Fix32.crem: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix32.bend as Fix32
Lines 176-183 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix32.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix32.bend&start_line=176&end_line=183

import Base
import bend-collections-laws-math@1.0.0.0/spec/math/generic.bend as SG
import bend-collections-laws-math@1.0.0.0/spec/math/fixed.bend as SF
import bend-collections-laws-math@1.0.0.0/src/math/fixed.bend as F
import bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend as UD
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/fixgen.bend as G

def crem(+a: U32, +b: U32, +z: Bool, +hz: {U32.is_zero(b) == z : Bool}, +ez: {Nat.is_eq(SG.u32_val(b), 0n) == z : Bool}) -> {SF.mval(~U32, ~SG.u32_val, F.opt(U32, z, U32.mod(a, b))) == SF.keep(Bool.not(Nat.is_eq(SG.u32_val(b), 0n)), Nat.mod(SG.u32_val(a), SG.u32_val(b))) : Maybe<&2, Nat>}:
  match z:
    case False{}:
      %Equal.sym(Bool, Nat.is_eq(SG.u32_val(b), 0n), False{}, ez) : {SF.mval(~U32, ~SG.u32_val, F.opt(U32, False{}, U32.mod(a, b))) == SF.keep(Bool.not(_), Nat.mod(SG.u32_val(a), SG.u32_val(b))) : Maybe<&2, Nat>}
      G.some_of(~U32, ~SG.u32_val, False{}, {==}, U32.mod(a, b), Nat.mod(SG.u32_val(a), SG.u32_val(b)), UD.mod_nat(a, b, hz))
    case True{}:
      %Equal.sym(Bool, Nat.is_eq(SG.u32_val(b), 0n), True{}, ez) : {SF.mval(~U32, ~SG.u32_val, F.opt(U32, True{}, U32.mod(a, b))) == SF.keep(Bool.not(_), Nat.mod(SG.u32_val(a), SG.u32_val(b))) : Maybe<&2, Nat>}
      {==}

Uses (16):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Maybe
  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_eq
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.is_zero
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend&name=mod_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/fixgen.bend&name=some_of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/fixed.bend&name=keep
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/fixed.bend&name=mval
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/generic.bend&name=u32_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/fixed.bend&name=opt