Float.m_val: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend as Float
Lines 320-331 of bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend&start_line=320&end_line=331

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u32laws.bend as LW
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bits.bend as FB

def m_val(+x: WU.U64) -> {SW.value(mm(x)) == C.low(53n, SW.value(x)) : Nat}:
  match x:
    case WU.U64{+l, +h}:
      +n = Nat.add(v(l), C.shift(32n, v(h)))
      Equal.trans(Nat, Nat.add(v(l), C.shift(32n, v(U32.and(h, 2097151)))), Nat.add(v(l), C.shift(32n, C.low(21n, v(h)))), C.low(53n, n),
        Equal.cong(Nat, Nat, z => Nat.add(v(l), C.shift(32n, z)), v(U32.and(h, 2097151)), C.low(21n, v(h)), FB.andm(h, 21n, 2097151, {==})),
        Equal.sym(Nat, C.low(53n, n), Nat.add(v(l), C.shift(32n, C.low(21n, v(h)))),
          Equal.trans(Nat, C.low(53n, n), Nat.add(C.low(32n, n), C.shift(32n, C.low(21n, C.high(32n, n)))), Nat.add(v(l), C.shift(32n, C.low(21n, v(h)))),
            FB.low_split(32n, 21n, n),
            Equal.trans(Nat, Nat.add(C.low(32n, n), C.shift(32n, C.low(21n, C.high(32n, n)))), Nat.add(v(l), C.shift(32n, C.low(21n, C.high(32n, n)))), Nat.add(v(l), C.shift(32n, C.low(21n, v(h)))),
              Equal.cong(Nat, Nat, z => Nat.add(z, C.shift(32n, C.low(21n, C.high(32n, n)))), C.low(32n, n), v(l), WW.low_u(32n, v(l), v(h), LW.vb(l))),
              Equal.cong(Nat, Nat, z => Nat.add(v(l), C.shift(32n, C.low(21n, z))), C.high(32n, n), v(h), WW.high_u(32n, v(l), v(h), LW.vb(l)))))))

Uses (18):
  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=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=U32.and
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend&name=mm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bits.bend&name=andm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bits.bend&name=low_split
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u32laws.bend&name=vb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=high_u
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=low_u
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=high
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/w64.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/u64.bend&name=U64