const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const int : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const eps_ : set set term diadic_rational_p = \x:set.?y:set.y iIn omega & ?z:set.z iIn int & x = eps_ y * z term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom int_add_SNo: !x:set.x iIn int -> !y:set.y iIn int -> x + y iIn int const SNo : set prop axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) axiom int_SNo: !x:set.x iIn int -> SNo x const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const ordsucc : set set const Empty : set axiom mul_SNo_oneL: !x:set.SNo x -> ordsucc Empty * x = x const exp_SNo_nat : set set set axiom mul_SNo_eps_power_2: !x:set.nat_p x -> eps_ x * exp_SNo_nat (ordsucc (ordsucc Empty)) x = ordsucc Empty axiom mul_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * y * z = (x * y) * z axiom mul_SNo_com_4_inner_mid: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x * y) * z * w = (x * z) * y * w axiom mul_SNo_distrL: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * (y + z) = x * y + x * z axiom mul_SNo_eps_eps_add_SNo: !x:set.x iIn omega -> !y:set.y iIn omega -> eps_ x * eps_ y = eps_ (x + y) var x:set var y:set var z:set var w:set var u:set var v:set hyp z iIn omega hyp SNo (eps_ z) hyp w iIn int hyp x = eps_ z * w hyp u iIn omega hyp SNo (eps_ u) hyp v iIn int hyp y = eps_ u * v hyp SNo (eps_ u * v) hyp exp_SNo_nat (ordsucc (ordsucc Empty)) u iIn int hyp exp_SNo_nat (ordsucc (ordsucc Empty)) u * w iIn int hyp exp_SNo_nat (ordsucc (ordsucc Empty)) z iIn int claim exp_SNo_nat (ordsucc (ordsucc Empty)) z * v iIn int -> exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v iIn int & x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)