ProofShift.ws.g1w: def of bend-osdev-word64@0.1.1.1
Import: import bend-osdev-word64@0.1.1.1/proof_shift.bend as ProofShift
Lines 369-374 of bend-osdev-word64@0.1.1.1/proof_shift.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-word64@0.1.1.1/proof_shift.bend&start_line=369&end_line=374

import Base
import bend-osdev-u32@0.1.0.0/u32.bend as U
import bend-osdev-u32@0.1.0.0/LAWS.bend as UL

# The bits a shift left by k carries out are the shift right by n - k:
# 2^k * w = shl(w, k) + 2^n * shr(w, n - k), and shl(w, k) is 2^k times
# the remainder of w by 2^(n - k)
def ws.g1w(+n: Nat, +k: Nat, +w: Word(n), hk: {Nat.is_le(k, n) == True{} : Bool}) -> {U.U32Nat.dbl(k, Word.to_nat(n, w)) == Nat.add(U.U32Nat.dbl(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), U.U32Nat.dbl(n, Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))) : Nat}:
  +er = UL.nat_le_witness(U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))), Word.to_nat(n, w), ws.shr_le(n, Nat.sub(n, k), w))
  Equal.trans(Nat, U.U32Nat.dbl(k, Word.to_nat(n, w)), U.U32Nat.dbl(k, Nat.add(Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), Nat.add(U.U32Nat.dbl(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), U.U32Nat.dbl(n, Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), Equal.cong(Nat, Nat, v => U.U32Nat.dbl(k, v), Word.to_nat(n, w), Nat.add(Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), Equal.sym(Nat, Nat.add(Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), Word.to_nat(n, w), er)), Equal.trans(Nat, U.U32Nat.dbl(k, Nat.add(Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), Nat.add(U.U32Nat.dbl(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), Nat.add(U.U32Nat.dbl(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), U.U32Nat.dbl(n, Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), UL.dbl_add(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), Equal.cong(Nat, Nat, v => Nat.add(U.U32Nat.dbl(k, Nat.sub(Word.to_nat(n, w), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))))), v), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)))), U.U32Nat.dbl(n, Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w))), ns.dkm(n, k, Word.to_nat(n, ws.shr(n, Nat.sub(n, k), w)), 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.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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.to_nat
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=dbl_add
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_le_witness
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.dbl
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_shift.bend&name=ns.dkm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_shift.bend&name=ws.shr
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_shift.bend&name=ws.shr_le