PROOF.un.shlv: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 1361-1368 of bend-osdev-u32@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&start_line=1361&end_line=1368

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

def un.shlv(n: Nat, +w: Word(n)) -> {Nat.add(Word.to_nat(n, Word.shl(n, w)), U.U32Nat.dbl(n, U.U32Nat.bit(un.top(n, w)))) == Nat.double(Word.to_nat(n, w)) : Nat}:
  match n:
    case 0n:
      {==}
    case 1n+ +p:
      match w:
        case WCon{+x, +t}:
          Equal.trans(Nat, Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, x, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(un.topc(p, x, t))))), Nat.double(Nat.add(U.U32Nat.bit(x), Nat.double(Word.to_nat(p, t)))), Nat.double(Word.to_nat(1n+p, WCon{x, t})), Equal.trans(Nat, Nat.add(Nat.double(Word.to_nat(p, Word.shl.put(p, x, t))), Nat.double(U.U32Nat.dbl(p, U.U32Nat.bit(un.topc(p, x, t))))), Nat.double(Nat.add(Word.to_nat(p, Word.shl.put(p, x, t)), U.U32Nat.dbl(p, U.U32Nat.bit(un.topc(p, x, t))))), Nat.double(Nat.add(U.U32Nat.bit(x), Nat.double(Word.to_nat(p, t)))), ur.r13(Word.to_nat(p, Word.shl.put(p, x, t)), p, U.U32Nat.bit(un.topc(p, x, t))), Equal.cong(Nat, Nat, zz => Nat.double(zz), Nat.add(Word.to_nat(p, Word.shl.put(p, x, t)), U.U32Nat.dbl(p, U.U32Nat.bit(un.topc(p, x, t)))), Nat.add(U.U32Nat.bit(x), Nat.double(Word.to_nat(p, t))), un.putv(p, x, t))), Equal.cong(Nat, Nat, zz => Nat.double(zz), Nat.add(U.U32Nat.bit(x), Nat.double(Word.to_nat(p, t))), Word.to_nat(1n+p, WCon{x, t}), Equal.sym(Nat, Word.to_nat(1n+p, WCon{x, t}), Nat.add(U.U32Nat.bit(x), Nat.double(Word.to_nat(p, t))), un.wc(p, x, t))))

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=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.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/PROOF.bend&name=un.putv
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.top
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.topc
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.wc
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=ur.r13
  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