Canon.low_bit_val: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/canon.bend as Canon
Lines 184-194 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/canon.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/canon.bend&start_line=184&end_line=194

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend as M
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend as LM

# the value mod 2 is the low limb mod 2
def low_bit_val(+one: Nat, +h1: {one == 1n : Nat}, +ys: List<&2, U32>, +e: {LM.okb(32n, ys, 255n) == True{} : Bool}) -> {v(F.low_bit(ys)) == Nat.mod(FS.value(ys), 2n) : Nat}:
  match ys:
    case Nil{}:
      Empty.absurd({0n == Nat.mod(0n, 2n) : Nat}, L.false_true(e))
    case Con{+l, +t}:
      +V = FS.value(t)
      +q = WD.sc(7n, V)
      +e1 = Equal.trans(Nat, Nat.add(v(l), C.shift(8n, V)), Nat.add(C.shift(8n, V), v(l)), Nat.add(Nat.mul(q, 2n), v(l)), NA.add_comm(v(l), C.shift(8n, V)), M.cong_l(C.shift(8n, V), Nat.mul(q, 2n), v(l), Equal.trans(Nat, Nat.double(q), Nat.mul(2n, q), Nat.mul(q, 2n), NA.double_mul(q), NA.mul_comm(2n, q))))
      +e2 = Equal.trans(Nat, Nat.mod(Nat.add(v(l), C.shift(8n, V)), 2n), Nat.mod(Nat.add(Nat.mul(q, 2n), v(l)), 2n), Nat.mod(v(l), 2n), M.cong_mod(2n, Nat.add(v(l), C.shift(8n, V)), Nat.add(Nat.mul(q, 2n), v(l)), e1), NR.absorb(1n, q, v(l)))
      Equal.trans(Nat, v(U32.and(l, 1)), Nat.mod(v(l), 2n), Nat.mod(Nat.add(v(l), C.shift(8n, V)), 2n), and1(one, h1, l), Equal.sym(Nat, Nat.mod(Nat.add(v(l), C.shift(8n, V)), 2n), Nat.mod(v(l), 2n), e2))

Uses (26):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  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=List
  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=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.and
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/canon.bend&name=and1
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/canon.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=okb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend&name=cong_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend&name=cong_mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=double_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=absorb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=low_bit