const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f const binunion : set set set axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y const SNoL : set set const SNoLev : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoR : set set axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P axiom binunionI2: !x:set.!y:set.!z:set.z iIn y -> z iIn binunion x y const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoCut : set set set axiom add_SNo_eq: !x:set.SNo x -> !y:set.SNo y -> x + y = SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) lemma !x:set.!y:set.!z:set.SNoCutP (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) -> z iIn SNoL x -> z + y iIn binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x)) -> (z + y) < SNoCut (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) lemma !x:set.!y:set.!z:set.SNoCutP (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) -> z iIn SNoR x -> z + y iIn binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x)) -> SNoCut (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) < z + y lemma !x:set.!y:set.!z:set.SNoCutP (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) -> z iIn SNoL y -> x + z iIn binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x)) -> (x + z) < SNoCut (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) lemma !x:set.!y:set.!z:set.SNoCutP (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) -> z iIn SNoR y -> x + z iIn binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x)) -> SNoCut (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) < x + z var x:set var y:set hyp SNo x hyp SNo y hyp SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) claim SNo (SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x)))) -> SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x)))