[1] | 1 | ;;; -*- Mode: Lisp; Syntax: Common-Lisp; Package: Grobner; Base: 10 -*-
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| 2 | #|
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| 3 | $Id: prover.lisp,v 1.3 2009/01/22 04:06:51 marek Exp $
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| 4 | *--------------------------------------------------------------------------*
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| 5 | | Copyright (C) 1994, Marek Rychlik (e-mail: rychlik@math.arizona.edu) |
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| 6 | | Department of Mathematics, University of Arizona, Tucson, AZ 85721 |
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| 7 | | |
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| 8 | | Everyone is permitted to copy, distribute and modify the code in this |
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| 9 | | directory, as long as this copyright note is preserved verbatim. |
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| 10 | *--------------------------------------------------------------------------*
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| 11 | |#
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| 12 | (defpackage "PROVER"
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| 13 | (:use "COMMON-LISP" "GROBNER" "PARSE" "ORDER" "COEFFICIENT-RING" "PRINTER")
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| 14 | (:export identical-points perpendicular parallel collinear translate-theorem prove-theorem
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| 15 | equidistant midpoint translate-statements real-identical-points
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| 16 | euclidean-distance
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| 17 | *prover-order*))
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| 18 |
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| 19 | (in-package "PROVER")
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| 20 |
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[51] | 21 | (proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
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[1] | 22 |
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| 23 | (defvar *prover-order* #'grevlex>
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| 24 | "Admissible monomial order used internally in the proofs of theorems.")
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| 25 |
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| 26 | ;; Translate a geometric theorem specification into a statement of the
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| 27 | ;; form
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| 28 | ;; for all u1, u2, ... , us f1=f2=...=fn=0 => g=0
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| 29 |
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| 30 | (defun csym (symbol number)
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| 31 | "Return symbol whose name is a concatenation of (SYMBOL-NAME SYMBOL)
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| 32 | and a number NUMBER."
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| 33 | (intern (concatenate 'string (symbol-name symbol) (format nil "~d" number))))
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| 34 |
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| 35 | (defmacro real-identical-points (A B)
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| 36 | "Return [ (A1-B1)**2 + (A2-B2)**2 ] in lisp (prefix) notation.
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| 37 | The second value is the list of variables (A1 B1 A2 B2). Note that
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| 38 | if the distance between two complex points A, B is zero, it does not
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| 39 | mean that the points are identical. Use IDENTICAL-POINTS to express
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| 40 | the fact that A and B are really identical. Use this macro in conclusions
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| 41 | of theorems, as it may not be possible to prove that A and B are trully
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| 42 | identical in the complex domain."
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| 43 | (let ((A1 (csym A 1))
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| 44 | (A2 (csym A 2))
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| 45 | (B1 (csym B 1))
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| 46 | (B2 (csym B 2)))
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| 47 | `(list
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| 48 | '((+ (expt (- ,A1 ,B1) 2) (expt (- ,A2 ,B2) 2)))
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| 49 | '(,A1 ,A2 ,B1 ,B2))))
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| 50 |
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| 51 | (defmacro identical-points (A B)
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| 52 | "Return [ A1-B1, A2-B2 ] in lisp (prefix) notation.
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| 53 | The second value is the list of variables (A1 B1 A2 B2). Note that
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| 54 | sometimes one is able to prove only that (A1-B1)**2 + (A2-B2)**2 = 0.
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| 55 | This equation in the complex domain has solutions with A and B distinct.
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| 56 | Use REAL-IDENTICAL-POINTS to express the fact that the distance between
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| 57 | two points is 0. Use this macro in assumptions of theorems, although this
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| 58 | is seldom necessary because we assume most of the time that in assumptions
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| 59 | all points are distinct if they are denoted by different symbols."
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| 60 | (let ((A1 (csym A 1))
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| 61 | (A2 (csym A 2))
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| 62 | (B1 (csym B 1))
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| 63 | (B2 (csym B 2)))
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| 64 | `(list
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| 65 | '((- ,A1 ,B1) (- ,A2 ,B2))
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| 66 | '(,A1 ,A2 ,B1 ,B2))))
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| 67 |
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| 68 | (defmacro perpendicular (A B C D)
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| 69 | "Return [ (A1-B1) * (C1-D1) + (A2-B2) * (C2-D2) ] in lisp (prefix) notation.
