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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-syntax-rule (define-proof/is name p t1 t2)
(define-proof name
(((2) if-same (set x (p x)))
((2 2 1) t1)
((2 2) equal-same)
((2) t2)
(() if-same))))
(define-system prelude/macros ()
(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-system prelude/equal (prelude/macros)
(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-system prelude/if (prelude/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/if)
(define-core-function number? (x) number?)
(define-core-function rational? (x) rational?)
(define-core-function integer? (x) integer?)
(define-function zero? (x)
(equal? x 0))
(define-core-function < (x y)
(lambda (x y)
(if (rational? x)
(if (rational? y)
(< x y)
(< x 0))
(if (rational? y)
(< 0 y)
#f))))
(define-axiom axiom-< (x y)
(if (rational? x)
(if (rational? y)
#t
(equal? (< x y) (< x 0)))
(if (rational? y)
(equal? (< x y) (< 0 y))
(equal? (< x y) #f))))
(define-function natural? (x)
(if (integer? x)
(if (zero? x)
#t
(< 0 x))
#f))
(define-axiom rational-is-number (x y z)
(implies (rational? x) (equal? (if (number? x) y z) y)))
(define-axiom integer-is-rational (x y z)
(implies (integer? x) (equal? (if (rational? x) y z) y)))
(define-theorem integer-is-number (x y z)
(implies (integer? x) (equal? (if (number? x) y z) y)))
(define-theorem natural-is-integer (x y z)
(implies (natural? x) (equal? (if (integer? x) y z) y)))
(define-theorem natural-is-rational (x y z)
(implies (natural? x) (equal? (if (rational? x) y z) y)))
(define-theorem natural-is-number (x y z)
(implies (natural? x) (equal? (if (number? x) y z) y)))
(define-core-function + (x y)
(lambda (x y)
(if (number? x)
(if (number? y)
(+ x y)
(+ x 0))
(if (number? y)
(+ 0 y)
0)))))
(define-system prelude/measure (prelude/number)
(set-measure-predicate natural?)
(set-measure-less-than <))
(define-system prelude/pair (prelude/measure)
(define-core-function pair? (x) pair?)
(define-core-function cons (x y) cons)
(define-core-function car (x)
(lambda (x) (if (pair? x) (car x) '())))
(define-core-function cdr (x)
(lambda (x) (if (pair? x) (cdr x) '()))))
(define-system prelude/tree (prelude/pair)
(define-trivial-total-function size (x)
(if (pair? x)
(+ 1
(+ (size (car x))
(size (cdr x))))
0))
(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-function tree-induction (x)
(if (not (pair? x))
x
(cons (tree-induction (car x))
(tree-induction (cdr x))))))
(define-system prelude/proofs (prelude/tree)
(define-proof/is integer-is-number
rational?
rational-is-number
integer-is-rational)
(define-proof natural-is-integer
((() if-same (set x (integer? x)))
((2 2 1) if-nest)
((2 2) equal-same)
((2) if-same)
((3 1) natural?)
((3 1) if-nest)
((3) if-false)
(() if-same)))
(define-proof/is natural-is-rational
integer?
integer-is-rational
natural-is-integer)
(define-proof/is natural-is-number
rational?
rational-is-number
natural-is-rational)
(define-proof tree-induction
(size x)
(((2 3 1) size/car)
((2 3 2) size/cdr)
((2 3) if-true)
((2) if-same)
((1) natural/size)
(() if-true))))
(define-system prelude (prelude/proofs))
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