path: root/examples/prelude.scm

;;; 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-syntax-rule (define-axiom/is p1 p2)
  (define-axiom (is p1 p2) (x) (is (p1 x) (p2 x))))

(define-syntax-rule (define-theorem/is p1 p2)
  (define-theorem (is p1 p2) (x) (is (p1 x) (p2 x))))

(define-syntax-rule (define-proof/is p1 p2 pc)
  (define-proof (is p1 p2)
    ((rewrite (2) if-same (set x (pc x)))
     (rewrite (2 2 1) (is pc p2))
     (eval (2 2))
     (rewrite (2 1) (is p1 pc))
     (rewrite (2) if-true)
     (rewrite () if-same))))

(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 predicate ()
    ((predicate x) (implies x (equal? x #t))))
  (define-syntax-rules is ()
    ((is x y) (implies x (equal? y #t)))
    ((is x y . ys) (implies x (is y . ys)))))

(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 (predicate number?) (x) (predicate (number? x)))

  (define-core-function real? (x) real?)
  (define-axiom (predicate real?) (x) (predicate (real? x)))
  (define-axiom/is real? number?)

  (define-core-function integer? (x) integer?)
  (define-axiom (predicate integer?) (x) (predicate (integer? x)))
  (define-axiom/is integer? real?)
  (define-theorem/is integer? number?)
  (define-proof/is integer? number? real?)

  (define-core-function/guard exact? (x) (number? x) exact?)
  (define-axiom (predicate exact?) (x) (predicate (exact? x)))
  (define-function/guard inexact? (x)
    (number? x)
    (not (exact? x)))
  (define-theorem (predicate inexact?) (x) (predicate (inexact? x)))

  (define-core-function exact-integer? (x) exact-integer?)
  (define-axiom (predicate exact-integer?) (x) (predicate (exact-integer? x)))
  (define-axiom/is exact-integer? integer?)
  (define-axiom/is exact-integer? exact?)
  (define-theorem/is exact-integer? real?)
  (define-proof/is exact-integer? real? integer?)
  (define-theorem/is exact-integer? number?)
  (define-proof/is exact-integer? number? real?)
  (define-core-function/guard < (x y) (and (real? x) (real? y)) <)
  (define-axiom (predicate <) (x y) (predicate (< x y)))

  (define-function/guard positive? (x)
    (real? x)
    (< 0 x))
  (define-theorem (predicate positive?) (x) (predicate (positive? x)))
  (define-function/guard negative? (x)
    (real? x)
    (< x 0))
  (define-theorem (predicate negative?) (x) (predicate (negative? x)))
  (define-function exact-positive-integer? (x)
    (and (exact-integer? x)
         (positive? x)))
  (define-theorem (predicate exact-positive-integer?) (x)
    (predicate (exact-positive-integer? x)))
  (define-theorem/is exact-positive-integer? exact-integer?)
  (define-theorem/is exact-positive-integer? exact?)
  (define-proof/is exact-positive-integer? exact? exact-integer?)
  (define-theorem/is exact-positive-integer? integer?)
  (define-proof/is exact-positive-integer? integer? exact-integer?)
  (define-theorem/is exact-positive-integer? real?)
  (define-proof/is exact-positive-integer? real? exact-integer?)
  (define-theorem/is exact-positive-integer? number?)
  (define-proof/is exact-positive-integer? number? exact-integer?)

  (define-function exact-zero? (x) (equal? x 0))
  (define-theorem (predicate exact-zero?) (x)
    (predicate (exact-zero? x)))
  (define-theorem/is exact-zero? exact-integer?)
  (define-theorem/is exact-zero? integer?)
  (define-proof/is exact-zero? integer? exact-integer?)
  (define-theorem/is exact-zero? exact?)
  (define-proof/is exact-zero? exact? exact-integer?)
  (define-theorem/is exact-zero? real?)
  (define-proof/is exact-zero? real? exact-integer?)
  (define-theorem/is exact-zero? number?)
  (define-proof/is exact-zero? number? real?)

  (define-function natural? (x)
    (if (exact-zero? x)
        #t
        (exact-positive-integer? x)))

  (define-theorem (predicate natural?) (x)
    (predicate (natural? x)))
  (define-theorem/is natural? exact-integer?)
  (define-theorem/is natural? integer?)
  (define-proof/is natural? integer? exact-integer?)
  (define-theorem/is natural? exact?)
  (define-proof/is natural? exact? exact-integer?)
  (define-theorem/is natural? real?)
  (define-proof/is natural? real? exact-integer?)
  (define-theorem/is natural? number?)
  (define-proof/is natural? number? real?)

