PROOF.un.shl_w: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 766-781 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=766&end_line=781

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

def un.shl_w(n: Nat, w: Word(n), h: {Nat.is_lt(Nat.double(Word.to_nat(n, w)), U.U32Nat.dbl(n, 1n)) == True{} : Bool}) -> {Word.to_nat(n, Word.shl(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}:
          +k = U.U32Nat.dbl(p, 1n)
          +x = Word.to_nat(p, t)
          un.double_cong(Word.to_nat(p, Word.shl.put(p, False{}, t)), Nat.double(x), un.shlput(p, False{}, t, Equal.trans(Bool, Nat.is_lt(Nat.double(x), k), Nat.is_lt(Nat.double(Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(Nat.double(Nat.double(x)), Nat.double(k)), Nat.is_lt(Nat.double(x), k), un.lt_dbl(Nat.double(x), k)), h)))
        case WCon{True{}, +t}:
          +k = U.U32Nat.dbl(p, 1n)
          +x = Word.to_nat(p, t)
          un.double_cong(Word.to_nat(p, Word.shl.put(p, True{}, t)), 1n+Nat.double(x), un.shlput(p, True{}, t, Equal.trans(Bool, Nat.is_lt(1n+Nat.double(x), k), Nat.is_lt(Nat.double(1n+Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(Nat.double(1n+Nat.double(x)), Nat.double(k)), Nat.is_lt(1n+Nat.double(x), k), un.lt_dbl(1n+Nat.double(x), k)), h)))

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.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.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  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/PROOF.bend&name=un.double_cong
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.lt_dbl
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&name=un.shlput
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.dbl