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'}) ) | suc n =  let open @$\equiv$@-Reasoning  in
          begin
             n' + (vari env + 1)
          @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ n' + z ) ( +-sym  {vari env} {1} )  @$\rangle$@
             n' + (1 + vari env )
          @$\equiv$@@$\langle$@ sym ( +-assoc (n')  1 (vari env) ) @$\rangle$@
             (n' + 1) + vari env
          @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ z + vari env )  +1@$\equiv$@suc  @$\rangle$@
             (suc n' ) + vari env
          @$\equiv$@@$\langle$@@$\rangle$@
             varn env + vari env
          @$\equiv$@@$\langle$@ proof  @$\rangle$@
             c10
          @$\blacksquare$@