-- Nomal CodeGear
whileLoop!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc)
  !$\rightarrow$! (n !$\equiv$! varn env)
  !$\rightarrow$! (next : Envc !$\rightarrow$! t)
  !$\rightarrow$! (exit : Envc !$\rightarrow$! t) !$\rightarrow$! t
whileLoop!$\prime$! zero env refl _ exit = exit env
whileLoop!$\prime$! (suc n) env refl next _ = next (record env {varn = pred (varn env) ; vari = suc (vari env) })

-- Hoare Logic base CodeGear
whileLoopPwP!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc )
  !$\rightarrow$! (n !$\equiv$! varn env) !$\rightarrow$! (pre : varn env + vari env !$\equiv$! c10 env)
  !$\rightarrow$! (next : (env : Envc ) !$\rightarrow$! (pred n !$\equiv$! varn env) !$\rightarrow$! (post : varn env + vari env !$\equiv$! c10 env) !$\rightarrow$! t)
  !$\rightarrow$! (exit : (env : Envc ) !$\rightarrow$! (fin : vari env !$\equiv$! c10 env)  !$\rightarrow$! t) !$\rightarrow$! t
whileLoopPwP!$\prime$! zero env refl refl next exit = exit env refl
whileLoopPwP!$\prime$! (suc n) env refl refl next exit = next (record env {varn = pred (varn env) ; vari = suc (vari env) }) refl (+-suc n (vari env))