loopPPSem : (input output : Envc ) @$\rightarrow$@  output @$\equiv$@ loopPP (varn input)  input refl
  @$\rightarrow$@ (whileTestStateP s2 input ) @$\rightarrow$@ (whileTestStateP s2 input ) implies (whileTestStateP sf output)
loopPPSem input output refl s2p = loopPPSemInduct (varn input) input  refl refl s2p
  where
    lem : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ n + suc (vari env) @$\equiv$@ suc (n + vari env)
    lem n env = +-suc (n) (vari env)
    loopPPSemInduct : (n : @$\mathbb{N}$@) @$\rightarrow$@ (current : Envc) @$\rightarrow$@ (eq : n @$\equiv$@ varn current) @$\rightarrow$@  (loopeq : output @$\equiv$@ loopPP n current eq)
      @$\rightarrow$@ (whileTestStateP s2 current ) @$\rightarrow$@ (whileTestStateP s2 current ) implies (whileTestStateP sf output)
    loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (@$\lambda$@ x @$\rightarrow$@ refl)
    loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
        whileLoopPSem current refl
            (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
            (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)


whileLoopPSemSound : {l : Level} @$\rightarrow$@ (input output : Envc )
  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input)
  @$\rightarrow$@  output @$\equiv$@ loopPP (varn input) input refl
  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input) implies (vari output @$\equiv$@ c10 output)
whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre