Fortran: modules and array size

Question

I have three pieces of code

1. Everything in the main.f90 file

   program main
      implicit none
   
      integer, parameter :: N = 10
      real :: x(N), y(N), z(N)
      integer :: i
   
      do i = 1, N
         x(i) = i
         y(i) = 2 * i
      end do
   
      call sum_v(x, y, z)
   
      print *, x
      print *, y
      print *, z
   
   contains
      subroutine sum_v(x, y, z)
         implicit none
   
         real, intent(in) :: x(:), y(:)
         real, intent(out) :: z(:)
   
         integer :: nx, ny, nz
   
         nx = size(x, 1)
         ny = size(y, 1)
         nz = size(z, 1)
   
         if ((nx == ny) .AND. (nx == nz)) then
            z = x + y
         else
            print *, nx, ny, nz
         end if
      end subroutine sum_v
   end program main
   

2. A main.f90 like: program main implicit none

       integer, parameter :: N = 10
       real :: x(N), y(N), z(N)
       integer :: i
   
       do i = 1, N
          x(i) = i
          y(i) = 2 * i
       end do
   
       call sum_v(x, y, z)
   
       print *, x
       print *, y
       print *, z
   end program main
   

   and a separate file sum.f90

   subroutine sum_v(x, y, z)
      implicit none
   
      real, intent(in) :: x(:), y(:)
      real, intent(out) :: z(:)
   
      integer :: nx, ny, nz
   
      nx = size(x, 1)
      ny = size(y, 1)
      nz = size(z, 1)
   
      if ((nx == ny) .AND. (nx == nz)) then
         z = x + y
      else
         print *, nx, ny, nz
      end if
   end subroutine sum_v
   

3. Finally, a main.f90 like

   program main
      use v_sum
   
      implicit none
   
      integer, parameter :: N = 10
      real :: x(N), y(N), z(N)
      integer :: i
   
      do i = 1, N
          x(i) = i
          y(i) = 2 * i
      end do
   
      call sum_v2(x, y, z)
   
      print *, x
      print *, y
      print *, z
   end program main
   

   and a sum.f90 like

   module v_sum
      implicit none
   
   contains
      subroutine sum_v(x, y, z)
          implicit none
   
          real, intent(in) :: x(:), y(:)
          real, intent(out) :: z(:)
   
          integer :: nx, ny, nz
   
          nx = size(x, 1)
          ny = size(y, 1)
          nz = size(z, 1)
   
          if ((nx == ny) .AND. (nx == nz)) then
              z = x + y
          else
              print *, nx, ny, nz
          end if
       end subroutine sum_v
   end module v_sum
   

In cases 1 and 3 I observe the correct behavior: a sum of vector is carried out, while in case 2, the other branch of if is take, that is the function is not called correctly. I compile with gfortran using flags -Wall -Wextra -pedantic and no error is given.

three questions

1. What is going on?
2. Is there a way to make case 2 work as well without modify the function?
3. What is the correct way to proceed if one want to compile sum.f90 as as shared library? (what to do with .mod files?)