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|
; $Id: extraction.scm 2156 2008-01-25 13:25:12Z schimans $
; (load "~/minlog/init.scm")
; (set! DOT-NOTATION #f)
; (set! COMMENT-FLAG #f)
; (libload "nat.scm")
; (libload "numbers.scm")
; (load "real.scm")
; (load "cont.scm")
; (set! COMMENT-FLAG #t)
(time (pp (nt (pt "(IntN 2#9)+(2*((IntP 7#9)-(IntN 2#9)))/4"))))
; "810#2916" 30 ms
(time (pp (nt (pt "(IntN 2#9)+((IntP 7#9)-(IntN 2#9))/2")))) ;"405#1458" 26 ms
(time (pp (nt (pt "((IntN 2#9)+(7#9))/2")))) ;"45#162" 15 ms
(time (pp (nt (pt "((IntN 2#9)+(7#9))")))) ;"45#81" 11 ms
; We deanimate what was left animated in cont.scm
; (deanimate "ApproxSplit")
; (deanimate "IVTAux")
; (deanimate "IVTcds")
; Now we check whether indeed all lets (i.e., cAp/cId) are not unfolded in
; the present rules.
; (display-program-constants)
; We now animate all theorems, working from root to leaves
(animate "IVTApprox")
(animate "RealApprox")
(animate "IVTFinal")
(animate "IVTcds")
(animate "IVTAux")
(animate "ApproxSplit")
; (animate "GCDGInd")
; (animate "QRCorGInd")
; (animate "QRPos")
; (animate "QR")
; (animate "PosInt")
; For further speed-up we provide an external version of +
; We now want to view RatPlus as a program constant with external
; code, using add-external-code. The external code - after evaluation
; and application to tsubst and the arguments - should give either the
; final value (e.g., the numeral for the sum) or else #f, in which
; case the rules are tried next on the arguments.
(define ratplus-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(sum (+ (/ numer1 denom1) (/ numer2 denom2)))
(numer (numerator sum))
(denom (denominator sum))
(numer-term (int-to-int-term numer))
(denom-term (make-numeric-term denom))
(constr (constr-name-to-constr "RatConstr"))
(term (mk-term-in-app-form
(make-term-in-const-form constr)
numer-term denom-term)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(define ratminus-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(diff (- (/ numer1 denom1) (/ numer2 denom2)))
(numer (numerator diff))
(denom (denominator diff))
(numer-term (int-to-int-term numer))
(denom-term (make-numeric-term denom))
(constr (constr-name-to-constr "RatConstr"))
(term (mk-term-in-app-form
(make-term-in-const-form constr)
numer-term denom-term)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(define rattimes-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(prod (* (/ numer1 denom1) (/ numer2 denom2)))
(numer (numerator prod))
(denom (denominator prod))
(numer-term (int-to-int-term numer))
(denom-term (make-numeric-term denom))
(constr (constr-name-to-constr "RatConstr"))
(term (mk-term-in-app-form
(make-term-in-const-form constr)
numer-term denom-term)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(define ratdiv-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(quot (/ (/ numer1 denom1) (/ numer2 denom2)))
(numer (numerator quot))
(denom (denominator quot))
(numer-term (int-to-int-term numer))
(denom-term (make-numeric-term denom))
(constr (constr-name-to-constr "RatConstr"))
(term (mk-term-in-app-form
(make-term-in-const-form constr)
numer-term denom-term)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(define ratlt-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(res (< (/ numer1 denom1) (/ numer2 denom2)))
(const (if res true-const false-const))
(term (make-term-in-const-form const)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(define ratle-code
'(lambda (tsubst objs)
(let ((val1 (nbe-object-to-value (car objs)))
(val2 (nbe-object-to-value (cadr objs))))
(and (nbe-constr-value? val1) (nbe-constr-value? val2)
(let* ((args1 (nbe-constr-value-to-args val1))
(args2 (nbe-constr-value-to-args val2))
(vals1 (map nbe-object-to-value args1))
(vals2 (map nbe-object-to-value args2)))
