whileTest : {l : Level} {t : Set l}  !$\rightarrow$! {c10 : !$\mathbb{N}$! } !$\rightarrow$! (Code : (env : Env)  !$\rightarrow$!
            ((vari env) !$\equiv$! 0) !$\wedge$! ((varn env) !$\equiv$! c10) !$\rightarrow$! t) !$\rightarrow$! t
whileTest {_} {_} {c10} next = next env proof2
  where
    env : Env
    env = record {vari = 0 ; varn = c10}
    proof2 : ((vari env) !$\equiv$! 0) !$\wedge$! ((varn env) !$\equiv$! c10)
    proof2 = record {pi1 = refl ; pi2 = refl}    

conversion1 : {l : Level} {t : Set l } !$\rightarrow$! (env : Env) !$\rightarrow$! {c10 : !$\mathbb{N}$! } !$\rightarrow$! ((vari env) !$\equiv$! 0) !$\wedge$! ((varn env) !$\equiv$! c10)
               !$\rightarrow$! (Code : (env1 : Env) !$\rightarrow$! (varn env1 + vari env1 !$\equiv$! c10) !$\rightarrow$! t) !$\rightarrow$! t
conversion1 env {c10} p1 next = next env proof4
   where
      proof4 : varn env + vari env !$\equiv$! c10
      proof4 = let open !$\equiv$!-Reasoning  in
          begin
            varn env + vari env
          !$\equiv$!!$\langle$! cong ( !$\lambda$! n !$\rightarrow$! n + vari env ) (pi2 p1 ) !$\rangle$!
            c10 + vari env
          !$\equiv$!!$\langle$! cong ( !$\lambda$! n !$\rightarrow$! c10 + n ) (pi1 p1 ) !$\rangle$!
            c10 + 0
          !$\equiv$!!$\langle$! +-sym {c10} {0} !$\rangle$!
            c10
          !$\blacksquare$!

{-!$\#$! TERMINATING !$\#$!-}
whileLoop : {l : Level} {t : Set l} !$\rightarrow$! (env : Env) !$\rightarrow$! {c10 : !$\mathbb{N}$! } !$\rightarrow$! ((varn env) + (vari env) !$\equiv$! c10) !$\rightarrow$! (Code : Env !$\rightarrow$! t) !$\rightarrow$! t
whileLoop env proof next with  ( suc zero  !$\leq$!? (varn  env) )
whileLoop env proof next | no p = next env
whileLoop env {c10} proof next | yes p = whileLoop env1 (proof3 p ) next
    where
      env1 = record {varn = (varn  env) - 1 ; vari = (vari env) + 1}
      1<0 : 1 !$\leq$! zero !$\rightarrow$! !$\bot$!
      1<0 ()
      proof3 : (suc zero  !$\leq$! (varn  env))  !$\rightarrow$! varn env1 + vari env1 !$\equiv$! c10
      proof3 (s!$\leq$!s lt) with varn  env
      proof3 (s!$\leq$!s z!$\leq$!n) | zero = !$\bot$!-elim (1<0 p)
      proof3 (s!$\leq$!s (z!$\leq$!n {n!$\prime$!}) ) | suc n =  let open !$\equiv$!-Reasoning  in
          begin
             n!$\prime$! + (vari env + 1)
          !$\equiv$!!$\langle$! cong ( !$\lambda$! z !$\rightarrow$! n!$\prime$! + z ) ( +-sym  {vari env} {1} )  !$\rangle$!
             n!$\prime$! + (1 + vari env )
          !$\equiv$!!$\langle$! sym ( +-assoc (n!$\prime$!)  1 (vari env) ) !$\rangle$!
             (n!$\prime$! + 1) + vari env
          !$\equiv$!!$\langle$! cong ( !$\lambda$! z !$\rightarrow$! z + vari env )  +1!$\equiv$!suc  !$\rangle$!
             (suc n!$\prime$! ) + vari env
          !$\equiv$!!$\langle$!!$\rangle$!
             varn env + vari env
          !$\equiv$!!$\langle$! proof  !$\rangle$!
             c10
          !$\blacksquare$!