Quelle trig_inverses.pvs Sprache: PVS

trig_inverses: THEORY
%----------------------------------------------------------------------------
% Interface to Inverse Trig Function
%
% Rick Butler 1/8/08
%----------------------------------------------------------------------------
BEGIN
   IMPORTING trig_basic, asin, acos, atan, atan2

  a: VAR real

% nnreal_quad1_closed: NONEMPTY_TYPE = {x:nnreal | x <= pi/2}
% nnreal_quad1_open:   NONEMPTY_TYPE = {x:nnreal | x <  pi/2}
% real_abs_lt_pi:      NONEMPTY_TYPE = {x:real   | -pi/2 < x AND x < pi/2}
% posreal_lt_pi:       NONEMPTY_TYPE = {x:posreal| x < pi}
% nnreal_le_pi :       NONEMPTY_TYPE = {x:nnreal | x <= pi}
% real_abs_le1:        NONEMPTY_TYPE = {x:real | -1 <= x AND x <= 1}
% real_abs_lt1:        NONEMPTY_TYPE = {x:real | -1 < x  AND x <  1}
% real_abs_le_pi2:     NONEMPTY_TYPE = {x:real | -pi/2 <= x AND x <= pi/2}
% real_abs_lt_pi:      NONEMPTY_TYPE = {x:real   | -pi/2 < x AND x < pi/2}

% ---------- ArcSine ----------------- (See asin.pvs)
%
%   asin(x:real_abs_le1): real_abs_le_pi2
%
%   asin_0:                 LEMMA asin(0)  = 0
%   asin_sqrt_half:         LEMMA asin(sqrt(1/2)) = pi/4
%   asin_1:                 LEMMA asin(1)  = pi/2
%   asin_neg:               LEMMA asin(-x) = -asin(x)
%   asin_minus1:            LEMMA asin(-1) = -pi/2
%   asin_minus_sqrt_half:   LEMMA asin(-sqrt(1/2)) = -pi/4

    AUTO_REWRITE+ asin_0
    AUTO_REWRITE+ asin_1

% ---------- ArcCosine ----------------- (See acos.pvs)
%
%   acos(x:real_abs_le1): nnreal_le_pi = pi/2 - asin(x)
%
%   acos_neg:               LEMMA acos(-x) = pi-acos(x)
%   acos_0:                 LEMMA acos(0)  = pi/2
%   acos_sqrt_half:         LEMMA acos(sqrt(1/2)) = pi/4
%   acos_1:                 LEMMA acos(1)  = 0
%   acos_minus1:            LEMMA acos(-1) = pi
%   acos_minus_sqrt_half:   LEMMA acos(-sqrt(1/2)) = 3*pi/4

    AUTO_REWRITE+ acos_0
    AUTO_REWRITE+ acos_1

% ---------- ArcTangent ----------------- (See atan.pvs)
%
%  atan(x:real): real_abs_lt_pi2
%
%  atan_0                : LEMMA atan(0) = 0
%  atan_inv              : LEMMA atan(1/px) = pi/2-atan(px)
%  atan_inv_neg          : LEMMA atan(1/nx) = -pi/2-atan(nx)
%  atan_neg              : LEMMA atan(-x)   = -atan(x)
%  acot_neg              : LEMMA acot(-nzx) = -acot(nzx)

    AUTO_REWRITE+ atan_0

%  --------- Inverse Relationships (See sincos_def)

%   sin_asin: LEMMA sin(asin(x)) = x
%   cos_acos: LEMMA cos(acos(x)) = x
%   tan_atan: LEMMA tan(atan(a)) = a

%   asin_sin: LEMMA FORALL (x:real_abs_le_pi2): asin(sin(x)) = x
%   acos_cos: LEMMA FORALL (x:nnreal_le_pi):   acos(cos(x)) = x
%   atan_tan: LEMMA FORALL (x:real_abs_lt_pi2): atan(tan(x)) = x

%  --- The following provide additional names for the inverse functions
%  --- that include their basic property in the type.  These are included
%  --- for upward compatibility.

%  -------------------- Arcsin --------------------

   arcsin(y: real_abs_le1): {x: real_abs_le_pi2 | y = sin(x)} = asin(y)

   sin_arcsin: LEMMA (FORALL (y: real_abs_le1): sin(arcsin(y)) = y)
   arcsin_sin: LEMMA (FORALL (x: real_abs_lt_pi2): arcsin(sin(x)) = x)

%  -------------------- Arccos --------------------

   arccos(y: real_abs_le1): {x: nnreal_le_pi | y = cos(x)} = acos(y)

   cos_arccos: LEMMA (FORALL (y: real_abs_le1): cos(arccos(y)) = y)
   arccos_cos: LEMMA (FORALL (x: nnreal_le_pi): arccos(cos(x)) = x)

%  -------------------- Arctan --------------------

   arctan_prep: LEMMA FORALL (x: real_abs_lt_pi2): Tan?(x);

   arctan(y: real): {x: real_abs_lt_pi2 | y = tan(x)} = atan(y)

   tan_arctan: LEMMA (FORALL (y: real): tan(arctan(y)) = y)
   arctan_tan: LEMMA (FORALL (x: real_abs_lt_pi2): arctan(tan(x)) = x)

END trig_inverses

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