Quellcode-Bibliothek hyperbolic.pvs   Sprache: PVS

java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
%------------------------------------------------------------------------
%   Definition of Hyperbolic Trig Functions
%
%   Version 1.0    12/3/03 
%   Version 1.1    8/25/04  
%   Version 1.2    10/27/04        added exp_approx, ln_approx
%
%   Author:  David Lester
%
%   Formula labels are from Handbook of Mathematical Functions
%                           by Abramowitz and Stegun
%
%------------------------------------------------------------------------
%

  (:real)+(x))2                   
 @,analysis,@
            analysis@nth_derivatives, analysis(:nzreal          /(n0x% 4.5.4
            analysispolynomial_deriv
            ln_expc(n0xnzreal:real_abs_gt1=/tanhn0x% 4.5.6
            reals



  posreal_ge1
  :N ={xreal|x            } 
  posreal_le1(:posrealposreal      = (px
 (:nnreal:posreal_le1  = sechnnx
posreal_cothpx:)posreal_gt1=java.lang.StringIndexOutOfBoundsException: Range [46, 42) out of bounds for length 50


c: (0  1                                            
:  ()                                             
  conn_abs_lt1        : LEMMA% Relations between Hyperbolic Functions
               connectedreal
  deriv_domain_abs_lt1: LEMMAtanh_sech_one   LEMMAsqtanh)+ sq((x)   =               4.5.17
  deriv_domain_posreal_gt1: LEMMA deriv_domain?[posreal_gt1]

  AUTO_REWRITE+ noa_abs_lt1    
  AUTO_REWRITE+ noa_posreal_gt1
  AUTO_REWRITE+ conn_abs_lt1   
  AUTO_REWRITE+ conn_real
  AUTO_REWRITE+ deriv_domain_abs_lt1
  AUTO_REWRITE+ deriv_domain_posreal_gt1



  x,y:     VAR real
  pxle1:   VAR posreal_le1
  pxge1:   VAR posreal_ge1
  xalt1:   VAR real_abs_lt1
  n0x,n0y: VAR nzreal
  n,m:     VAR nat

% A&S Section 4.5 Hyperbolic Functions

  sinh(x:  real)  :real         = (exp(x)-exp(-x))/2                   % 4.5.1
  (x: )  posreal_ge1=(()+exp-x))/2% 45.
  tanh(x:  real)  :real_abs_lt1 = sinh(x)/cosh(x)cosh_plus_sinh   coshx        (x= exp)        
  cschn0x:):          1/sinh()                          % 4.5.4
  sech(x  :real)  :posreal_le1  = 
  coth(:nzreal:=1tanhn0x                           .56

% Restrictions for Branch Properties

  nnreal_coshcsch_neg  cschn0x)=cschn0x
    sech_neg LEMMA(x    sechx)
  nnreal_sech(:):   sechnnx)
  posreal_coth(px:posreal

% Monotonicity Properties

  sinh_strict_increasing: LEMMA strict_increasing?(sinh)  
  cosh_strict_increasing: LEMMAsinh_diffLEMMA  sinh)cosh(  cosh()sinhy)
  tanh_strict_increasing: LEMMA strict_increasing?(tanh)
  sch_strict_decreasing LEMMA strict_decreasing()
  sech_strict_decreasing: LEMMA strict_decreasing?(  cosh_diff: LEMMA coshx-y =coshx)*cosh(y - sinhsinh(x)sinh(y)
  coth_strict_decreasing: LEMMA strict_decreasing?(posreal_coth)

% Special Values of the Hyperbolic Functions

  sinh_0: LEMMA sinh(0) = 0                                            tanh_sum: LEMMA (x+y) ==((x)+tanhy))(+tanhx*tanhy) % 4.5.26
  cothn0x
  tanh_0=1+()())/cothn0x+coth))
  sech_0: LEMMA sech(0) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

% Relations between Hyperbolic Functions

  cosh_sinh_one:   LEMMAcosh_half:  EMMA(2 = sqrt(cosh)+1/2% 4.5.29
  : LEMMAtanhx2   sqrt(coshx-1/coshx)1)
  coth_csch_one:   LEMMA sq(INIFx=  THENELSE-java.lang.StringIndexOutOfBoundsException: Range [65, 60) out of bounds for length 65
  cosh_plus_sinh:    :  (x/2)=sinh(x)/(cosh(x)+1)                    
  cosh_minus_sinhjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

% Negative Angle FormulasMA FORALL (k:{i:nat| i<n}): C(n,k+1) = C(n,k)*((n-k)/(k+1))


  sinh_neg: LEMMA sinh(-x)   = -sinh(x)                                % 4.5.21
  cosh_neg: LEMMA cosh(-x)   = cosh(x)                                 % 4.5.22
: tanh(x    tanh)                                
  : LEMMA(-n0x =-csch()
  : sech-)=sech()
  coth_neg:   cosh2x_C    (2*x)= sqcoshx)java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 79

