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hyperbolic:hyperbolic: THEORY
%------------------------------------------------------------------------
%   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
%
%------------------------------------------------------------------------
BEGIN

  IMPORTING reals@sq, reals@sqrt,
     analysis@sqrt_derivative, analysis@restrict2_deriv, analysis@deriv_domains,
            analysis@nth_derivatives, analysis@taylors,
            analysis@polynomial_deriv,
            ln_exp,  %% RWB
            reals@binomial



  posreal_ge1:  NONEMPTY_TYPE = {x:real    | x >= 1         } CONTAINING 1
  posreal_gt1:  NONEMPTY_TYPE = {x:real    | x >  1         } CONTAINING 2
  posreal_le1:  NONEMPTY_TYPE = {x:posreal | x <= 1         } CONTAINING 1/2
  real_abs_lt1: NONEMPTY_TYPE = {x:real    | -1 < x &  x < 1} CONTAINING 0
  real_abs_gt1: NONEMPTY_TYPE = {x:real    | x < -1 OR 1 < x} CONTAINING 2


  noa_abs_lt1         : LEMMA not_one_element?[real_abs_lt1];
  noa_posreal_gt1     : LEMMA not_one_element?[posreal_gt1];
  conn_abs_lt1        : LEMMA connected?[real_abs_lt1];
  conn_real           : LEMMA connected?[real]
  deriv_domain_abs_lt1: LEMMA deriv_domain?[real_abs_lt1]
  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+ %   Definition of Hyperbolic Trig Functions
  AUTO_REWRITE+ deriv_domain_posreal_gt1



%   Version 1.1    8/25/04  
  pxle1:   VAR posreal_le1
  pxge1:   VAR posreal_ge1
  xalt1:   VAR%   Author:  David Lester
  n0x,n0y: VAR nzreal
  n,m:%                           by Abramowitz and Stegun

% A&S Section 4.5 Hyperbolic Functions

  
  coshcoshx  real)  :posreal_ge1  = (exp(x)+exp-x)/2% 4.5.2
  tanh(x:  real)  :real_abs_lt1 = sinh(x)/cosh(    analysissqrt_derivative analysis@restrict2_deriv analysisderiv_domains,
  cschn0xnzreal):real         =1sinh(n0x)                          
  sech            @polynomial_deriv,
  othn0x:):  1tanh()                          

% Restrictions for Branch Properties

  nnreal_coshposreal_gt1  ONEMPTY_TYPE= x:     >1          CONTAINING2
  posreal_cschpx):posreal       csch)
  nnreal_sech(nnx):=sech()
  (pxposreal:   coth(px)

% Monotonicity Properties

  sinh_strict_increasing: LEMMA strict_increasing?(sinh)  
  cosh_strict_increasing: LEMMA strict_increasing?(nnreal_cosh)
  tanh_strict_increasing: LEMMA strict_increasing?(tanh)
  csch_strict_decreasing: LEMMA strict_decreasing?(posreal_csch)
  sech_strict_decreasing: LEMMA strict_decreasing?(nnreal_sech)
  coth_strict_decreasing: LEMMA strict_decreasing?(posreal_coth)

% Special Values of the Hyperbolic Functions

  sinh_0
  cosh_0:LEMMA cosh)=                                             % 4.5.61
  tanh_0 LEMMAtanh0 =0% 4.5.61
  sech_0: LEMMA sech(0) = 1                                            % 4.5.61

% Relations between Hyperbolic Functions

  cosh_sinh_one:   LEMMA sq(cosh(x))   - sq(sinh(x))  conn_real:LEMMAconnected?[real]
  tanh_sech_one:   LEMMA ((x))    sq(sech)   1%java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
  coth_csch_onecosh real  :   expx)(-))/                   ..
  cosh_plus_sinh:LEMMA()+sinh)        (x)% 4.5.19
  cosh_minus_sinh: (n0x:nzreal:real= 1sinhn0x

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


  cothn0x)real_abs_gt1  1/tanh()% 45.
  cosh_neg: LEMMA cosh(-x)   = cosh
  tanh_neg
  :LEMMA(-n0x  -()
   sech_neg:L sech-)   =(x
  coth_neg: LEMMA coth(-n0x (nxnnreal:posreal_le1= ()

% Addition Formulas

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  :  sinh(x-y)=(x*(y)- coshx)(java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
  cosh_sum:  LEMMAc:?posreal_csch
cosh_diff LEMMA()= ()( (*java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
   tanh  tanh+())/(tanh(x)*tanh(y)  % 4.5.26
  coth_sum:  LEMMA n0x+n0y /= 0 IMPLIES                                % 4.5.27
                   oth(n0x+n0y)
                             = (1+cothn0x)cothn0y/(coth()+(n0y

