Nat.sub_lt: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as Nat
Lines 307-315 of bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&start_line=307&end_line=315

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

def sub_lt(+a: Nat, +b: Nat, +c: Nat, le: {Nat.is_le(b, a) == True{} : Bool}, lt: {Nat.is_lt(a, Nat.add(b, c)) == True{} : Bool}) -> {Nat.is_lt(Nat.sub(a, b), c) == True{} : Bool}:
  match a b:
    case _ 0n:
      %Equal.sym(Nat, Nat.sub(a, 0n), a, sub_zero(a)) : {Nat.is_lt(_, c) == True{} : Bool}
      lt
    case 0n 1n+q:
      Empty.absurd({Nat.is_lt(Nat.sub(0n, 1n+q), c) == True{} : Bool}, L.false_true(le))
    case 1n+p 1n+q:
      sub_lt(p, q, c, le, lt)

Uses (10):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=sub_zero