% Addition Formulas

  sinh_sum:  LEMMA sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)       % 4.5.24
  inh_diff: LEMMA sinh(x-y) = sinhx)*coshcosh(y) -coshx*sinhyjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
  osh_sum  LEMMAcosh(+y))= (x)*cosh(y + sinhx)*sinh(y)% 4.5.25
  cosh_diff: LEMMA   :      sinh(4                                          
  tanh_sum  LEMMAtanhx+)=  (tanh()+tanh(y))/(1+tanh(x)*(y))   % 4.5.26
  coth_sum:  LEMMA n0x+n0y /= 0 IMPLIES                                % 4.5.27
                   coth(n0x+n0y)
                             = (1+coth(n0x)*coth(n0y))/(coth(n0x)+coth(n0y))

% Half-angle Formulas

  sinh_half:  LEMMA sinh(x/2)                                (x)^46sqsinh)coshx)sinh)4
                                IF x >= 0 THEN y ELSE -y ENDIF
  cosh_half:  LEMMA cosh(x/2) = sqrt((cosh(x)+1)/2)                    % 4.5.29
  tanh_half1
                                N  x> 0 THEN y ELSE -y ENDIF
  tanh_half2: LEMMA tanh(n0x/2) = (cosh(n0x)-1)/sinh(n0x)              % 4.5.30
  tanh_half3  cosh_times_cosh LEMMAcosh(x)*cosh(y) =cosh(+y+())/% 4.5.39

% Multiple-angle Formulas

  sinh2x:     LEMMA sinh(2*x)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  sinh2x_B:LEMMA sinh2*x)  *(x/(1(tanhx)              % 4.5.31
  cosh2x:     LEMMA  :  LEMMAsinh()-(y) = 2*((xy)2*sinh(xy/2    % 4.5.42
  cosh2x_B:   LEMMA cosh(2*x) = 2*  :   LEMMAcoshx+(y 2cosh(x+)2*(x-y/)    % 4.5.43
  :    cosh2x)= sqcoshx)+sqsinhx))              
  tanh2x:     LEMMA tanh(2*x) = :   LEMMA ()tanh) sinhx)(()*(y)% 4.5.45
:L sinh3x == 3*sinhx)+  4sinhx^3                % 4.5.34
  cosh3x:     LEMMA cosh(3*x)= sinh(n0x+)/sinhn0x*sinhn0y)
  sinh4x:     LEMMA sinh(4*x)                                          % 4.5.36
  : sq(x)sq(()  (x+)sinhx-y)% 4.5.47
  cosh4x:     LEMMAsum_cosh_sinh_sq:LEMMAsq(sinhx)+sq(coshy)                       4.548
                              = cosh(x))^+*sq((x*coshx)+sinh(x^4

% Products

  sinh_times_sinh: LEMMA sinh(x)*sinh(y) = 
  cosh_times_cosh: LEMMA()cosh(y) = cosh+)+coshx-y)/2     
  sinh_times_cosh:% Derivatives

% Addition and Subtraction

  sum_sinh:  LEMMA?()
  diff_sinh:  LEMMA sinh(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  sum_cosh:   LEMMA cosh(x)+cosh(y) = 2*cosh((x+y)/2  :LEMMAd(cosh =                           
  diff_cosh:  LEMMA cosh(x)-cosh(y) = 2*sinh((x+y)/2)*sinh((x-y)/2)    
     (,, (x(*+1factorial*+)java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
  sum_coth:   LEMMA coth(n0x)+coth(n0y)                                % 4.5.46
                                    = sinh(n0x+n0y)/(sinh(n0x)*sinh(n0y))

% Relations between squares of hyperbolic sines and cosines

  diff_sinh_sq: LEMMA sq(sinh(x))-sq(sinh(y)) = sinh(x+y)*sinh(x-y)    % 4.5.47
sinhysinh)    
  sum_cosh_sinh_sq
                                              

% De Moivre's Theorem

  hyperbolic_deMoivre: LEMMA

% Derivatives

  sinh_derivable2:  LEMMA derivable?(sinh)
  cosh_derivable2  LEMMA derivable()

  :  LEMMAjava.lang.StringIndexOutOfBoundsException: Range [35, 26) out of bounds for length 42

  deriv_sinh:       LEMMA deriv(sinh) = cosh                           % 4.5.71
  deriv_cosh:       LEMMA deriv(cosh) = sinh                           % 4.5.72
  eriv_tanh       LEMMAderiv() =sechs                      

% Series expansions

  sinh_series_nsech_bij:LEMMA bijective??nnrealposreal_le1(nnreal_sech)
    = sigma(0,n,LAMBDA i:nat x(2i+)/factorial2*i+1)

:LEMMAEXISTS( [real(,):                 
h_series_nxn)  coshc)x(2n)/(2n3







% Logarithmic representations of inverse hyperbolics

  asinh(x:real):                                ( (:eal) 1sqrt1(x)java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
(:osreal_ge1  (+(sqx))                    
  atanh(x:real_abs_lt1:LEMMA[]atanh                   
   