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| 70 | The second value is the list of variables (A1 A2 B1 B2 C1 C2 D1 D2)."
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| 71 | (let ((A1 (csym A 1))
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| 72 | (A2 (csym A 2))
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| 73 | (B1 (csym B 1))
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| 74 | (B2 (csym B 2))
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| 75 | (C1 (csym C 1))
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| 76 | (C2 (csym C 2))
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| 77 | (D1 (csym D 1))
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| 78 | (D2 (csym D 2)))
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| 79 | `(list
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| 80 | '((+ (* (- ,A1 ,B1) (- ,C1 ,D1))
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| 81 | (* (- ,A2 ,B2) (- ,C2 ,D2))))
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| 82 | '(,A1 ,A2 ,B1 ,B2 ,C1 ,C2 ,D1 ,D2))))
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| 83 |
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| 84 | (defmacro parallel (A B C D)
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| 85 | "Return [ (A1-B1) * (C2-D2) - (A2-B2) * (C1-D1) ] in lisp (prefix) notation.
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| 86 | The second value is the list of variables (A1 A2 B1 B2 C1 C2 D1 D2)."
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| 87 | (let ((A1 (csym A 1))
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| 88 | (A2 (csym A 2))
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| 89 | (B1 (csym B 1))
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| 90 | (B2 (csym B 2))
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| 91 | (C1 (csym C 1))
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| 92 | (C2 (csym C 2))
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| 93 | (D1 (csym D 1))
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| 94 | (D2 (csym D 2)))
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| 95 | `(list
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| 96 | '((- (* (- ,A1 ,B1) (- ,C2 ,D2))
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| 97 | (* (- ,A2 ,B2) (- ,C1 ,D1))))
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| 98 | '(,A1 ,A2 ,B1 ,B2 ,C1 ,C2 ,D1 ,D2))))
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| 99 |
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| 100 |
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| 101 | (defmacro collinear (A B C)
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| 102 | "Return the determinant det([[A1,A2,1],[B1,B2,1],[C1,C2,1]]) in lisp (prefix) notation.
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| 103 | The second value is the list of variables (A1 A2 B1 B2 C1 C2)."
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| 104 | (let ((A1 (csym A 1))
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| 105 | (A2 (csym A 2))
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| 106 | (B1 (csym B 1))
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| 107 | (B2 (csym B 2))
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| 108 | (C1 (csym C 1))
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| 109 | (C2 (csym C 2)))
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| 110 | `(list
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| 111 | '((+ (- (* ,B1 ,C2) (* ,B2 ,C1))
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| 112 | (- (* ,A2 ,C1) (* ,A1 ,C2))
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| 113 | (- (* ,A1 ,B2) (* ,A2 ,B1))))
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| 114 | '(,A1 ,A2 ,B1 ,B2 ,C1 ,C2))))
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| 115 |
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| 116 | (defmacro equidistant (A B C D)
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| 117 | "Return the polynomial [(A1-B1)**2+(A2-B2)**2-(C1-D1)**2-(C2-D2)**2] in lisp (prefix)
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| 118 | notation. The second value is the list of variables (A1 A2 B1 B2 C1 C2 D1 D2)."
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| 119 | (let ((A1 (csym A 1))
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| 120 | (A2 (csym A 2))
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| 121 | (B1 (csym B 1))
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| 122 | (B2 (csym B 2))
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| 123 | (C1 (csym C 1))
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| 124 | (C2 (csym C 2))
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| 125 | (D1 (csym D 1))
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| 126 | (D2 (csym D 2)))
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| 127 | `(list
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| 128 | '((- (+ (expt (- ,A1 ,B1) 2) (expt (- ,A2 ,B2) 2))
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| 129 | (+ (expt (- ,C1 ,D1) 2) (expt (- ,C2 ,D2) 2))))
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| 130 | '(,A1 ,A2 ,B1 ,B2 ,C1 ,C2 ,D1 ,D2))))
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| 131 |
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| 132 | (defmacro euclidean-distance (A B R)
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| 133 | "Return the polynomial [(A1-B1)**2+(A2-B2)**2-R^2] in lisp (prefix)
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| 134 | notation. The second value is the list of variables (A1 A2 B1 B2 R)."
