ProofShift.ws.gshr_lo_lt: def of bend-osdev-word64@0.1.1.1
Import: import bend-osdev-word64@0.1.1.1/proof_shift.bend as ProofShift
Lines 441-445 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=441&end_line=445

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
import bend-osdev-word64@0.1.1.1/proof_value.bend as P

def ws.gshr_lo_lt(+n: Nat, +k: Nat, +hw: Word(n), +lw: Word(n), +hk: {Nat.is_lt(k, n) == True{} : Bool}) -> {Nat.is_lt(P.wv.V(n, hw, lw), U.U32Nat.dbl(k, 1n+P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw))))) == True{} : Bool}:
  +hkle = P.wn.lt_le(k, n, hk)
  +s1 = Equal.trans(Bool, Nat.is_lt(Nat.add(Word.to_nat(n, lw), U.U32Nat.dbl(n, Word.to_nat(n, hw))), Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw)))), Nat.is_lt(Word.to_nat(n, lw), U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw)))), True{}, P.wn.lt_add_r(Word.to_nat(n, lw), U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), ws.shr_lt(n, k, lw))
  +e = Equal.trans(Nat, Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), U.U32Nat.dbl(k, 1n+P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw)))), Equal.cong(Nat, Nat, v => Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), v), U.U32Nat.dbl(n, Word.to_nat(n, hw)), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw))), Equal.sym(Nat, U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw))), U.U32Nat.dbl(n, Word.to_nat(n, hw)), ns.dkm(n, k, Word.to_nat(n, hw), hkle))), Equal.trans(Nat, Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), U.U32Nat.dbl(k, Nat.add(1n+Word.to_nat(n, ws.shr(n, k, lw)), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), U.U32Nat.dbl(k, 1n+P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw)))), Equal.sym(Nat, U.U32Nat.dbl(k, Nat.add(1n+Word.to_nat(n, ws.shr(n, k, lw)), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(k, U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), UL.dbl_add(k, 1n+Word.to_nat(n, ws.shr(n, k, lw)), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw)))), Equal.cong(Nat, Nat, v => U.U32Nat.dbl(k, 1n+v), Nat.add(Word.to_nat(n, ws.shr(n, k, lw)), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw))), P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw))), Equal.sym(Nat, P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw))), Nat.add(Word.to_nat(n, ws.shr(n, k, lw)), U.U32Nat.dbl(Nat.sub(n, k), Word.to_nat(n, hw))), ws.gshr_v(n, k, hw, lw, hk)))))
  P.wn.lt_rw2(P.wv.V(n, hw, lw), Nat.add(U.U32Nat.dbl(k, 1n+Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), U.U32Nat.dbl(k, 1n+P.wv.V(n, ws.shr(n, k, hw), Word.or(n, ws.shr(n, k, lw), ws.shl(n, Nat.sub(n, k), hw)))), e, s1)

Uses (22):
  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_lt
  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.or
  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/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.gshr_v
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_shift.bend&name=ws.shl
  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_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_value.bend&name=wn.lt_add_r
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_value.bend&name=wn.lt_le
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_value.bend&name=wn.lt_rw2
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.1/proof_value.bend&name=wv.V