Nat.le_trans: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as Nat
Lines 143-152 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&start_line=143&end_line=152

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

def le_trans(+a: Nat, +b: Nat, +c: Nat, ab: {Nat.is_le(a, b) == True{} : Bool}, bc: {Nat.is_le(b, c) == True{} : Bool}) -> {Nat.is_le(a, c) == True{} : Bool}:
  match a b c:
    case 0n _ _:
      zero_le(c)
    case 1n+p 0n _:
      Empty.absurd({Nat.is_le(1n+p, c) == True{} : Bool}, L.false_true(ab))
    case 1n+p 1n+q 0n:
      Empty.absurd({Nat.is_le(1n+p, 0n) == True{} : Bool}, L.false_true(bc))
    case 1n+p 1n+q 1n+r:
      le_trans(p, q, r, ab, bc)

Uses (6):
  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=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=zero_le