whileLoopPwP!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc )
    !$\rightarrow$! (n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env
    !$\rightarrow$! (next : (env : Envc ) !$\rightarrow$! (pred n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env  !$\rightarrow$! t)
    !$\rightarrow$! (exit : (env : Envc ) !$\rightarrow$! whileTestStateP sf env  !$\rightarrow$! t) !$\rightarrow$! t
whileLoopPwP!$\prime$! zero env refl refl _ exit = exit env refl
whileLoopPwP!$\prime$! (suc n) env refl refl next _ =
    next (record env {varn = pred (varn env) ; vari = suc (vari env) }) refl (+-suc n (vari env))


loopPwP!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc )
    !$\rightarrow$! (n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env
    !$\rightarrow$! (exit : (env : Envc ) !$\rightarrow$! whileTestStateP sf env !$\rightarrow$! t) !$\rightarrow$! t
loopPwP!$\prime$! zero env refl refl exit = exit env refl
loopPwP!$\prime$! (suc n) env refl refl exit
    = whileLoopPwP!$\prime$! (suc n) env refl refl (!$\lambda$! env x y !$\rightarrow$! loopPwP!$\prime$! n env x y exit) exit