Trac #22155: Doctest: Add more logic flags to more functions

Some functions are handled with some flags in
https://github.com/pynac/pynac/blob/master/ginac/function.cpp#L1581

This should be extended (in Pynac) and doctests added (here). This
ticket should list which questions can be answered definitely to the
positive. To start:

 - f(real).is_real() for f in sin/cos/tan/cot/sec/csc

URL: https://trac.sagemath.org/22155
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Vincent Delecroix

src/sage/symbolic/expression.pyx

@@ -1977,11 +1977,31 @@ cdef class Expression(CommutativeRingElement):
             False
             sage: (t0*t1).is_real()
             True
-            sage: t2 = SR.symbol("t1", domain='positive')
+            sage: t2 = SR.symbol("t2", domain='positive')
             sage: (t1**t2).is_real()
             True
             sage: (t0*x).is_real()
             False
+            sage: (t0^t1).is_real()
+            False
+            sage: (t1^t2).is_real()
+            True
+            sage: gamma(pi).is_real()
+            True
+            sage: cosh(-3).is_real()
+            True
+            sage: cos(exp(-3) + log(2)).is_real()
+            True
+            sage: gamma(t1).is_real()
+            True
+            sage: (x^pi).is_real()
+            False
+            sage: (cos(exp(t0) + log(t1))^8).is_real()
+            True
+            sage: cos(I).is_real()
+            False
+            sage: sin(2 - I).is_real()
+            False
             sage: (2^t0).is_real()
             True
 
@@ -1996,6 +2016,8 @@ cdef class Expression(CommutativeRingElement):
             sage: assume(x, 'real')
             sage: x.is_real()
             True
+            sage: cosh(x).is_real()
+            True
             sage: forget()
 
         The real domain is also set with the integer domain::
@@ -2039,11 +2061,42 @@ cdef class Expression(CommutativeRingElement):
             sage: assume(x>0)
             sage: x.is_positive()
             True
+            sage: cosh(x).is_positive()
+            True
             sage: f = function('f')(x)
             sage: assume(f>0)
             sage: f.is_positive()
             True
             sage: forget()
+
+        ::
+
+            sage: cosh(x).is_positive()
+            False
+            sage: cosh(real(x)).is_positive()
+            True
+            sage: (cosh(real(x))^2).is_positive()
+            True
+            sage: ((real(x))^2).is_positive()
+            False
+            sage: gamma(x^2).is_positive()
+            False
+            sage: gamma(x^2+1).is_positive()
+            False
+            sage: gamma(cosh(real(x))).is_positive()
+            True
+            sage: (real(x)^2).is_positive()
+            False
+            sage: (real(x)^2+1).is_positive()
+            True
+            sage: (abs(x)^2+1).is_positive()
+            True
+            sage: gamma(real(x)^2+1).is_positive()
+            True
+            sage: cos(I).is_positive()
+            False
+            sage: sin(2 - I).is_positive()
+            False
         """
         return self._gobj.info(info_positive)