Edit detail for Piecewise Functions revision 2 of 2

Consider the following function, given in recursive manner:

(1) -> N0(t|(t<0) or (t>1))==0

N0(t|(t>=0) and (t<=1))==1

N(t,i,0)==N0(t-i)

N(t,i,p|p>0)==(t-i)/p*N(t,i,p-1)+(i+1-t)/p*N(t,i+1,p-1)

This is a way to create (uniform) bsplines. Now try to differentiate

D(N(t,0,3),t)

   There are 3 exposed and 1 unexposed library operations named < 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op <
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named < 
      with argument type(s) 
                             Polynomial(Integer)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 3 exposed and 1 unexposed library operations named < 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op <
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named < 
      with argument type(s) 
                             Polynomial(Integer)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 3 exposed and 1 unexposed library operations named < 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op <
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named < 
      with argument type(s) 
                             Polynomial(Integer)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Yack!!! This does not work. The map

N(t,0,3)

   There are 3 exposed and 1 unexposed library operations named < 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op <
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named < 
      with argument type(s) 
                             Polynomial(Integer)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 3 exposed and 1 unexposed library operations named < 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op <
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named < 
      with argument type(s) 
                             Polynomial(Integer)
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

On the other hand the function can be evaluated for specific numerial arguments. See this:

N0(t|(t<0) or (t>1))==0;

   Compiled code for N0 has been cleared.
   Compiled code for N has been cleared.
   1 old definition(s) deleted for function or rule N0

N0(t|(t>=0) and (t<=1))==1;

   1 old definition(s) deleted for function or rule N0

N(t,i,0)==N0(t-i);

   1 old definition(s) deleted for function or rule N

N(t,i,p|p>0)==(t-i)/p*N(t,i,p-1)+(i+1-t)/p*N(t,i+1,p-1);

   1 old definition(s) deleted for function or rule N

for x in -5..15 repeat output N(x/10,0,3)

Compiling function N0 with type Fraction(Integer) -> 
      NonNegativeInteger

Compiling function N with type (Fraction(Integer), Integer, Integer)
       -> Fraction(Integer) 
   0
   0
   0
   0
   0
   0
     1
   ----
   6000
    1
   ---
   750
     9
   ----
   2000
    4
   ---
   375
    1
   --
   48
    9
   ---
   250
    343
   ----
   6000
    32
   ---
   375
    243
   ----
   2000
   1
   -
   6
      133
   - ----
     6000
      43
   - ---
     750
      217
   - ----
     2000
      67
   - ---
     375
     13
   - --
     48

Drawing the plot (unfortunately not available here) would show it even more clearly.

But in D(N(t,0,3),t) you are not calling the function N with numeric parameters. In N(t,0,3) the type of t is Variable t. Ultimately N(t,0,3) fails because of your function definition N0(t|(t<0) or (t>1))==0. This is because t>1 fails. Namely, FriCAS interprets both t and 1 as being of type POLY INT and there is no order on the polynomials. You can see why if you use the option )set message bottomup on to see the mode map selection

)set message bottomup on

   Your user access level is compiler and this set option is therefore 
      not available. See the )set userlevel command for more 
      information.
t>1

   There are 1 exposed and 2 unexposed library operations named > 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op >
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named > 
      with argument type(s) 
                                 Variable(t)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

This result is counter-intuitive for people expecting symbolic computation, but once you understand why FriCAS handles this as above then you will be in a good position to understand the rest of FriCAS type system!

It is possible to write the function N0 so that it returns the desired result (using abs).