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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 (atom? size implies natural? prelude)
#:use-module (vikalpa))
(define (atom? x)
(not (pair? x)))
(define (natural? x)
(and (integer? x)
(not (negative? x))))
(define (size x)
(if (pair? x)
(+ 1
(size (car x))
(size (cdr x)))
0))
(define-syntax implies
(syntax-rules ()
((_ x y) (if x y #t))
((_ x y z rest ...)
(if x (implies y z rest ...) #t))))
(define-system prelude ()
(define-function atom? (x)
(not (pair? x)))
(define-syntax-rules list ()
((list) '())
((list x . y) (cons x (list . y))))
(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 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 atom/cons (x y)
(equal? (atom? (cons x y)) '#f))
(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 cons/car+cdr (x)
(if (atom? x)
'#t
(equal? (cons (car x) (cdr x)) x)))
(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 natural?/size (x)
(equal? (natural? (size x))
'#t))
(define-axiom </size/car (x)
(if (atom? x)
'#t
(equal? (< (size (car x)) (size x))
'#t)))
(define-axiom </size/cdr (x)
(if (atom? x)
'#t
(equal? (< (size (cdr x)) (size x))
'#t)))
(define-axiom if-not (x y z)
(equal? (if (not x) y z)
(if x z y)))
(define-axiom sub1/add1 (x)
(implies (natural? x)
(equal? (sub1 (add1 x)) x)))
(define-axiom natural?/0 (x)
(equal? (natural? '0) '#t))
(define-axiom natural?/add1 (x)
(implies (natural? x)
(equal? (natural? (add1 x)) '#t)))
(define-axiom sub1/add1 (x)
(implies (natural? x)
(equal? (sub1 (add1 x)) x)))
(define-axiom </sub1 (x)
(implies (natural? x)
(equal? (< (sub1 x) x) '#t)))
(define-axiom common-add1 (x y)
(implies (natural? x)
(natural? y)
(equal? (equal? (add1 x) (add1 y))
(equal? x y))))
(define-axiom false-if (x)
(if x '#t (equal? x '#f)))
(define-axiom equal?/01 (x)
(equal? (equal? '0 '1) #f))
(define-axiom natural?/sub1 (x)
(if (natural? x)
(if (equal? '0 x)
'#t
(equal? (natural? (sub1 x)) '#t))
'#t))
(define-axiom add1/sub1 (x)
(if (natural? x)
(if (equal? '0 x)
'#t
(equal? (add1 (sub1 x)) x))
'#t))
(define-primitive-function + (x y)
(if (natural? x)
(if (equal? '0 x)
y
(add1 (+ (sub1 x) y)))
'undefined))
(define-proof +
(total/natural? x)
(((2) if-nest)
((2 3) </sub1)
((2) if-same)
(() if-same)))
(define-theorem thm-1+1=2 ()
(equal? (+ 1 1) 2))
(define-function natural-induction (n)
(if (natural? n)
(if (equal? '0 n)
'0
(add1 (natural-induction (sub1 n))))
'undefined))
(define-proof natural-induction
(total/natural? n)
(((2) if-nest)
((2 3) </sub1)
((2) if-same)
(() if-same)))
(define-theorem equal?/add1 (n)
(if (natural? n)
(equal? (equal? n (add1 n)) #f)
#t))
(define-proof equal?/add1
(induction (natural-induction n))
(((2 2 2 1 1) equal-if)
((2 2 2 1 2 1) equal-if)
((2 2 2 1) equal?/01)
((2 2 2) equal-same)
((2 2) if-same)
((2 3 1 1) natural?/sub1)
((2 3 1) if-true)
((2 3 2 2 1 1) add1/sub1)
((2 3 2 2 1 2 1) add1/sub1)
((2 3 2 2) if-same (set x (natural? (add1 (sub1 n)))))
((2 3 2 2) if-same (set x (natural? (sub1 n))))
((2 3 2 2 2 2 1) common-add1
;; ルール探索のアルゴリズムにバグがある
(set x (sub1 n))
(set y (add1 (sub1 n))))
((2 3 2 2 2 2 1) equal-if)
((2 3 2 2 2 2) equal-same)
((2 3 2 2 2 1) natural?/add1)
((2 3 2 2 2) if-true)
((2 3 2 2 1) natural?/sub1)
((2 3 2 2) if-true)
((2 3 2) if-same)
((2 3) if-same)
((2) if-same)
((3) if-nest)
(() if-same)))
(define-proof thm-1+1=2
()
((() if-same (set x (natural? '0)))
((2 1) +)
((2 1 2 2 1) equal-if)
((2 1 2 3 1 1) sub1/add1)
((2 1 2 3 1) +)
((2 1 2 3 1 2 1) equal-same)
((2 1 2 3 1 2) if-true)
((2 1 2 3 1 1) natural?/0)
((2 1 2 3 1) if-true)
((2 1 2) if-same)
((2 1 1) natural?/add1)
((2 1) if-true)
((2) equal-same)
((1) natural?/0)
(() if-true))))
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