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| 135 | (let ((A1 (csym A 1))
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| 136 | (A2 (csym A 2))
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| 137 | (B1 (csym B 1))
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| 138 | (B2 (csym B 2)))
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| 139 | `(list
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| 140 | '((- (+ (expt (- ,A1 ,B1) 2) (expt (- ,A2 ,B2) 2))
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| 141 | (expt ,R 2)))
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| 142 | '(,A1 ,A2 ,B1 ,B2 ,R))))
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| 143 |
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| 144 | (defmacro midpoint (A B C)
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| 145 | "Express the fact that C is a midpoint of the segment AB.
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| 146 | Returns the list [ 2*C1-A1-B1, 2*C2-A2-B2 ]. The second value returned is the list
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| 147 | of variables (A1 A2 B1 B2 C1 C2)."
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| 148 | (let ((A1 (csym A 1))
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| 149 | (A2 (csym A 2))
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| 150 | (B1 (csym B 1))
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| 151 | (B2 (csym B 2))
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| 152 | (C1 (csym C 1))
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| 153 | (C2 (csym C 2)))
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| 154 | `(list
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| 155 | '((- (* 2 ,C1) ,A1 ,B1) (- (* 2 ,C2) ,A2 ,B2))
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| 156 | '(,A1 ,A2 ,B1 ,B2 ,C1 ,C2))))
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| 157 |
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| 158 | (defmacro translate-statements (&rest statements)
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| 159 | `(list (mapcar #'car (list ,@statements))
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| 160 | (remove-duplicates (apply #'append (mapcar #'cadr (list ,@statements))))))
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| 161 |
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| 162 | (defmacro translate-assumptions (&rest assumptions)
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| 163 | (let ((x (gensym)))
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| 164 | `(let ((,x (translate-statements ,@assumptions)))
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| 165 | (list (apply #'append (car ,x))
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| 166 | (cadr ,x)))))
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| 167 |
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| 168 | (defmacro translate-conclusions (&rest conclusions)
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| 169 | `(translate-statements ,@conclusions))
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| 170 |
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| 171 | (defmacro translate-theorem ((&rest assumptions) (&rest conclusions))
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| 172 | "Translates a planar geometry theorem into a system of polynomial equations.
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| 173 | Each assumption or conclusion takes form of a declaration (RELATION-NAME A B C ...)
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| 174 | where A B C are points, entered as symbols and RELATION-NAME is a name of
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| 175 | a geometric relation, for example, (COLLINEAR A B C) means that points A, B, C
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| 176 | are all collinear. The translated equations use the convention that (A1,A2)
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| 177 | are the coordinates of the point A. This macro returns multiple values.
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| 178 | The first value is a list of polynomial expressions and the second value
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| 179 | is an automatically generated list of variables from points A, B, C, etc.
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| 180 | For convenience, several macros have been defined in order to make specifying
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| 181 | common geometric relations easy."
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| 182 | `(values
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| 183 | (translate-assumptions ,@assumptions)
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| 184 | (translate-conclusions ,@conclusions)))
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| 185 |
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| 186 | (defmacro prove-theorem ((&rest assumptions) (&rest conclusions)
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| 187 | &key (order *prover-order*))
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| 188 | "Proves a geometric theorem, specified in the same manner as in
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| 189 | the macro TRANSLATE-THEOREM. The proof is achieved by a call to
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| 190 | IDEAL-POLYSATURATION. The theorem is true if the returned value
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| 191 | is a trivial ideal containing 1."
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| 192 | (let ((vars (gensym))
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| 193 | (ideal (gensym))
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| 194 | (assump (gensym))
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| 195 | (concl (gensym)))
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| 196 | `(multiple-value-bind (,assump ,concl)
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| 197 | (translate-theorem ,assumptions ,conclusions)
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| 198 | (let* ((,vars (union (second ,assump) (second ,concl)))
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| 199 | (,ideal (ideal-polysaturation
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| 200 | (cdr (poly-eval `(:[ ,@(car ,assump)) ,vars
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| 201 | ,order *ring-of-integers*))
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| 202 | (mapcar #'(lambda (x)
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| 203 | (cdr (poly-eval (cons :[ x) ,vars
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| 204 | ,order *ring-of-integers*)))
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| 205 | (car ,concl))
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| 206 | ,order
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| 207 | 0
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| 208 | nil
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| 209 | *ring-of-integers*)))
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| 210 | (poly-print (cons '[ ,ideal) ,vars)
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| 211 | (values)))))
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