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jsops.lisp
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;;;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-Constant
((|$Derivative| SIMP) ?t ?var)
:TEST (not (occurs-in? ?var ?t))
:RESULT (* 0 ?var))
;;;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-Self
((|$Derivative| SIMP) ?exp ?exp)
:RESULT 1)
;;;;;;;;;;;;;;;;
;;in case constant arrives before variable, (* $a $x) for x
(defDifferentiation Move-Constant-outside
((|$Derivative| SIMP) (* ?const ?nonconst) ?var)
:TEST (and (occurs-in? ?var ?const)
(not (occurs-in? ?var ?nonconst)))
:SUBPROBLEMS ((?int ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?const ?var))))
:RESULT (* ?nonconst ?int))
;;in case constant arrives after variable, (* $x $z) for x
(defDifferentiation Move-Constant-outside-1
((|$Derivative| SIMP) (* ?const ?nonconst) ?var)
:TEST (and (not (occurs-in? ?var ?const))
(occurs-in? ?var ?nonconst))
:SUBPROBLEMS ((?int ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?nonconst ?var))))
:RESULT (* ?const ?int))
;;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-Product
((|$Derivative| SIMP) (* ?const ?nonconst) ?var)
:TEST (and (occurs-in? ?var ?const)
(occurs-in? ?var ?nonconst))
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?const ?var)))
(?int2 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?nonconst ?var))))
:RESULT (+ (* ?nonconst ?int1) (* ?const ?int2)))
;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-Sum
((|$Derivative| SIMP) (+ ?t1 ?t2) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var)))
(?int2 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t2 ?var))))
:RESULT (+ ?int1 ?int2))
;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;
(defIntegration Integral-of-Nary-sum
(Integral (+ ?t1 ?t2 . ?trest) ?var)
:SUBPROBLEMS ((?int1 (Integrate (Integral ?t1 ?var)))
(?int2 (Integrate (Integral ?t2 ?var)))
(?intr (Integrate (Integral (+ . ?trest) ?var))))
:TEST (not (null ?trest))
:RESULT (+ ?int1 ?int2 ?intr))
;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-Nary-sum
((|$Derivative| SIMP) (+ ?t1 ?t2 . ?trest) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var)))
(?int2 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t2 ?var)))
(?intr ((|$Differentiate| SIMP) ((|$Derivative| SIMP) (+ . ?trest) ?var))))
:TEST (not (null ?trest))
:RESULT (+ ?int1 ?int2 ?intr))
;;;;;;;;;;;;;;;
(defIntegration Integral-of-uminus
(Integral (- ?term) ?var)
:SUBPROBLEMS ((?int (Integrate (Integral ?term ?var))))
:RESULT (- ?int))
;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-uminus
((|$Derivative| SIMP) (* -1 ?term) ?var)
:SUBPROBLEMS ((?int ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?term ?var))))
:RESULT (* -1 ?int))
;;;;;;;;;;;;;;
(defIntegration Integral-of-minus
(Integral (- ?t1 ?t2) ?var)
:SUBPROBLEMS ((?int1 (Integrate (Integral ?t1 ?var)))
(?int2 (Integrate (Integral ?t2 ?var))))
:RESULT (- ?int1 ?int2))
;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-minus
((|$Derivative| SIMP) (- ?t1 ?t2) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var)))
(?int2 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t2 ?var))))
:RESULT (- ?int1 ?int2))
;;;;;;;;;;;;;;
(defIntegration Integral-of-polyterm
(Integral (expt ?var ?n) ?var)
:TEST (not (same-constant? ?n -1))
:RESULT (/ (expt ?var (+ 1 ?n)) (+ 1 ?n)))
(defDifferentiation Differentiation-of-nonconstant-polyterm
((|$Derivative| SIMP) (expt ?var ?n) ?var)
:TEST (not (occurs-in? ?n ?var))
:RESULT (* (expt ?var (+ -1 ?n)) ?n))
(defDifferentiation Differentiation-of-constant-polyterm
