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macsymar.lisp
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macsymar.lisp
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;;;; -*- Mode: Lisp; Syntax: Common-Lisp -*-
;;;; Code from Paradigms of AI Programming
;;;; Copyright (c) 1991 Peter Norvig
;;;; File macsymar.lisp: The rewrite rules for MACSYMA in Chapter 8
(requires "macsyma")
(setf *simplification-rules* (mapcar #'simp-rule '(
(x + 0 = x)
(0 + x = x)
(x + x = 2 * x)
(x - 0 = x)
(0 - x = - x)
(x - x = 0)
(- - x = x)
(x * 1 = x)
(1 * x = x)
(x * 0 = 0)
(0 * x = 0)
(x * x = x ^ 2)
(x / 0 = undefined)
(0 / x = 0)
(x / 1 = x)
(x / x = 1)
(0 ^ 0 = undefined)
(x ^ 0 = 1)
(0 ^ x = 0)
(1 ^ x = 1)
(x ^ 1 = x)
(x ^ -1 = 1 / x)
(x * (y / x) = y)
((y / x) * x = y)
((y * x) / x = y)
((x * y) / x = y)
(x + - x = 0)
((- x) + x = 0)
(x + y - x = y)
)))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule
'((s * n = n * s)
(n * (m * x) = (n * m) * x)
(x * (n * y) = n * (x * y))
((n * x) * y = n * (x * y))
(n + s = s + n)
((x + m) + n = x + n + m)
(x + (y + n) = (x + y) + n)
((x + n) + y = (x + y) + n)))))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule '(
(log 1 = 0)
(log 0 = undefined)
(log e = 1)
(sin 0 = 0)
(sin pi = 0)
(cos 0 = 1)
(cos pi = -1)
(sin(pi / 2) = 1)
(cos(pi / 2) = 0)
(log (e ^ x) = x)
(e ^ (log x) = x)
((x ^ y) * (x ^ z) = x ^ (y + z))
((x ^ y) / (x ^ z) = x ^ (y - z))
(log x + log y = log(x * y))
(log x - log y = log(x / y))
((sin x) ^ 2 + (cos x) ^ 2 = 1)
))))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule '(
(d x / d x = 1)
(d (u + v) / d x = (d u / d x) + (d v / d x))
(d (u - v) / d x = (d u / d x) - (d v / d x))
(d (- u) / d x = - (d u / d x))
(d (u * v) / d x = u * (d v / d x) + v * (d u / d x))
(d (u / v) / d x = (v * (d u / d x) - u * (d v / d x))
/ v ^ 2) ; [This corrects an error in the first printing]
(d (u ^ n) / d x = n * u ^ (n - 1) * (d u / d x))
(d (u ^ v) / d x = v * u ^ (v - 1) * (d u / d x)
+ u ^ v * (log u) * (d v / d x))
(d (log u) / d x = (d u / d x) / u)
(d (sin u) / d x = (cos u) * (d u / d x))
(d (cos u) / d x = - (sin u) * (d u / d x))
(d (e ^ u) / d x = (e ^ u) * (d u / d x))
(d u / d x = 0)))))
(integration-table
'((Int log(x) d x = x * log(x) - x)
(Int exp(x) d x = exp(x))
(Int sin(x) d x = - cos(x))
(Int cos(x) d x = sin(x))
(Int tan(x) d x = - log(cos(x)))
(Int sinh(x) d x = cosh(x))
(Int cosh(x) d x = sinh(x))
(Int tanh(x) d x = log(cosh(x)))
))
;;; Some examples to try (from an integration table):
; (simp '(int sin(x) / cos(x) ^ 2 d x))
; (simp '(int sin(x / a) d x))
; (simp '(int sin(a + b * x) d x))
; (simp '(int sin x * cos x d x))
; (simp '(Int log x / x d x))
; (simp '(Int 1 / (x * log x) d x))
; (simp '(Int (log x) ^ 3 / x d x))
; (simp '(Int exp(a * x) d x))