% Half-angle Formulas

  sinh_half:  LEMMA sinh(x/2) = LET y = sqrt((cosh(x)-1)/2) IN         % 4.5.28
                                IF x >= 0 THEN y ELSE -y ENDIF
  cosh_half:L coshx/) sqrt((x))2                    
tanh_half1 LEMMA (x/)= LETy =sqrt(cosh(x))(()+)  % 4.5.30
                                   >=0  y  y ENDIF
  tanh_half2: LEMMA tanh(n0x/2) = (cosh(n0x)-1)/sinh(n0x)              % 4.5.30
  tanh_half3LEMMAtanhx)  inhx1)                    % 4.5.30

% Multiple-angle Formulas

  sinh2xjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
  sinh2x_B:   LEMMA sinh(2*x) = 2*tanh(  tanh_neg LEMMA(-)= -(x)% 4.5.23
    csch_neg csch-n0x)= cschn0x
  sech_neg:LEMMA(x   =sechx
  cosh2x_C:   LEMMALEMMAcosh2x sq(() + sq(sinh(x))              % 4.5.32
  tanh2x:     LEMMA tanh(2*x) = 2*tanh(x)/(1+sq(tanh(x)))              % 4.5.33
sLEMMAx-y(*cosh  ()()
  cosh3x:     LEMMA cosh(3*x) = 4*cosh(x)^3-3*c: (xy)coshcoshy) (*sinhy       
sinh4xLEMMA(*x)% 4.5.36
                              = 4*sinh(x)*cosh(x)*(sq: (+ =(tanh(+tanh*tanhy))   
  cosh4x:     LEMMA cosh(4*x)                                          % 4.5.37
=cosh^+*((x*()+(x^java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

% Products

IIF =java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
: (x*cosh)  (cosh(xy)coshx-y/     
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

% Addition and Subtraction

  sum_sinh:   LEMMA sinh(x)+sinh(y) = 2*sinh(  sinh2x_B   EMMA(2)=2tanhx)(1sqtanh())% 4.5.31
diff_sinh   sinh(x)-sinhy  2cosh(x+/)sinh(-)2)java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
sum_cosh    ()+coshy) =*((y/)cosh()2    
  diff_cosh:  LEMMA cosh(x)-cosh(y) = cosh2x_CLEMMA(2) sq(()  (())% 4.5.32
  sum_tanh   tanhx+(y =(+y/coshx)cosh)      
  sum_coth:   LEMMA coth(n0x)+  sinh3x     EMMA(*) =3*(x + *()3                
                                    n0x+n0y(()sinh()java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

% 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
  diff_cosh_sq LEMMA(coshx)-sqcoshy) =sinhy*(x-y    
  : ()()% .
                                              = cosh(x+y)*cosh                              (x)4+sqsinhx)cosh()+sinh(x)4

% De Moivre's Theorem

cosh_times_cosh  coshx*(y) =((xy)+())2     % 4.5.39

% Derivatives

  % Addition and Subtraction
  cosh_derivable2  LEMMA derivable(osh

  tanh_derivable2:  LEMMA derivable?(tanh)

  deriv_sinh:       LEMMA deriv(sinh) = cosh                           % 4.5.71
  deriv_cosh        erivcosh) sinh                          % 4.5.72
  deriv_tanh:       LEMMA deriv(tanh) = sech*sech                      % 4.5.73

% Series expansions

  sinh_series_n(x:real,n:nat):real
=sigma0nLAMBDAi:nat): ^(*)/(2*i1)

  sinh_taylors: LEMMA EXISTS (c: between[real]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
             sinh(x) = sinh_series_n(x,n) + cosh(x+)*(x-y    % 4.5.47







% Logarithmic representations of inverse hyperbolics

  asinhjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  acosh(x:posreal_ge1):nnreal = cosh_derivable2: LEMMA?cosh
  tanh_derivable2   derivable?(tanh)
   
% Bijection relations

  sinh_bij: LEMMA bijective?[realjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  d: (tanh *ech% 4.5.73
  tanh_bij: LEMMA 
  csch_bij
  sech_bij  [,])
sigma,LAMBDA(): ^2*1)factorial(*ii))

  

  asinh_sinh:              LEMMA asinh(sinh(x)) = x
  sinh_asinh:              LEMMA sinh(asinh(x)) = x
  asinh_strict_increasing: LEMMA strict_increasing?(asinh)
  asinh_bij:               LEMMA bijective?[real,real](asinh)

