Sel.sel0: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/sel.bend as Sel
Lines 32-54 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/sel.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/sel.bend&start_line=32&end_line=54

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/secp256k1/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/limbs.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as Lg
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

def sel0(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(0n, xs, ys) == ys : List<&2, Nat>}:
  match n xs:
    case 0n Nil{}:
      match ys:
        case Nil{}:
          {==}
        case y <> t:
          Empty.absurd({L.select(0n, [], y <> t) == y <> t : List<&2, Nat>}, Lg.false_true(hy))
    case 0n x <> t:
      Empty.absurd({L.select(0n, x <> t, ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+k Nil{}:
      Empty.absurd({L.select(0n, [], ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+ +k +x <> +xt:
      match ys:
        case Nil{}:
          Empty.absurd({L.select(0n, x <> xt, []) == [] : List<&2, Nat>}, Lg.false_true(hy))
        case +y <> +yt:
          +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
          +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
          %Equal.sym(List<&2, Nat>, L.select(0n, xt, yt), yt, sel0(one, k, xt, yt, hxt, hyt)) : {Nat.add(Nat.mul(x, 0n), Nat.mul(y, 1n)) <> _ == y <> yt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(x, 0n), 0n, NA.mul_zero(x)) : {Nat.add(_, Nat.mul(y, 1n)) <> yt == y <> yt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(y, 1n), y, NA.mul_one(y)) : {Nat.add(0n, _) <> yt == y <> yt : List<&2, Nat>}
          {==}

Uses (15):
  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=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.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.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_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=and_right
  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/spec/crypto/secp256k1/field.bend&name=limbs16
  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/secp256k1/limbs.bend&name=select