ProofPcg.or_value: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend as ProofPcg
Lines 51-64 of bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend&start_line=51&end_line=64

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/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
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

# lo | 1
def or_value(+lo: WU.U64) -> {SW.value(WU.U64{U32.or(X.lo(lo), 1), X.hi(lo)}) == Nat.add(Nat.double(C.half(SW.value(lo))), 1n) : Nat}:
  match lo:
    case WU.U64{+l, +h}:
      +vl = v(l)
      +vh = v(h)
      +D = Nat.add(Nat.double(C.half(vl)), C.shift(32n, vh))
      +e1 = Equal.cong(Nat, Nat, z => Nat.add(z, C.shift(32n, vh)), v(U32.or(l, 1)), 1n+Nat.double(C.half(vl)), SH.or1(l))
      +e2 = Equal.cong(Nat, Nat, z => 1n+z, D, Nat.double(Nat.add(C.half(vl), C.shift(31n, vh))), Equal.sym(Nat, Nat.double(Nat.add(C.half(vl), C.shift(31n, vh))), D, NA.double_add(C.half(vl), C.shift(31n, vh))))
      +e3 = Equal.cong(Nat, Nat, z => 1n+Nat.double(z), Nat.add(C.half(vl), C.shift(31n, vh)), C.half(Nat.add(vl, C.shift(32n, vh))), Equal.sym(Nat, C.half(Nat.add(vl, C.shift(32n, vh))), Nat.add(C.half(vl), C.shift(31n, vh)), WW.half_dbl(vl, C.shift(31n, vh))))
      +e4 = Equal.sym(Nat, Nat.add(Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), 1n), 1n+Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), plus1(Nat.double(C.half(Nat.add(vl, C.shift(32n, vh))))))
      Equal.trans(Nat, Nat.add(v(U32.or(l, 1)), C.shift(32n, vh)), 1n+D, Nat.add(Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), 1n), e1,
        Equal.trans(Nat, 1n+D, 1n+Nat.double(Nat.add(C.half(vl), C.shift(31n, vh))), Nat.add(Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), 1n), e2,
          Equal.trans(Nat, 1n+Nat.double(Nat.add(C.half(vl), C.shift(31n, vh))), 1n+Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), Nat.add(Nat.double(C.half(Nat.add(vl, C.shift(32n, vh)))), 1n), e3, e4)))

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=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.or
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=double_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend&name=plus1
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64sh.bend&name=or1
  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
  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
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=hi
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=lo