PROOFDiv.ud.dm_small: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_div.bend as PROOFDiv
Lines 119-129 of bend-osdev-u32@0.1.0.0/PROOF_div.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&start_line=119&end_line=129

import Base
import bend-osdev-u32@0.1.0.0/LAWS.bend as UL

# fewer steps than m left in the round: the remainder grows by n
def ud.dm_small(+n: Nat, +m: Nat, +d: Nat, +r: Nat) -> {Nat.is_lt(n, 1n+m) == True{} : Bool} -> {Nat.divmod.go(n, m, d, r) == (d, Nat.add(n, r)) : Nat & Nat}:
  match n m:
    case 0n _:
      h => {==}
    case 1n+np 0n:
      h => ud.ex_falso({Nat.divmod.go(1n+np, 0n, d, r) == (d, Nat.add(1n+np, r)) : Nat & Nat}, ud.no_lt_one(np, h))
    case 1n+np 1n+mp:
      h =>
        %UL.nat_add_succ(np, r) : {Nat.divmod.go(1n+np, 1n+mp, d, r) == (d, _) : Nat & Nat}
        ud.dm_small(np, mp, d, 1n+r, h)

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=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.divmod.go
  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=Pair
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Sigma
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.ex_falso
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.no_lt_one