loopPPSem : (input output : Envc ) @$\rightarrow$@  output @$\equiv$@ loopPP (varn input)  input refl
  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input ) 
  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input ) implies (vari output @$\equiv$@ c10 output)
loopPPSem input output refl s2p = loopPPSemInduct (varn input) input  refl refl s2p