ProofShift.ws.gshr_lo_le: def of bend-osdev-word64@0.1.1.0
Import: import bend-osdev-word64@0.1.1.0/proof_shift.bend as ProofShift
Lines 436-439 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=436&end_line=439

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@0.1.1.0/proof_value.bend as P

def ws.gshr_lo_le(+n: Nat, +k: Nat, +hw: Word(n), +lw: Word(n), +hk: {Nat.is_lt(k, n) == True{} : Bool}) -> {Nat.is_le(U.U32Nat.dbl(k, 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)))), P.wv.V(n, hw, lw)) == True{} : Bool}:
  +hkle = P.wn.lt_le(k, n, hk)
  +e = Equal.trans(Nat, U.U32Nat.dbl(k, 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)))), U.U32Nat.dbl(k, Nat.add(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, Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), Equal.cong(Nat, Nat, v => U.U32Nat.dbl(k, v), 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)), Equal.trans(Nat, U.U32Nat.dbl(k, Nat.add(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, 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)))), Nat.add(U.U32Nat.dbl(k, Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), UL.dbl_add(k, 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 => Nat.add(U.U32Nat.dbl(k, Word.to_nat(n, ws.shr(n, k, lw))), v), 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))))
  P.wn.le_rw(Nat.add(U.U32Nat.dbl(k, Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), U.U32Nat.dbl(k, 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)))), P.wv.V(n, hw, lw), Equal.sym(Nat, U.U32Nat.dbl(k, 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(U.U32Nat.dbl(k, Word.to_nat(n, ws.shr(n, k, lw))), U.U32Nat.dbl(n, Word.to_nat(n, hw))), e), ns.le_add_mono_r(U.U32Nat.dbl(k, Word.to_nat(n, ws.shr(n, k, lw))), Word.to_nat(n, lw), U.U32Nat.dbl(n, Word.to_nat(n, hw)), ws.shr_le(n, k, lw)))

Uses (23):
  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.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@0.1.1.0/proof_shift.bend&name=ns.dkm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ns.le_add_mono_r
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.gshr_v
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.shl
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.shr
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_shift.bend&name=ws.shr_le
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_value.bend&name=wn.le_rw
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_value.bend&name=wn.lt_le
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-word64@0.1.1.0/proof_value.bend&name=wv.V