Arith.pow2_add: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend as Arith
Lines 99-104 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&start_line=99&end_line=104

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/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

def pow2_add(+a: Nat, +b: Nat) -> {Nat.mul(C.pow2(a), C.pow2(b)) == C.pow2(Nat.add(a, b)) : Nat}:
  +pa = C.pow2(a)
  +pb = C.pow2(b)
  tr(Nat.mul(pa, pb), Nat.mul(pb, pa), C.pow2(Nat.add(a, b)), NA.mul_comm(pa, pb),
    tr(Nat.mul(pb, pa), Nat.mul(pb, C.shift(a, 1n)), C.pow2(Nat.add(a, b)), Equal.cong(Nat, Nat, t => Nat.mul(pb, t), pa, C.shift(a, 1n), sy(C.shift(a, 1n), pa, WW.shift_one(a))),
      tr(Nat.mul(pb, C.shift(a, 1n)), C.shift(a, pb), C.pow2(Nat.add(a, b)), sy(C.shift(a, pb), Nat.mul(pb, C.shift(a, 1n)), WW.shift_mul(a, pb)), WW.shift_pow2(a, b))))

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&name=sy
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&name=tr
  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/math/typed/width.bend&name=shift_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=shift_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=shift_pow2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=pow2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=shift