Wrappers.shr_half: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/wrappers.bend as Wrappers
Lines 83-91 of bend-collections-laws-math@1.0.0.0/proofs/math/random/wrappers.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/wrappers.bend&start_line=83&end_line=91

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/proofs/lib/u32.bend as U
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW

# x >> 1 is half of x
def shr_half(+x: U32) -> {v(U32.shr(x)) == C.half(v(x)) : Nat}:
  +b = S.bit_value(U.low_bit(x))
  +y = v(U32.shr(x))
  Equal.sym(Nat, C.half(v(x)), y,
    Equal.trans(Nat, C.half(v(x)), C.half(Nat.add(b, Nat.double(y))), y,
      Equal.cong(Nat, Nat, z => C.half(z), v(x), Nat.add(b, Nat.double(y)), U.shr_split(x)),
      Equal.trans(Nat, C.half(Nat.add(b, Nat.double(y))), Nat.add(C.half(b), y), y, WW.half_dbl(b, y),
        Equal.cong(Nat, Nat, z => Nat.add(z, y), C.half(b), 0n, half_bv(U.low_bit(x))))))

Uses (15):
  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=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.shr
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend&name=low_bit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend&name=shr_split
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/wrappers.bend&name=half_bv
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/wrappers.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=half_dbl
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=half