whileLoopPSem : {l : Level} {t : Set l}   → (input : Envc ) → (vari input) + (varn input) ≡ (c10 input)
→ (next : (output : Envc ) → ((vari input) + (varn input) ≡ (c10 input) ) implies ((vari output) + (varn output) ≡ (c10 output))  → t)
→ (exit : (output : Envc ) → ((vari input) + (varn input) ≡ (c10 input) ) implies ((vari output ≡ c10 output))  → t) → t
whileLoopPSem env s next exit with varn env | s
... | zero | _ = exit env (proof (λ z → z))
... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof λ x → +-suc varn (vari env) )


loopPPSem : (input output : Envc ) →  output ≡ loopPP (varn input)  input refl
  → (vari input) + (varn input) ≡ (c10 input) → ((vari input) + (varn input) ≡ (c10 input) ) implies ((vari output ≡ c10 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)
      → ((vari current) + (varn current) ≡ (c10 current) ) → ((vari current) + (varn current) ≡ (c10 current) ) implies ((vari output ≡ c10 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)