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;;; Vikalpa --- Proof Assistant
;;; Copyright © 2020, 2021 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 (exact-positive-integer? x)
(and (exact-integer? x)
(positive? x)))
(define (exact-negative-integer? x)
(and (exact-integer? x)
(negative? x)))
(define (succ x)
(if (number? x)
(+ x 1)
1))
(define (pred x)
(if (number? x)
(- x 1)
-1))
(define (negate x)
(if (exact-integer? x)
(- x)
0))
(define-syntax implies
(syntax-rules ()
((_ x y) (if x y #t))
((_ x y z ...) (if x (implies y z ...) #t))))
(define-system prelude/macro ()
(define-syntax-rules cond (else)
((cond (else e)) e)
((cond (test then) . rest)
(if test then (cond . rest))))
(define-syntax-rules or ()
((or) '#f)
((or x) x)
((or x . xs) (if x x (or . xs))))
(define-syntax-rules implies ()
((implies x y) (if x y #t))
((implies x y z . rest)
(if x (implies y z . rest) #t)))
(define-syntax-rules is-a ()
((is-a x y) (implies x (equal? y #t)))))
(define-system prelude/axiom/equal (prelude/macro)
(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 equal-if-false (x y)
(if x #t (equal? x #f))))
(define-system prelude/axiom/if (prelude/axiom/equal)
(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-system prelude/number (prelude/axiom/if)
(define-core-function number? (x) number?)
(define-axiom number-predicate (x)
(is-a (number? x) (number? x)))
(define-core-function real? (x) real?)
(define-axiom real-predicate (x)
(implies (real? x) (equal? (real? x) #t)))
(define-axiom real-is-a-numbers (x)
(implies (real? x) (equal? (number? x) #t)))
(define-core-function integer? (x) integer?)
(define-axiom integer-predicate (x)
(implies (integer? x) (equal? (integer? x) #t)))
(define-axiom integer-is-a-real (x)
(implies (integer? x) (equal? (number? x) #t)))
(define-core-function/guard exact? (x) (number? x) exact?)
(define-axiom exact-predicate (x)
(implies (exact? x) (equal? (exact? x) #t)))
(define-function/guard inexact? (x)
(number? x)
(not (exact? x)))
(define-theorem inexact-predicate (x)
(implies (inexact? x) (equal? (inexact? x) #t)))
(define-core-function exact-integer? (x) exact-integer?)
(define-axiom exact-integer-predicate (x)
(implies (exact-integer? x) (equal? (exact-integer? x) #t)))
(define-axiom exact-integer-is-a-integer (x)
(implies (exact-integer? x) (equal? (integer? x) #t)))
(define-axiom exact-integer-is-a-exact (x)
(implies (exact-integer? x) (equal? (exact? x) #t)))
(define-core-function/guard < (x y) (and (real? x) (real? y)) <)
(define-axiom less-than-predicate (x y)
(implies (< x y) (equal? (< x y) #t)))
(define-function/guard positive? (x)
(real? x)
(< 0 x))
(define-theorem positive-predicate (x)
(implies (positive? x) (equal? (positive? x) #t)))
(define-function/guard negative? (x)
(real? x)
(< x 0))
(define-theorem negative-predicate (x)
(implies (negative? x) (equal? (negative? x) #t)))
(define-function exact-positive-integer? (x)
(and (exact-integer? x)
(positive? x)))
(define-theorem exact-positive-integer-predicate (x)
(implies (exact-positive-integer? x)
(equal? (exact-positive-integer? x) #t)))
(define-function exact-zero? (x) (equal? x 0))
(define-theorem exact-zero-predicate (x)
(implies (exact-zero? x) (equal? (exact-zero? x) #t)))
(define-function natural? (x)
(if (exact-zero? x)
#t
(exact-positive-integer? x)))
(define-theorem (predicate natural?) (x)
(implies (natural? x) (equal? (natural? x) #t))))
(define-system prelude/measure/natural (prelude/number)
(set-measure-predicate natural?)
(set-measure-predicate <))
(define-system prelude (prelude/measure/natural)
(define-proof inexact?
((rewrite (2) if-nest)
(rewrite () if-same)))
(define-proof inexact-predicate
((rewrite (1) inexact?)
(rewrite () if-not)
(rewrite (3 1) inexact?)
(rewrite (3 1 1) equal-if-false)
(eval (3))
(rewrite () if-same)))
(define-proof positive?
((eval (2 1 1))
(rewrite (2 1) if-true)
(rewrite (2) if-nest)
(rewrite () if-same)))
(define-proof positive-predicate
((rewrite (1) positive?)
(rewrite (2 1) positive?)
(rewrite (2 1) less-than-predicate)
(eval (2))
(rewrite () if-same)))
(define-proof negative?
((rewrite (2 1) if-nest)
(eval (2 1))
(rewrite (2) if-true)
(rewrite () if-same)))
(define-proof negative-predicate
((rewrite (1) negative?)
(rewrite (2 1) negative?)
(rewrite (2 1) less-than-predicate)
(eval (2))
(rewrite () if-same))))
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