  (define-theorem/is exact-zero? natural?)
  (define-theorem/is exact-positive-integer? natural?)
  (set-measure-predicate natural?)
  (set-measure-less-than <)

(define-core-function/guard + (x y) (and (number? x) (number? y)) +)
  (define-core-function/guard - (x y) (and (number? x) (number? y)) -)
  (define-function/guard succ (x) (natural? x) (+ x 1))
  (define-function/guard pred (x) (exact-positive-integer? x) (- x 1))
  (define-function/guard natural-induction (x)
    (natural? x)
    (if (exact-zero? x)
        #t
        (succ (natural-induction (pred x)))))
  (admit natural-induction)

  (define-axiom (+ natural? natural?) (x y)
    (implies (natural? x)
             (natural? y)
             (equal? (+ x y)
                     (if (exact-zero? x)
                         y
                         (succ (+ (pred x) y))))))
  (define-axiom pred/succ (x)
    (implies (natural? x)
             (equal? (pred (succ x)) x)))
  (define-theorem succ/pred (x)
    (implies (exact-positive-integer? x)
             (equal? (succ (pred x)) x)))
  (define-theorem (is natural? (o natural? succ)) (x)
    (is (natural? x) (natural? (succ x))))

  (define-theorem (is (o natural? pred) natural?) (x)
    (is (natural? (pred x))
        (natural? x)))

  (define-axiom pred/zero (x)
    (equal? (natural? (pred 0)) #f))

  (define-theorem (is (o not natural?) (o not natural? pred)) (x)
    (is (not (natural? x))
        (not (natural? (pred x)))))

  (define-theorem (is (o natural?)
                      (o not exact-zero?)
                      exact-positive-integer?)
    (x)
    (is (not (natural? x))
        (not (exact-zero? x))
        (exact-positive-integer? x)))

  (define-function/guard natural-induction-+ (x y)
    (and (natural? x) (natural? y))
    (if (exact-zero? x)
        y
        (succ (natural-induction-+ (pred x) y))))
  (admit natural-induction-+)

  (define-theorem (is natural? (o not exact-zero?) exact-positive-integer?) (x)
    (is (natural? x)
        (not (exact-zero? x))
        (exact-positive-integer? x)))

  (define-theorem (implies natural?
                           (equal? (o natural? pred)
                                   exact-positive-integer?))
    (x)
    (implies (natural? x)
             (equal? (natural? (pred x))
                     (exact-positive-integer? x))))

  (define-theorem (+ (-> natural? natural? natural?)) (x y)
    (implies (natural? x)
             (natural? y)
             (natural? (+ x y)))))

(define-system prelude/measure/natural (prelude/number)
  (set-measure-predicate natural?)
  (set-measure-predicate <))

(define-system prelude/pair (prelude/measure/natural)
  (define-core-function pair? (x) pair?)
  (define-core-function cons (x y) cons)
  (define-core-function/guard car (x) (pair? x) car)
  (define-core-function/guard cdr (x) (pair? x) cdr)
  (define-function size (x)
    (if (pair? x)
        (+ (size (car x))
           (size (cdr x)))
        1))
  (admit size))


