Fix64.crem: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix64.bend as Fix64
Lines 150-159 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix64.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/fix64.bend&start_line=150&end_line=159

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/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
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/math/typed/w64add.bend as WA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64dmrem.bend as DR

def crem(+a: WU.U64, +b: WU.U64, +z: Bool, +hz: {X.is_zero(b) == z : Bool}) -> {SF.mval(~WU.U64, ~SG.u64_val, F.u64_crem(z, a, b)) == SF.keep(Bool.not(Nat.is_eq(SG.u64_val(b), 0n)), Nat.mod(SG.u64_val(a), SG.u64_val(b))) : Maybe<&2, Nat>}:
  match z:
    case False{}:
      +ez = Equal.trans(Bool, Nat.is_eq(SG.u64_val(b), 0n), X.is_zero(b), z, Equal.sym(Bool, X.is_zero(b), Nat.is_eq(SG.u64_val(b), 0n), WA.is_zero_value(b)), hz)
      %Equal.sym(Bool, Nat.is_eq(SG.u64_val(b), 0n), False{}, ez) : {SF.mval(~WU.U64, ~SG.u64_val, F.u64_crem(False{}, a, b)) == SF.keep(Bool.not(_), Nat.mod(SG.u64_val(a), SG.u64_val(b))) : Maybe<&2, Nat>}
      Equal.cong(Nat, Maybe<&2, Nat>, t => Some{t}, SG.u64_val(X.rem(a, b)), Nat.mod(SG.u64_val(a), SG.u64_val(b)), DR.divmod_rem(a, b, hz))
    case True{}:
      +ez = Equal.trans(Bool, Nat.is_eq(SG.u64_val(b), 0n), X.is_zero(b), z, Equal.sym(Bool, X.is_zero(b), Nat.is_eq(SG.u64_val(b), 0n), WA.is_zero_value(b)), hz)
      %Equal.sym(Bool, Nat.is_eq(SG.u64_val(b), 0n), True{}, ez) : {SF.mval(~WU.U64, ~SG.u64_val, F.u64_crem(True{}, a, b)) == SF.keep(Bool.not(_), Nat.mod(SG.u64_val(a), SG.u64_val(b))) : Maybe<&2, Nat>}
      {==}

Uses (18):
  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.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  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=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend&name=is_zero_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64dmrem.bend&name=divmod_rem
  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=u64_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/fixed.bend&name=u64_crem
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/u64.bend&name=U64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=is_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=rem