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;;; Vikalpa --- Proof Assistant
;;; Copyright © 2020 Masaya Tojo <masaya@tojo.tokyo>
;;;
;;; This file is part of Vikalpa.
;;;
;;; Vikalpa is free software; you can redistribute it and/or modify it
;;; under the terms of the GNU General Public License as published by
;;; the Free Software Foundation; either version 3 of the License, or
;;; (at your option) any later version.
;;;
;;; Vikalpa is distributed in the hope that it will be useful, but
;;; WITHOUT ANY WARRANTY; without even the implied warranty of
;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
;;; General Public License for more details.
;;;
;;; You should have received a copy of the GNU General Public License
;;; along with Vikalpa. If not, see <http://www.gnu.org/licenses/>.
(define-module (vikalpa prelude)
#:export (prelude)
#:use-module (vikalpa))
(define-system prelude ()
(define-core-function natural? (x) (lambda (x)
(and (integer? x)
(< 0 x))))
(define-core-function < (x y) (natural? x) (natural? y) <)
(define-core-function pair? (x) pair?)
(define-core-function cons (x y) cons)
(define-core-function car (x) (pair? x)
(lambda (x) (if (pair? x) (car x) '())))
(define-core-function cdr (x) (pair? x)
(lambda (x) (if (pair? x) (cdr x) '())))
(define-core-function + (x y)
(lambda (x y)
(if (number? x)
(if (number? y)
(+ x y)
x)
(if (number? y)
y
0))))
(set-measure-predicate natural?)
(set-measure-less-than <)
(define-trivial-total-function list-induction (x)
(if (not (pair? x))
x
(cons (car x)
(list-induction (cdr x)))))
(define-trivial-total-function size (x)
(if (not (pair? x))
0
(+ 1
(+ (size (car x))
(size (cdr x))))))
(define-syntax-rules and ()
((and) '#t)
((and x) x)
((and x . xs) (if x (and . xs) #f)))
(define-syntax-rules or ()
((or) '#f)
((or x) x)
((or x . xs) (if x x (or . xs))))
(define-syntax-rules cond (else)
((cond (else e)) e)
((cond (test then) . rest)
(if test then (cond . rest))))
(define-syntax-rules implies ()
((implies x y) (if x y #t))
((implies x y z . rest)
(if x (implies y z . rest) #t)))
(define-axiom if-nest (x y z)
(if x
(equal? (if x y z) y)
(equal? (if x y z) z)))
(define-axiom if-true (x y)
(equal? (if '#t x y) x))
(define-axiom if-false (x y)
(equal? (if '#f x y) y))
(define-axiom if-same (x y)
(equal? (if x y y) y))
(define-axiom if-not (x y z)
(equal? (if (not x) y z)
(if x z y)))
(define-axiom equal-same (x)
(equal? (equal? x x) '#t))
(define-axiom equal-swap (x y)
(equal? (equal? x y) (equal? y x)))
(define-axiom equal-if (x y)
(implies (equal? x y) (equal? x y)))
(define-axiom pair/cons (x y)
(equal? (pair? (cons x y)) '#t))
(define-axiom cons/car+cdr (x)
(implies (pair? x)
(equal? (cons (car x) (cdr x)) x)))
(define-axiom car/cons (x y)
(equal? (car (cons x y)) x))
(define-axiom cdr/cons (x y)
(equal? (cdr (cons x y)) y))
(define-axiom natural/size (x)
(equal? (natural? (size x)) #t))
(define-axiom size/car (x)
(equal? (< (size (car x)) (size x)) #t))
(define-axiom size/cdr (x)
(equal? (< (size (cdr x)) (size x)) #t))
(define-axiom equal-car (x1 y1 x2 y2)
(implies (equal? (cons x1 y1) (cons x2 y2))
(equal? x1 x2)))
(define-theorem caar-cons2 (x y z)
(equal? (car (car (cons (cons x y) z))) x))
(define-function app (x y)
(if (not (pair? x))
y
(cons (car x)
(app (cdr x) y))))
(define-theorem assoc-app (x y z)
(equal? (app x (app y z))
(app (app x y) z)))
(define-proof caar-cons2
(((1 1) car/cons)
((1) car/cons)
(() equal-same)))
(define-proof app
(size x)
(((2 3) size/cdr)
((2) if-same)
((1) natural/size)
(() if-true)))
(define-proof assoc-app
(list-induction x)
()))
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