ProofShift.ws.shlv: def of bend-osdev-word64@0.1.1.0
Import: import bend-osdev-word64@0.1.1.0/proof_shift.bend as ProofShift
Lines 277-289 of bend-osdev-word64@0.1.1.0/proof_shift.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&start_line=277&end_line=289

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

# Word.shl: the doubled value, less 2^n for the top bit
def ws.shlv(n: Nat, w: Word(n)) -> {Nat.add(Word.to_nat(n, Word.shl(n, w)), U.U32Nat.dbl(n, U.U32Nat.bit(ws.top(n, w)))) == Nat.double(Word.to_nat(n, w)) : Nat}:
  match n:
    case 0n:
      match w:
        case WNil{}:
          {==}
    case 1n+ +p:
      match w:
        case WCon{False{}, +t}:
          Equal.trans(Nat, Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, False{}, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t))))), Nat.double(Nat.add(Word.to_nat(p, Word.shl.put(p, False{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t))))), Nat.double(Nat.add(U.U32Nat.bit(False{}), Nat.double(Word.to_nat(p, t)))), Equal.sym(Nat, Nat.double(Nat.add(Word.to_nat(p, Word.shl.put(p, False{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t))))), Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, False{}, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t))))), ns.double_add(Word.to_nat(p, Word.shl.put(p, False{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t))))), Equal.cong(Nat, Nat, v => Nat.double(v), Nat.add(Word.to_nat(p, Word.shl.put(p, False{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, False{}, t)))), Nat.add(U.U32Nat.bit(False{}), Nat.double(Word.to_nat(p, t))), ws.putv(p, False{}, t)))
        case WCon{True{}, +t}:
          Equal.trans(Nat, Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, True{}, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t))))), Nat.double(Nat.add(Word.to_nat(p, Word.shl.put(p, True{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t))))), Nat.double(Nat.add(U.U32Nat.bit(True{}), Nat.double(Word.to_nat(p, t)))), Equal.sym(Nat, Nat.double(Nat.add(Word.to_nat(p, Word.shl.put(p, True{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t))))), Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, True{}, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t))))), ns.double_add(Word.to_nat(p, Word.shl.put(p, True{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t))))), Equal.cong(Nat, Nat, v => Nat.double(v), Nat.add(Word.to_nat(p, Word.shl.put(p, True{}, t)), U.U32Nat.dbl(p, U.U32Nat.bit(ws.topc(p, True{}, t)))), Nat.add(U.U32Nat.bit(True{}), Nat.double(Word.to_nat(p, t))), ws.putv(p, True{}, t)))

Uses (19):
  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.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Con
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Nil
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.shl
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.shl.put
  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/u32.bend&name=U32Nat.bit
  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@0.1.1.0/proof_shift.bend&name=ns.double_add
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.putv
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.top
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.topc