Fieldpow.exp_inv: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend as Fieldpow
Lines 193-199 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&start_line=193&end_line=199

import Base
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/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
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/secp256k1/consts.bend as K
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsv.bend as BV

def exp_inv(+one: Nat, +h1: {one == 1n : Nat}) -> {BV.hv(F.inv_bits(), one, 0n) == Nat.sub(1n+K.pp(one), 2n) : Nat}:
  +h = BV.hv(F.inv_bits(), one, 0n)
  +e = NR.add_cancel(FS.cp(one), Nat.add(h, 2n), 1n+K.pp(one), inv_h2(one, h1))
  %Equal.sym(Nat, 1n+K.pp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n), Equal.sym(Nat, Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n), 1n+K.pp(one), e)) : {BV.hv(F.inv_bits(), one, 0n) == Nat.sub(_, 2n) : Nat}
  %Equal.sym(Nat, Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n), Nat.add(2n, BV.hv(F.inv_bits(), one, 0n)), NA.add_comm(BV.hv(F.inv_bits(), one, 0n), 2n)) : {BV.hv(F.inv_bits(), one, 0n) == Nat.sub(_, 2n) : Nat}
  %Equal.sym(Nat, Nat.sub(Nat.add(2n, BV.hv(F.inv_bits(), one, 0n)), 2n), BV.hv(F.inv_bits(), one, 0n), N.add_sub_cancel(2n, BV.hv(F.inv_bits(), one, 0n))) : {BV.hv(F.inv_bits(), one, 0n) == _ : Nat}
  {==}

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsv.bend&name=hv
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/consts.bend&name=pp
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=inv_h2
  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/nat.bend&name=add_sub_cancel
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=add_cancel
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend&name=cp
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend&name=inv_bits