(and (int-numeral-value? (car vals1))
(pos-numeral-value? (cadr vals1))
(int-numeral-value? (car vals2))
(pos-numeral-value? (cadr vals2))
(let* ((numer1 (int-numeral-value-to-number (car vals1)))
(denom1 (pos-numeral-value-to-number (cadr vals1)))
(numer2 (int-numeral-value-to-number (car vals2)))
(denom2 (pos-numeral-value-to-number (cadr vals2)))
(res (<= (/ numer1 denom1) (/ numer2 denom2)))
(const (if res true-const false-const))
(term (make-term-in-const-form const)))
(nbe-term-to-object
term (nbe-make-bindings '() '())))))))))
(add-external-code "RatPlus" ratplus-code)
(add-external-code "RatMinus" ratminus-code)
(add-external-code "RatTimes" rattimes-code)
(add-external-code "RatDiv" ratdiv-code)
(add-external-code "RatLt" ratlt-code)
(add-external-code "RatLe" ratle-code)
; Finally we animate Id, to enable numeric calculations
(animate "Id")
; Here is a list of the used theorems. Their formulas can be printed
; using (pp "IVTApprox") und their extracted terms via
; (define IVTApprox-neterm
; (nt (proof-to-extracted-term (theorem-name-to-proof "IVTApprox"))))
; (pp IVTApprox-neterm)
"RealApprox"
"ApproxSplit"
"IVTAux"
"IVTcds"
"IVTFinal"
"IVTApprox"
; Now the crucial test
(define a-sq-minus-two
(pt "ContConstr 1 2([a0,n1]a0*a0-2)([k]Zero)([k]k+3)"))
(time (pp (nt
(apply mk-term-in-app-form
(list (proof-to-extracted-term (theorem-name-to-proof "IVTApprox"))
a-sq-minus-two
(pt "IntN One") ;-1 is the modulus of increase
(pt "IntZero") ;1 <= b-a
(pt "IntZero") ;b-a <= 1
(pt "20"))))))
; 17193534846817967675#12157665459056928801
; 767 ms
(exact->inexact (/ 17193534846817967675 12157665459056928801))
1.4142135186002784
(sqrt 2)
1.4142135623730951
; Check accuracy
; (define diff (- 1.4142135186002784 1.4142135623730951))
; diff
; -4.3772816704645834e-8
; (exact->inexact (expt 2 -20))
; 9.5367431640625e-7
(deanimate "Id")
(pp (nt (proof-to-extracted-term (theorem-name-to-proof "IVTApprox"))))
#|
[f0,k1,k2,k3,k4]
left((cDC rat@@rat)(f0 doml@f0 domr)
([n5]
(cId rat@@rat=>rat@@rat)
([cd7]
[let cd8
((2#3)*left cd7+(1#3)*right cd7@(1#3)*left cd7+(2#3)*right cd7)
[if (0<=
(f0 approx left cd8
(f0 uMod(IntS(IntS(IntS(IntS(IntS(IntS(k2+n5+k1))))))))+
f0 approx right cd8
(f0 uMod(IntS(IntS(IntS(IntS(IntS(IntS(k2+n5+k1)))))))))/
2)
(left cd7@right cd8)
(left cd8@right cd7)]]))
(IntToNat(2*(k4+k3))))
|#
(define sqrt-two-approx
(nt (apply mk-term-in-app-form
(list (proof-to-extracted-term
(theorem-name-to-proof "IVTApprox"))
a-sq-minus-two
(pt "IntN One") ;-1 is the modulus of increase
(pt "IntZero") ;1 <= b-a
(pt "IntZero") ;b-a <= 1
))))
(pp sqrt-two-approx)
#|
[k0]
left((cDC rat@@rat)(1@2)
([n1]
(cId rat@@rat=>rat@@rat)
([cd3]
[let cd4
((2#3)*left cd3+(1#3)*right cd3@(1#3)*left cd3+(2#3)*right cd3)
[if (0<=(left cd4*left cd4-2+(right cd4*right cd4-2))/2)
(left cd3@right cd4)
(left cd4@right cd3)]]))
(IntToNat(2*k0)))
|#
(animate "Id")
(pp (nt (make-term-in-app-form sqrt-two-approx (pt "2"))))
; 107#81
(pp (nt (make-term-in-app-form sqrt-two-approx (pt "20"))))
; 17193534846817967675#12157665459056928801
(time (tag (nbe-normalize-term-without-eta
(make-term-in-app-form sqrt-two-approx (pt "20")))))
; 550 ms
(deanimate "Id")
; We now translate terms into scheme expressions, for faster
; evaluation (no conversions between internal and external numbers)
(term-to-expr sqrt-two-approx)
#|
(lambda (k0)
(car (((|cDC| (cons 1 2))
(lambda (n1)
(lambda (cd3)
(let ([cd4
(cons (+ (* 2/3 (car cd3)) (* 1/3 (cdr cd3)))
(+ (* 1/3 (car cd3)) (* 2/3 (cdr cd3))))])
(if (<= 0
(/ (+ (- (* (car cd4) (car cd4)) 2)
(- (* (cdr cd4) (cdr cd4)) 2))
2))
(cons (car cd3) (cdr cd4))
(cons (car cd4) (cdr cd3)))))))
(|IntToNat| (* 2 k0)))))
|#
(time ((ev (term-to-expr sqrt-two-approx)) 20))
; 8ms
; 1910392699673572643/1350851717672992089
(time ((ev (term-to-expr sqrt-two-approx)) 100))
; 136 ms
41737211713808721950509113461986613702889339109196103625535604673708288858253142530485267574435/29512665430652752148753480226197736314359272517043832886063884637676943433478020332709411004889
(time ((ev (term-to-expr sqrt-two-approx)) 300))
; 1387 ms
2944593304156165436102846247558257490730845085059145775348712785737552429558941055472664500523847993653960875075500489392461830532348741253430285578096821615417701491158209086792184369992128090401780684332213746112424235660933353732326055138766537198666286440104173743582833475176351331/2082141893205326654083779991150902602700941003443642395329656664801323440350862630969568906052114539645303398663539990042118787521457672342793285135263403898153882623763114393917433013110956461871522162788143751759237923280744039682511207437298831530097535001606799426410247097767236889
; Same for "Inv"
(deanimate "IVTApprox")
(deanimate "RealApprox")
(deanimate "IVTFinal")
(deanimate "IVTcds")
(deanimate "IVTAux")
(deanimate "ApproxSplit")
; We now animate all theorems, working from root to leaves
(animate "InvApprox")
(animate "RealApprox")
(animate "Inv")
(animate "IVTcds")
(animate "IVTAux")
(animate "ApproxSplit")
; We also need to animate "AC" "IP" with identities:
(animate "AC" (pt "[alpha1=>alpha2]alpha1=>alpha2"))
(animate "IP" (pt "[alpha]alpha"))
(define inv-approx-eterm
(nt (proof-to-extracted-term (theorem-name-to-proof "InvApprox"))))
(pp inv-approx-eterm)
#|
[f0,k1,k2,k3,a4,a5,a6,k7]
left((cDC rat@@rat)(f0 doml@f0 domr)
([n8]
(cId rat@@rat=>rat@@rat)
([cd10]
[let cd11
((2#3)*left cd10+(1#3)*right cd10@
(1#3)*left cd10+(2#3)*right cd10)
[if (0<=
(f0 approx left cd11
(f0 uMod(IntS(IntS(IntS(IntS(IntS(IntS(k2+n8+k1))))))))-
a6+
(f0 approx right cd11
(f0 uMod(IntS(IntS(IntS(IntS(IntS(IntS(k2+n8+k1))))))))-
a6))/
2)
(left cd10@right cd11)
(left cd11@right cd10)]]))
(IntToNat(2*f0 uModCont(IntS(IntS(IntS(IntS(k7+k1)))))+k3+k3)))
|#
(define sq (pt "ContConstr 1 2([a0,n1]a0*a0)([k]Zero)([k]k+3)"))
(define inv-sq-approx
(nt (apply mk-term-in-app-form
(list (proof-to-extracted-term
(theorem-name-to-proof "InvApprox"))
sq ;continuous function to be inverted
(pt "IntN One") ;uniform lower bound on the slope
(pt "IntZero") (pt "IntZero") ;bounds for b-a
(pt "1") (pt "4") ;interval in range
))))
(pp inv-sq-approx)
#|
[a0,k1]
left((cDC rat@@rat)(1@2)
([n2]
(cId rat@@rat=>rat@@rat)
([cd4]
[let cd5
((2#3)*left cd4+(1#3)*right cd4@(1#3)*left cd4+(2#3)*right cd4)
[if (0<=(left cd5*left cd5-a0+(right cd5*right cd5-a0))/2)
(left cd4@right cd5)
(left cd5@right cd4)]]))
(IntToNat(2*IntS(IntS(IntS(IntS(IntS(IntS k1))))))))
|#
; Finally we animate Id, to enable numeric calculations
(animate "Id")
(time
(pp (nbe-normalize-term-without-eta
(mk-term-in-app-form
inv-sq-approx
(pt "3") ;argument of inverted function
(pt "20") ;error bound (number of binary digits)
))))
; 3730307366945298869534434#2153693963075557766310747 in 803 ms
; (exact->inexact (/ 3730307366945298869534434 2153693963075557766310747))
; 1.7320508070785863
; (sqrt 3)
; 1.7320508075688772
; Difference at the 10th decimal digit
; We now translate terms into scheme expressions, for faster
; evaluation (no conversions between internal and external numbers)
(term-to-expr inv-sq-approx)
#|
(lambda (a0)
(lambda (k1)
(car (((|cDC| (cons 1 2))
(lambda (n2)
(lambda (cd4)
(let ([cd5
(cons (+ (* 2/3 (car cd4)) (* 1/3 (cdr cd4)))
(+ (* 1/3 (car cd4)) (* 2/3 (cdr cd4))))])
(if (<= 0
(/ (+ (- (* (car cd5) (car cd5)) a0)
(- (* (cdr cd5) (cdr cd5)) a0))
2))
(cons (car cd4) (cdr cd5))
(cons (car cd5) (cdr cd4)))))))
(|IntToNat|
(* 2
(|IntS|
(|IntS| (|IntS| (|IntS| (|IntS| (|IntS| k1))))))))))))
|#
(time (((ev (term-to-expr inv-sq-approx)) 3) 20))
; 13 ms
(time (((ev (term-to-expr inv-sq-approx)) 3) 100))
; 134 ms
(time (((ev (term-to-expr inv-sq-approx)) 3) 200))
; 560 ms
|
