Limbs.take_drop_v: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as Limbs
Lines 137-157 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&start_line=137&end_line=157

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/src/crypto/secp256k1/limbs.bend as L
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/typed/width.bend as WW

# ev(xs) = ev(take n xs) + 2^(16 n) ev(drop n xs)
def take_drop_v(n: Nat, xs: List<&2, Nat>) -> {ev(xs) == Nat.add(ev(L.take(n, xs)), pos(n, ev(L.drop(n, xs)))) : Nat}:
  match n xs:
    case 0n Nil{}:
      {==}
    case 0n x <> t:
      {==}
    case 1n+ +k Nil{}:
      %Equal.sym(Nat, pos(k, 0n), 0n, pos_zero(k)) : {0n == Nat.add(0n, C.shift(16n, _)) : Nat}
      %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {0n == Nat.add(0n, _) : Nat}
      %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {0n == _ : Nat}
      {==}
    case 1n+ +k +x <> +t:
      %Equal.sym(Nat, ev(t), Nat.add(ev(L.take(k, t)), pos(k, ev(L.drop(k, t)))), take_drop_v(k, t)) : {Nat.add(x, C.shift(16n, _)) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
      %Equal.sym(Nat, C.shift(16n, Nat.add(ev(L.take(k, t)), pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t))))), WW.shift_add(16n, ev(L.take(k, t)), pos(k, ev(L.drop(k, t))))) : {Nat.add(x, _) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
      %Equal.sym(Nat, Nat.add(x, Nat.add(C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t)))))), Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))), NA.add_swap(x, C.shift(16n, ev(L.take(k, t))), C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {_ == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
      %Equal.sym(Nat, Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x), NA.add_comm(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), _) == Nat.add(Nat.add(x, C.shift(16n, ev(L.take(k, t)))), C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
      %Equal.sym(Nat, Nat.add(x, C.shift(16n, ev(L.take(k, t)))), Nat.add(C.shift(16n, ev(L.take(k, t))), x), NA.add_comm(x, C.shift(16n, ev(L.take(k, t))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == Nat.add(_, C.shift(16n, pos(k, ev(L.drop(k, t))))) : Nat}
      %Equal.sym(Nat, Nat.add(Nat.add(C.shift(16n, ev(L.take(k, t))), x), C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))), NA.add_assoc(C.shift(16n, ev(L.take(k, t))), x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == _ : Nat}
      %Equal.sym(Nat, Nat.add(x, C.shift(16n, pos(k, ev(L.drop(k, t))))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x), NA.add_comm(x, C.shift(16n, pos(k, ev(L.drop(k, t)))))) : {Nat.add(C.shift(16n, ev(L.take(k, t))), Nat.add(C.shift(16n, pos(k, ev(L.drop(k, t)))), x)) == Nat.add(C.shift(16n, ev(L.take(k, t))), _) : Nat}
      {==}

Uses (15):
  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=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&name=ev
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&name=pos
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&name=pos_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_assoc
  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=add_swap
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=shift_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=shift_zero
  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=drop
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/limbs.bend&name=take