;;; 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 (natural? size implies prelude)
  #:use-module (vikalpa))

(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-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-function zero? (x)
    (equal? 0 x))

  (define-function natural? (x)
    (and (integer? x)
         (or (zero? x)
             (< x 0))))

  (define-totality-claim natural? natural? <)

  (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 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-axiom pair/cons (x y)
    (equal? (pair? (cons x y)) '#t))

  (define-axiom cons/car+cdr (x)
    (implies (pair? x)
             (equal? (cons (car x) (cdr x)) x)))

  (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 equal-car (x1 y1 x2 y2)
    (implies (equal? (cons x1 y1) (cons x2 y2))
             (equal? x1 x2)))

  (define-axiom equal-cdr (x1 y1 x2 y2)
    (implies (equal? (cons x1 y1) (cons x2 y2))
             (equal? y1 y2)))

  (define-axiom pred/succ (x)
    (implies (natural? x)
             (equal? (pred (succ x)) x)))

  (define-axiom succ/pred (x)
    (implies (natural? x)
             (not (zero? x))
             (equal? (succ (pred x)) x)))

  (define-axiom less/pred (x)
    (implies (natural? x)
             (equal? (< (pred x) x) #t)))

  (define-function + (x y)
    (if (zero? 0 x)
        y
        (succ (+ (pred x) y))))

  ;; (define-proof if-implies
  ;;   ()
  ;;   (((1) if-same (set x x))
  ;;    ((1 2 1) if-nest)
  ;;    ((1 3 1) if-nest)
  ;;    ((1 3) if-true)
  ;;    (() equal-same)))

  ;; (define-proof less-than/pred
  ;;   (natural-induction x)
  ;;   (((2 2) if-nest)
  ;;    ((2 2) if-not)
  ;;    ((2 2) if-nest)
  ;;    ((2 3 2) if-nest)
  ;;    ((2 3 2) if-nest)
  ;;    ((2 3) if-implies)
  ;;    ((2 3 2) if-implies)
     
  ;;    (() error)
  ;;    (() error)))

  ;; (define-proof less-than/pred-1
  ;;   ()
  ;;   (((2 2) if-same (set x (natural? (pred x))))
  ;;    ((2 2 2) if-same (set x (not (zero? (pred x)))))
  ;;    ((2 2 2) if-same (set x (not (zero? (pred x)))))
  ;;    ;; ((2 2 3 1 1) zero?)
  ;;    ;; ((2 2 3 1 1) if-nest)
  ;;    ;; ((2 2 3 1) if-false)
  ;;    ;; ((2 2 3 1) if-false)
  ;;    ))
  


  ;; (define-theorem natural/add1 (x)
  ;;     (implies (natural? x)
  ;;              (equal? (natural? (add1 x)) #t)))

  #;
  (define-axiom natural/sub1 (x)
  (if (natural? x)
  (if (< '0 x)
  (equal? (natural? (sub1 x)) '#t)
  '#t)
  '#t))

  #;
  (define-theorem less-than/sub1 (x)
  (implies (natural? x)
  (< '0 x)
  (equal? (< (sub1 x) x) '#t)))

  #;
  (define-axiom add1/sub1 (x)
  (if (natural? x)
  (if (equal? '0 x)
  '#t
  (equal? (add1 (sub1 x)) x))
  '#t))

  #;
  (define-built-in-function + (x y)
  (if (natural? x)
  (if (equal? '0 x)
  y
  (add1 (+ (sub1 x) y)))
  'undefined))

  #;
  (define-axiom natural/size (x)
  (equal? (natural? (size x))
  '#t))

  #;
  (define-axiom size/car (x)
  (if (pair? x)
  (equal? (< (size (car x)) (size x))
  '#t)
  '#t))

  #;
  (define-axiom size/cdr (x)
  (if (pair? x)
  (equal? (< (size (cdr x)) (size x))
  '#t)
  '#t))
  
  #;
  (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)))
  )