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


whileLoopPSemSound : {l : Level} → (input output : Envc )
  → (varn input + vari input ≡ c10 input)
  →  output ≡ loopPP (varn input) input refl
  → (varn input + vari input ≡ c10 input) implies (vari output ≡ c10 output)
whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre