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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 the-little-prover)
#:export (atom? nat? the-little-prover)
#:use-module (vikalpa))
(define (atom? x)
(not (pair? x)))
(define (nat? x)
(and (integer? x)
(<= 0 x)))
(define-system the-little-prover ()
(define-core-function atom? (x) atom?)
(define-core-function nat? (x) nat?)
(define-core-function < (x y)
(lambda (x y)
(if (number? x)
(if (number? y)
(< x y)
(< x 0))
(if (number? y)
(< 0 y)
#f))))
(set-measure-predicate nat?)
(set-measure-less-than <)
(define-core-function + (x y)
(lambda (x y)
(if (number? x)
(if (number? y)
(+ x y)
x)
(if (number? y)
y
0))))
(define-core-function cons (x y) cons)
(define-core-function car (x)
(lambda (x)
(if (atom? x) '() (car x))))
(define-core-function cdr (x)
(lambda (x)
(if (atom? x) '() (cdr x))))
(define-trivial-total-function size (x)
(if (atom? x)
0
(+ 1
(+ (size (car x))
(size (cdr x))))))
;; Axioms of Equal
(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))
;; Axioms of Cons
(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)))
;; Axioms of If
(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-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)))
;; Axioms of Size
(define-axiom nat/size (x)
(equal? (nat? (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)))
;; Axioms of `+` and `<`
(define-axiom identity-+ (x)
(if (nat? 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 positive-+ (x y)
(if (< '0 x)
(if (< '0 y)
(equal? (< '0 (+ x y)) #t)
#t)
#t))
(define-axiom nat/+ (x y)
(if (nat? x)
(if (nat? y)
(equal? (nat? (+ x y)) #t)
#t)
#t))
(define-axiom common-addends-< (x y z)
(equal? (< (+ x z) (+ y z))
(< x y)))
;; Prelude
(define-function list-induction (x)
(if (atom? x)
x
(cons (car x)
(list-induction (cdr x)))))
(define-function star-induction (x)
(if (atom? x)
x
(cons (star-induction (car x))
(star-induction (cdr x)))))
(define-proof list-induction
(size x)
(((2 3) size/cdr)
((2) if-same)
((1) nat/size)
(() if-true)))
(define-proof star-induction
(size x)
(((2 3 1) size/car)
((2 3 2) size/cdr)
((2 3) if-true)
((2) if-same)
((1) nat/size)
(() if-true))))
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