((|$Derivative| SIMP) (expt ?r ?n) ?var)
:TEST (not (occurs-in? ?r ?var))
:RESULT (* (expt ?r ?n) (log ?r)))
(defDifferentiation Differentiation-of-polyterm
((|$Derivative| SIMP) (expt ?var ?n) ?var)
:TEST (not (same-constant? ?n -1))
:RESULT (* (expt ?var (- ?n 1)) ?n))
;;;; Some exponentials and trig functions
(defDifferentiation Differentiate-Simple-e-integral
((|$Derivative| SIMP) (expt $%e ?var) ?var)
:RESULT (expt $%e ?var))
(defDifferentiation Differentiation-complex-e-integral
((|$Derivative| SIMP) (expt $%e (* ?a ?var)) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?a ?var))))
:TEST (occurs-in? ?var ?a)
:RESULT (* (expt $%e (* ?a ?var)) ?int1))
(defDifferentiation Differentiation-e-constant-power
((|$Derivative| SIMP) (expt $%e (* ?a ?var)) ?var)
:TEST (not (occurs-in? ?var ?a))
:RESULT (* (expt $%e (* ?a ?var)) ?a))
(defIntegration non-e-power-integral
(Integral (expt ?b (* ?a ?var)) ?var)
:TEST (and (not (occurs-in? ?var ?a))
(not (occurs-in? ?var ?b)))
:RESULT (/ (expt ?b (* ?a ?var)) (* ?a (log ?b %e))))
(defDifferentiation Log-Differentiation
((|$Derivative| SIMP) (log ?var) ?var)
:RESULT (/ 1 ?var))
(defDifferentiation Log-power-Differentiation
((|$Derivative| SIMP) (expt (log ?r) ?n) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?r ?var)))
(?int2 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?n ?var))))
:RESULT (* (expt (log ?r) ?n) (+ (* ?int2 (log (log ?r))) (/ (* ?n ?int1) (* ?r (log ?r))))))
(defIntegration sin-integral
(Integral (sin (* ?a ?var)) ?var)
:TEST (not (occurs-in? ?var ?a))
:RESULT (- (/ (cos (* ?a ?var)) ?a)))
(defIntegration cos-integral
(Integral (cos (* ?a ?var)) ?var)
:TEST (not (occurs-in? ?var ?a))
:RESULT (/ (sin (* ?a ?var)) ?a))
(defIntegration sin-sqr-integral
(Integral (sqr (sin ?var)) ?var)
:RESULT (- (/ ?var 2) (/ (sin (* 2 ?var)) 4)))
(defIntegration cos-sqr-integral
(Integral (sqr (cos ?var)) ?var)
:RESULT (+ (/ ?var 2) (/ (sin (* 2 ?var)) 4)))
;;;; Some not-so-clever operators
(defIntegration SinToCosSqrSub
(Integral ?exp ?var)
:TEST (and (occurs-in? ?var ?exp)
(occurs-in? `(sin ,?var) ?exp))
:SUBPROBLEMS
((?Int (Integrate (Integral
(:EVAL (subst `(sqrt (- 1 (expt (cos ,?var) 2)))
`(sin ,?var)
?exp :TEST 'equal)) ?var))))
:RESULT ?Int)
(defIntegration CosToSinSqrSub
(Integral ?exp ?var)
:TEST (and (occurs-in? ?var ?exp)
(occurs-in? `(cos ,?var) ?exp))
:SUBPROBLEMS
((?Int (Integrate (Integral
(:EVAL (subst `(sqrt (- 1 (expt (sin ,?var) 2)))
`(cos ,?var)
?exp :TEST 'equal)) ?var))))
:RESULT ?Int)
(defIntegration SinSqrToTanCosSub
(Integral ?exp ?var)
:TEST (and (occurs-in? ?var ?exp)
(occurs-in? `(sin ,?var) ?exp))
:SUBPROBLEMS ((?int (Integrate (Integral
(:EVAL (subst `(* (sqr (tan ,?var))
(sqr (cos ,?var)))
`(sin ,?var)
?exp :TEST 'equal))
?var))))
:RESULT ?Int)
(defDifferentiation Differentiation-of-SQR
(Derivative (sqr ?var) ?var)
:RESULT (* ?var 2))
;;;;;;;;;;;;;;;;;;;;
(defDifferentiation Differentiation-of-plus-Bracketed
((|$Derivative| SIMP) (+ ?t1) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var))))
:RESULT ?int1)
(defDifferentiation Differentiation-of-plus-Bracketed
((|$Derivative| SIMP) ((MPLUS SIMP) ?t1) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var))))
:RESULT ?int1)
(defDifferentiation Differentiation-of-times-Bracketed
((|$Derivative| SIMP) (* ?t1) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var))))
:RESULT ?int1)
(defDifferentiation Differentiation-of-times-Bracketed
((|$Derivative| SIMP) ((MTIMES SIMP) ?t1) ?var)
:SUBPROBLEMS ((?int1 ((|$Differentiate| SIMP) ((|$Derivative| SIMP) ?t1 ?var))))
:RESULT ?int1)