whileConvPSemSound : {l : Level} !$\rightarrow$! (input : Envc) !$\rightarrow$! ((vari input !$\equiv$! 0) !$\wedge$! (varn input !$\equiv$! c)) implies (varn input + vari input !$\equiv$! c10 input)
whileConvPSemSound input = proof !$\lambda$! x !$\rightarrow$! (conversion input x) where
  conversion : (env : Envc ) !$\rightarrow$! (vari env !$\equiv$! 0) !$\wedge$! (varn env !$\equiv$! c10 env) !$\rightarrow$! varn env + vari env !$\equiv$! c10 env
  conversion e record { pi1 = refl ; pi2 = refl } = +zero