(define-system prelude (prelude/pair)
  (define-proof inexact?
    ((rewrite (2) if-nest)
     (rewrite () if-same)))
  (define-proof (predicate inexact?)
    ((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 (predicate positive?)
    ((rewrite (1) positive?)
     (rewrite (2 1) positive?)
     (rewrite (2 1) (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 (predicate negative?)
    ((rewrite (1) negative?)
     (rewrite (2 1) negative?)
     (rewrite (2 1) (predicate <))
     (eval (2))
     (rewrite () if-same)))
  (define-proof exact-positive-integer?
    ((rewrite (1 2) (is exact-integer? real?))
     (rewrite (1) if-same)
     (eval ())))
  (define-proof (predicate exact-positive-integer?)
    ((rewrite (1) exact-positive-integer?)
     (rewrite () if-same (set x (exact-integer? x)))
     (rewrite (3 1) if-nest)
     (rewrite (3) if-false)
     (rewrite (2 1) if-nest)
     (rewrite (2 2 1) exact-positive-integer?)
     (rewrite (2 2 1 1) (predicate exact-integer?))
     (rewrite (2 2 1 2) (predicate positive?))
     (eval (2 2))
     (rewrite (2) if-same)
     (rewrite () if-same)))
  (define-proof (predicate exact-zero?)
    ((rewrite (1) exact-zero?)
     (rewrite (2 1 1) equal-if)
     (eval (2))
     (rewrite () if-same)))
  (define-proof (predicate natural?)
    ((rewrite (1) natural?)
     (rewrite () if-same (set x (exact-zero? x)))
     (rewrite (2 1) if-nest)
     (rewrite (2) if-true)
     (rewrite (3 1) if-nest)
     (rewrite (1) exact-zero?)
     (rewrite (2 1 1) equal-if)
     (eval (2))
     (rewrite (1) exact-zero?)
     (rewrite (3 2 1) natural?)
     (rewrite (3 2 1) if-nest)
     (rewrite (3 2 1) (predicate exact-positive-integer?))
     (eval (3 2))
     (rewrite (3) if-same)
     (rewrite () if-same)))
  (define-proof (is exact-positive-integer? exact-integer?)
    ((rewrite (1) exact-positive-integer?)
     (rewrite () if-same (set x (exact-integer? x)))
     (rewrite (3 1) if-nest)
     (rewrite (2 1) if-nest)
     (rewrite (2 2 1) (predicate exact-integer?))
     (eval (2 2))
     (rewrite (2) if-same)
     (rewrite (3) if-false)
     (rewrite ()  if-same)))
  (define-proof (is exact-zero? exact-integer?)
    ((rewrite (1) exact-zero?)
     (rewrite (2 1 1) equal-if)
     (eval (2))
     (rewrite () if-same)))
  (define-proof (is natural? exact-integer?)
    ((rewrite (1) natural?)
     (rewrite () if-same (set x (exact-zero? x)))
     (rewrite (2 1) if-nest)
     (rewrite (2) if-true)
     (rewrite (2 1) (is exact-zero? exact-integer?))
     (eval (2))
     (rewrite (3 1) if-nest)
     (rewrite (3 2 1) (is exact-positive-integer? exact-integer?))
     (eval (3 2))
     (rewrite (3) if-same)
     (rewrite () if-same)))
  (define-proof succ
    ((rewrite (2 1 1) (is natural? number?))
     (eval (2))
     (rewrite ()  if-same)))
  (define-proof pred
    ((rewrite (2 1 1) (is exact-positive-integer? number?))
     (eval (2))
     (rewrite () if-same)))

  (define-proof (+ (-> natural? natural? natural?))
    ((induction (2 2) (natural-induction-+ x y))
     (rewrite (2 2 2 1) (+ natural? natural?))
     (rewrite (2 2 2 1) if-nest)
     (rewrite (2 2 2) (predicate natural?))
     (rewrite (2 2 3 2) if-same (set x (natural? (pred x))))
     (rewrite (2 2 3 2 2 1) (+ natural? natural?))
     (rewrite (2 2 3 2 2 1) if-nest)
     (rewrite (2 2 3 2 2) (is natural? (o natural? succ)))
     (rewrite (2 2 3 2 1) (implies natural?
                                   (equal? (o natural? pred)
                                           exact-positive-integer?)))
     (rewrite (2 2 3 2 1) (is natural? (o not exact-zero?)
                              exact-positive-integer?))
     (rewrite (2 2 3 2) if-true)
     (rewrite (2 2 3) if-same)
     (rewrite (2 2) if-same)
     (rewrite (2) if-same)
     (rewrite () if-same)))

  (define-proof (is exact-zero? natural?)
    ((rewrite (1) exact-zero?)
     (rewrite (2 1 1) equal-if)
     (eval (2))
     (rewrite () if-same)))

  (define-proof (is exact-positive-integer? natural?)
    ((rewrite (2 1) natural?)
     (rewrite (2 1 3) (predicate exact-positive-integer?))
     (rewrite (2 1) if-same)
     (eval (2))
     (rewrite () if-same)))

  (define-proof (is (o natural? pred) natural?)
    ((rewrite () if-same (set x (natural? x)))
     (rewrite (2 2 1) (predicate natural?))
     (eval (2 2))
     (rewrite (3 2 1) equal-if-false)
     (eval (3 2))
     (rewrite (2) if-same)))
  (admit (is natural? (o natural? succ))))