→ B C) (→ (→ A B) (→ A C)))))
|
||||
(check-type (λ (A B C)
|
||||
(λ (f g)
|
||||
(λ (x)
|
||||
(f (g x)))))
|
||||
: (∀ (A B C) (→ (→ B C) (→ A B) (→ A C))))
|
||||
;; TODO: partial annotations
|
||||
|
||||
;; booleans -------------------------------------------------------------------
|
||||
|
||||
;; Bool type
|
||||
(define-type-alias Bool (∀ (A) (→ A A A)))
|
||||
|
||||
;; Bool terms
|
||||
(define T (λ ([A : *]) (λ ([x : A][y : A]) x)))
|
||||
(define F (λ ([A : *]) (λ ([x : A][y : A]) y)))
|
||||
(check-type T : Bool)
|
||||
(check-type F : Bool)
|
||||
(define and (λ ([x : Bool][y : Bool]) ((x Bool) y F)))
|
||||
(check-type and : (→ Bool Bool Bool))
|
||||
|
||||
;; And type constructor, ie type-level fn
|
||||
(define-type-alias And
|
||||
(λ ([A : *][B : *])
|
||||
(∀ (C) (→ (→ A B C) C))))
|
||||
(check-type And : (→ * * *))
|
||||
|
||||
;; And type intro
|
||||
(define ∧
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([x : A][y : B])
|
||||
(λ ([C : *])
|
||||
(λ ([f : (→ A B C)])
|
||||
(f x y))))))
|
||||
(check-type ∧ : (∀ (A B) (→ A B (And A B))))
|
||||
|
||||
;; And type elim
|
||||
(define proj1
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e∧ : (And A B)])
|
||||
((e∧ A) (λ ([x : A][y : B]) x)))))
|
||||
(define proj2
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e∧ : (And A B)])
|
||||
((e∧ B) (λ ([x : A][y : B]) y)))))
|
||||
;; bad proj2: (e∧ A) should be (e∧ B)
|
||||
(typecheck-fail
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e∧ : (And A B)])
|
||||
((e∧ A) (λ ([x : A][y : B]) y))))
|
||||
#:verb-msg
|
||||
"expected (→ A B C), given (Π ((x : A) (y : B)) B)")
|
||||
(check-type proj1 : (∀ (A B) (→ (And A B) A)))
|
||||
(check-type proj2 : (∀ (A B) (→ (And A B) B)))
|
||||
|
||||
;((((conj q) p) (((proj2 p) q) a)) (((proj1 p) q) a)))))
|
||||
(define and-commutes
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e∧ : (And A B)])
|
||||
((∧ B A) ((proj2 A B) e∧) ((proj1 A B) e∧)))))
|
||||
;; bad and-commutes, dont flip A and B: (→ (And A B) (And A B))
|
||||
(typecheck-fail
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e∧ : (And A B)])
|
||||
((∧ A B) ((proj2 A B) e∧) ((proj1 A B) e∧))))
|
||||
#:verb-msg
|
||||
"#%app: type mismatch: expected A, given C") ; TODO: err msg should be B not C?
|
||||
(check-type and-commutes : (∀ (A B) (→ (And A B) (And B A))))
|
||||
|
||||
;; nats -----------------------------------------------------------------------
|
||||
(define-type-alias nat (∀ (A) (→ A (→ A A) A)))
|
||||
|
||||
(define-type-alias z (λ ([Ty : *]) (λ ([zero : Ty][succ : (→ Ty Ty)]) zero)))
|
||||
(define-type-alias s (λ ([n : nat])
|
||||
(λ ([Ty : *])
|
||||
(λ ([zero : Ty][succ : (→ Ty Ty)])
|
||||
(succ ((n Ty) zero succ))))))
|
||||
(check-type z : nat)
|
||||
(check-type s : (→ nat nat))
|
||||
|
||||
(define-type-alias one (s z))
|
||||
(define-type-alias two (s (s z)))
|
||||
(check-type one : nat)
|
||||
(check-type two : nat)
|
||||
|
||||
(define-type-alias plus
|
||||
(λ ([x : nat][y : nat])
|
||||
((x nat) y s)))
|
||||
(check-type plus : (→ nat nat nat))
|
||||
|
||||
;; equality -------------------------------------------------------------------
|
||||
|
||||
(check-type (eq-refl one) : (= one one))
|
||||
(check-type (eq-refl one) : (= (s z) one))
|
||||
(check-type (eq-refl two) : (= (s (s z)) two))
|
||||
(check-type (eq-refl two) : (= (s one) two))
|
||||
(check-type (eq-refl two) : (= two (s one)))
|
||||
(check-type (eq-refl two) : (= (s (s z)) (s one)))
|
||||
(check-type (eq-refl two) : (= (plus one one) two))
|
||||
(check-not-type (eq-refl two) : (= (plus one one) one))
|
||||
|
||||
;; symmetry of =
|
||||
(check-type
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e : (= A B)])
|
||||
(eq-elim A (λ ([W : *]) (= W A)) (eq-refl A) B e)))
|
||||
: (∀ (A B) (→ (= A B) (= B A))))
|
||||
(check-not-type
|
||||
(λ ([A : *][B : *])
|
||||
(λ ([e : (= A B)])
|
||||
(eq-elim A (λ ([W : *]) (= W A)) (eq-refl A) B e)))
|
||||
: (∀ (A B) (→ (= A B) (= A B))))
|
||||
|
||||
;; transitivity of =
|
||||
(check-type
|
||||
(λ ([X : *][Y : *][Z : *])
|
||||
(λ ([e1 : (= X Y)][e2 : (= Y Z)])
|
||||
(eq-elim Y (λ ([W : *]) (= X W)) e1 Z e2)))
|
||||
: (∀ (A B C) (→ (= A B) (= B C) (= A C))))
|
||||
Loading…
Reference in New Issue
Block a user