U32half.sub_le: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend as U32half
Lines 143-152 of bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend&start_line=143&end_line=152

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

# U32.sub is exact whenever the subtrahend is not larger (no borrow), so it
# is the Nat subtraction of the two indices.
def sub_le(+a: Nat, +b: Nat) -> {Nat.is_le(Nat.sub(a, b), a) == True{} : Bool}:
  match a b:
    case a0 0n:
      L.subst(Nat, z => {Nat.is_le(z, a0) == True{} : Bool}, a0, Nat.sub(a0, 0n), Equal.sym(Nat, Nat.sub(a0, 0n), a0, N.sub_zero(a0)), N.le_refl(a0))
    case 0n 1n+q:
      N.zero_le(Nat.sub(0n, 1n+q))
    case 1n+p 1n+q:
      N.le_trans(Nat.sub(p, q), p, 1n+p, sub_le(p, q), N.le_succ(p))

Uses (11):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_le
  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=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_refl
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=sub_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=zero_le