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-;;; 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 (exact-positive-integer? x)
-  (and (exact-integer? x)
-       (positive? x)))
-
-(define (exact-negative-integer? x)
-  (and (exact-integer? x)
-       (negative? x)))
-
-(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)))))
-
-(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-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-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-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-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)))
-  )