def sc_lt_cancel_g

bend-collections-laws-math@1.0.0.0 · proofs/lib/arith.bend:47

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L

47def sc_lt_cancel_g(+k: Nat, +a: Nat, +b: Nat, +c: Bool, +hc: {Nat.is_lt(a, b) == c : Bool}, +h: {Nat.is_lt(sc(k, a), sc(k, b)) == True{} : Bool}) -> {Nat.is_lt(a, b) == True{} : Bool}:
48  match c:
49    case True{}:
50      hc
51    case False{}:
52      Empty.absurd({Nat.is_lt(a, b) == True{} : Bool}, L.true_false(Equal.trans(Bool, True{}, Nat.is_le(sc(k, b), sc(k, a)), False{}, Equal.sym(Bool, Nat.is_le(sc(k, b), sc(k, a)), True{}, sc_le(k, b, a, N.not_lt_le(a, b, hc))), N.lt_not_le(sc(k, a), sc(k, b), h))))