% Derivatives

  asinh_derivable2: LEMMA derivable?(asinh)
  acosh_derivable2: LEMMA
  sinh_taylors:   c:between]0x) % 4.5.62
  atanh_derivable2: LEMMA             sinh(x) = sin(,n +cosh(*x^(*+3factorial2*+)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
=  LAMBDA(x:eal:1/sqrt(+sq))
  deriv_acosh:      LEMMA                                              % 4.6.38
              deriv[posreal_gt1](LAMBDA (x:posreal_gt1): acosh(x))
                                = (LAMBDA (x:posreal_gt1): 1/sqrt(sq(x)-1 acoshxposreal_ge1):nnreal= lnxsqrt(sq(x-1)                    % 4.6.21
  deriv_atanh:     LEMMA derivreal_abs_lt1()                   % 4.6.39
                                =  (java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 3

% Taylor Series for atanh

  z:  VARjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  pn: VAR posnat

  atanhF(n:nat)(i:nat):int
    = IF i > 2*n OR odd?(i) THEN 0 ELSE factorial(2*n)*C(2*  :  bijectiveposrealposreal_gt1()

  atanhD(n:nat)(x:real

  atanhN(sinh_asinh               (asinhx) x

  atanh_taylors_prep1: LEMMA
    derivable_n_times?(atanhN(n),m) AND derivable_n_times?(atanhD(n),m)

  atanh_taylors_prep2: LEMMA
    deriv(atanhN(n))
      = IF n = 0 THEN ing:LEMMAstrict_increasing?asinh)
                 ELSE polynomial(LAMBDA asinh_bij:               ?[,]asinh
        ENDIF

  atanh_taylors_prep3: LEMMA
    deriv(atanhD(n)) = (LAMBDA (z:real): -(4*n+2)*z*
                    F    THEN1 ELSE(-sq(z))*atanhD(n-1)()ENDIF)

  : LEMMA
    
      =IF n  0T (0java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
                 ELSE polynomial(LAMBDA (i:nat): (i+2deriv_acosh      EMMA% 4.6.38
        ENDIF

  atanh_taylors_prep5: LEMMA FORALL= ( x:osreal_gt1) 1sqrt(sqx-))
                                     gderiv_atanh      LEMMAderivreal_abs_lt1]atanh                   
derivable?[real_abs_lt1()A?[]g)IMPLIES
       derivable?[real_abs_lt1](f/g^pn) AND
       deriv[real_abs_lt1
          = (deriv

  atanhND(n:nat):[real_abs_lt1->real]
= ( xreal_abs_lt1) (n(x/atanhD()x)

  atanh_taylors_prep6: LEMMA
    nderiv[real_abs_lt1](2,LAMBDA (x:real_abs_lt1
      = (LAMBDA= IFi>2*  ?i HEN0ELSE factorial(2*n)*C(2*n+1,i) ENDIF

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

  atanh_taylors_prep8: java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

  atanh_nderiv: LEMMA
    nderivreal_abs_lt1(n,)
      = IF  atanh_tayl: LEMMA
        ELSIF even?n  derivreal_abs_lt1(atanhND(/-1))
                       ELSE atanhND

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

                    polynomialLAMBDA (:):(i+)atanhFn)+1,*n-1

  atanh_series_term(z:atanh_taylors_prep3:LEMMA
       ( (n:nat) z^2n+1)/2*+))
  atanh_series_n(z:real_abs_lt1,n:nat):real = sigma(0,IF nn= 0THEN1 LSE (-sq(z)atanhD()z )

  atanh_series_eqv: LEMMA
   atanh_series_n(z,n) = sigma  atanh_taylors_prep4:LEMMA
      LAMBDA (nn:nat): IF nn > 2*n+2 OR nn = 0 THEN 0 ELSE
                         nderiv[real_abs_lt1](nn    deriv(derivatanhN(n))java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
                                                     z^nn/factorial(                 ELSE polynomial(LAMBDA (:nat) (+)(i1*atanhF(n)(i2,*n-2)

  atanh_taylors: LEMMA (                                     :real_abs_lt1-nzreal):
               atanh(z) = atanh_series_n(z,n) + 
nderiv]2n+3,)c)*
                                                    z^(2*n+3)/factorial(2*n+3))

  atanh_series: LEMMA abs(atanh(z)-atanh_series_n(z,n)) <       derivable[real_abs_lt1(f/gpn) AND
                   ((1+z)^(2*n+3)+(1-z)^(2*n+3))/(2*(2*n+3)*(1-sq(z))^(2*n+3))

END hyperbolic