% Bijection relations

  sinh_bij: LEMMA bijective?[real,real](sinh)
% Taylor 
    z:  VAR :   real_abs_lt1
  csch_bij: LEMMA bijective?[posreal,posreal](posreal_csch)
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
  coth_bij LEMMA?[posreal,]posreal_coth

  asinh_alt_def: LEMMA asinh(x) = inverse(sinh

  asinh_sinh
  :LEMMAsinhasinh() =java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
ing  strict_increasing(asinhjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  sinh_bij:LEMMAbijectiverealreal()

% Derivatives

  
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                     derivableIn=0   1sq))(n-1)z 
  atanh_taylors_prep4LEMMA

         IFn=0 HENconst_fun)
                                =  (LAMBDA (x:real): 1/sqrt(1+sq(x)))
  eriv_acosh:L                                              
              deriv
                                 LAMBDA(:osreal_gt1:/sqrt(()1
  :       [real_abs_lt1])% 4.6.39
                                =  (LAMBDA (x:real_abs_lt1): 1/(    derivable[]f ND derivablereal_abs_lt1(g MPLIES

% Taylor Series for atanh

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
  pn      (AMBDA(:real_abs_lt1:atanhN)x)atanhDn))java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

  atanhF(n:nat)(i:nat):int
        2nORodd()T  ELSEjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

  atanhD(n:nat)(x:real):real = (1-sq(x))^(2*n+1)

  atanhN(nnderiv[](natanh)

ors_prep1 LEMMA
    derivable_n_times?ELSIF ()THEN[]atanhNDn21))

  atanh_taylors_prep2: LEMMA
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      = IF n = 0 THEN const_fun(0)
ELSEpolynomial( inat: i1*()i+)2)
        ENDIF

  tanh_taylors_prep3 LEMMA
    deriv      =LAMBDA:nat:z(*n+)(n1)
                     0  E 1sq()*n-1()ENDIF

 java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    ((atanhN)
      = IF n = 0 THEN const_fun(0)
ELSELAMBDAi:):i2*i+)atanhFn(+)2n-2
        ENDIF

  atanh_taylors_prep5: LEMMA FORALL 
                                     g[>nzreal])java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
    derivable?                                 [real_abs_lt1(*+3atanh(java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
      ?]f/^ java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
       deriv[real_abs_lt1](f/g^pn)
          = (deriv[real_abs_lt1](f)*g-pn*f*deriv[real_abs_lt1](g))/(g^(pn+1))

  atanhND(n:nat):[real_abs_lt1->real]
      = (LAMBDA (x:real_abs_lt1): atanhN(n)(x)/atanhD(n)(x))

  atanh_taylors_prep6: LEMMA
    nderiv[real_abs_lt1](2,LAMBDA (x:real_abs_lt1): atanhN(n)(x)/atanhD(n)(x))
      = (LAMBDA (x:real_abs_lt1): atanhN(n+1)(x)/atanhD(n+1)(x))

  atanh_taylors_prep7: LEMMA
    derivable_n_times?[real_abs_lt1](atanhND(n),2*m)

  atanh_taylors_prep8: LEMMA nderiv[real_abs_lt1](2*m,atanhND(n))=atanhND(n+m)

  atanh_nderiv: LEMMA
    nderiv[real_abs_lt1](n,atanh)
      = IF    n = 0    THEN atanh
        ELSIF even?(n) THEN deriv[real_abs_lt1](atanhND(n/2-1))
                       ELSE atanhND((n-1)/2) ENDIF

  atanh_nderiv_0: LEMMA
    nderiv[real_abs_lt1](n,atanh)(0)
      = IF even?(n) THEN 0 ELSE factorial(n-1) ENDIF

  atanh_n_times_derivable: LEMMA derivable_n_times?[real_abs_lt1](atanh,n)

  atanh_series_term(z:real_abs_lt1):[nat->real]
      = (LAMBDA (n:nat): z^(2*n+1)/(2*n+1))
  atanh_series_n(z:real_abs_lt1,n:nat):real = sigma(0,n,atanh_series_term(z))

  atanh_series_eqv: LEMMA
   atanh_series_n(z,n) = sigma(0,2*n+2,
      LAMBDA (nn:nat): IF nn > 2*n+2 OR nn = 0 THEN 0 ELSE
                         nderiv[real_abs_lt1](nn,atanh)(0)*
                                                     z^nn/factorial(nn) ENDIF)

  atanh_taylors: LEMMA (EXISTS (c: between[real_abs_lt1](0,z)): 
               atanh(z) = atanh_series_n(z,n) + 
                                 nderiv[real_abs_lt1](2*n+3,atanh)(c)*
                                                    z^(2*n+3)/factorial(2*n+3))

  atanh_series: LEMMA abs(atanh(z)-atanh_series_n(z,n)) <=
                   ((1+z)^(2*n+3)+(1-z)^(2*n+3))/(2*(2*n+3)*(1-sq(z))^(2*n+3))

END hyperbolic