Float.hv_one: 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 380-389 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=380&end_line=389

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/src/math/w64.bend as X
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/w64sh.bend as SH

# the value of the high word of 1.0 without its sign bit: k * 2^20 (k = 1023,
# kept a variable so no 30-bit constant is ever expanded)
def hv_one(+H: U32, +K: U32, +k: Nat, +hk: {C.fits(12n, k) == True{} : Bool}, +hH: {U32.and(H, 2147483647) == U32.mul(K, X.pow2(20n)) : U32}, +hK: {U32.to_nat(K) == k : Nat}) -> {v(U32.and(H, 2147483647)) == C.shift(20n, k) : Nat}:
  Equal.trans(Nat, v(U32.and(H, 2147483647)), v(U32.mul(K, X.pow2(20n))), C.shift(20n, k),
    Equal.cong(U32, Nat, z => v(z), U32.and(H, 2147483647), U32.mul(K, X.pow2(20n)), hH),
    Equal.trans(Nat, v(U32.mul(K, X.pow2(20n))), C.low(32n, C.shift(20n, v(K))), C.shift(20n, k),
      SH.mul_p2(K, 20n, {==}),
      Equal.trans(Nat, C.low(32n, C.shift(20n, v(K))), C.low(32n, C.shift(20n, k)), C.shift(20n, k),
        Equal.cong(Nat, Nat, z => C.low(32n, C.shift(20n, z)), v(K), k, hK),
        WW.low_fit(32n, C.shift(20n, k), fits_shift(20n, 12n, k, hk)))))

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=Equal.cong
  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=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.and
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/float.bend&name=fits_shift
  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/w64sh.bend&name=mul_p2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=low_fit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=fits
  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/src/math/w64.bend&name=pow2