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;;; Vikalpa --- Proof Assistant
;;; Copyright © 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 the-little-prover)
#:export (prelude)
#:use-module (vikalpa))
(define (bool->t-nil x)
(if x
't
'nil))
(define (size x)
(if (pair? x)
(+ (size (car x))
(size (cdr x)))
1))
(define-system axioms-of-equal (core-system/equal-t-nil)
(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)
(if (equal x y)
(equal x y)
't)))
(define-system axioms-of-cons (axioms-of-equal)
(define-core-function atom (x) (lambda (x) (bool->t-nil (not (pair? x)))))
(define-core-function cons (x y) cons)
(define-core-function car (x) (lambda (x) (if (not (pair? x)) '() (car x))))
(define-core-function cdr (x) (lambda (x) (if (not (pair? x)) '() (cdr x))))
(define-axiom atom/cons (x y)
(equal (atom (cons x y)) 'nil))
(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-system axioms-of-if (axioms-of-cons)
(define-axiom if-true (x y)
(equal (if 't x y) x))
(define-axiom if-false (x y)
(equal (if 'nil x y) y))
(define-axiom if-same (x y)
(equal (if x y y) y))
(define-axiom if-nest-A (x y z)
(if x
(equal (if x y z) y)
't))
(define-axiom if-nest-E (x y z)
(if x
't
(equal (if x y z) z))))
(define-system definitions-of-measure (axioms-of-if)
(define-core-function natp (x)
(lambda (x)
(bool->t-nil
(and (exact-integer? x)
(<= 0 x)))))
(set-measure-predicate natp)
(define-core-function < (x y)
(lambda (x y)
(bool->t-nil
(and (exact-integer? x)
(exact-integer? y)
(< x y)))))
(set-measure-less-than <))
(define-system axioms-of-size (definitions-of-measure)
(define-core-function + (x y)
(lambda (x y)
(if (and (exact-integer? x)
(exact-integer? y))
(+ x y)
0)))
(define-core-function size (x) size)
(define-axiom natp/size (x)
(equal (natp (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-system axioms-of-+-and-< (axioms-of-size)
(define-axiom identity-+ (x)
(if (natp x)
(equal (+ 0 x) x)
't))
(define-axiom commute-+ (x y)
(equal (+ x y) (+ y x)))
(define-axiom associate-+ (x y z)
(equal (+ (+ x y) z)
(+ x (+ y z))))
(define-axiom positives-+ (x y)
(if (< 0 x)
(if (< 0 x)
(equal (< 0 (+ x y)) 't)
't)
't))
(define-axiom natp/+ (x y)
(if (natp x)
(if (natp y)
(equal (natp (+ x y)) 't)
't)
't))
(define-axiom common-addends-< (x y z)
(equal (< (+ x z) (+ y z))
(< x y))))
(define-system inductions (axioms-of-+-and-<)
(define-function list-induction (x)
(if (atom x)
't
(list-induction (cdr x))))
(define-function star-induction (x)
(if (atom x)
't
(cons (star-induction (car x))
(star-induction (cdr x)))))
(define-proof list-induction
(size x)
((rewrite (1) natp/size)
(rewrite () if-true)
(rewrite (3) size/cdr)
(rewrite () if-same)))
(define-proof star-induction
(size x)
((rewrite (1) natp/size)
(rewrite () if-true)
(rewrite (3 1) size/car)
(rewrite (3 2) size/cdr)
(eval (3))
(rewrite () if-same))))
(define-system prelude (inductions))
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