PROOFDiv.ud.dm_rounds: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_div.bend as PROOFDiv
Lines 140-155 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=140&end_line=155

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

def ud.dm_rounds(+g: Nat, +i: Nat, +d: Nat, +bp: Nat) -> {Nat.is_lt(i, 1n+bp) == True{} : Bool} -> {Nat.divmod.go(Nat.add(Nat.mul(g, 1n+bp), i), bp, d, 0n) == (Nat.add(g, d), i) : Nat & Nat}:
  match g:
    case 0n:
      h =>
        %UL.nat_add_zero(i) : {Nat.divmod.go(i, bp, d, 0n) == (d, _) : Nat & Nat}
        ud.dm_small(i, bp, d, 0n, h)
    case 1n+gp:
      h =>
        +gg = Nat.mul(gp, 1n+bp)
        +y = Nat.add(gg, i)
        %Equal.sym(Nat, Nat.add(Nat.add(bp, gg), i), Nat.add(bp, y), UL.nat_add_assoc(bp, gg, i)) : {Nat.divmod.go(1n+_, bp, d, 0n) == (1n+Nat.add(gp, d), i) : Nat & Nat}
        %UL.nat_add_succ(bp, y) : {Nat.divmod.go(_, bp, d, 0n) == (1n+Nat.add(gp, d), i) : Nat & Nat}
        %Equal.sym(Nat & Nat, Nat.divmod.go(Nat.add(bp, 1n+y), bp, d, 0n), Nat.divmod.go(y, Nat.add(bp, 0n), 1n+d, 0n), ud.dm_round(bp, y, d, 0n)) : {_ == (1n+Nat.add(gp, d), i) : Nat & Nat}
        %Equal.sym(Nat, Nat.add(bp, 0n), bp, UL.nat_add_zero(bp)) : {Nat.divmod.go(y, _, 1n+d, 0n) == (1n+Nat.add(gp, d), i) : Nat & Nat}
        %UL.nat_add_succ(gp, d) : {Nat.divmod.go(y, bp, 1n+d, 0n) == (_, i) : Nat & Nat}
        ud.dm_rounds(gp, i, 1n+d, bp, h)

Uses (14):
  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.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=Nat.mul
  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_assoc
  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/LAWS.bend&name=nat_add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.dm_round
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_div.bend&name=ud.dm_small