def inv_h2

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldpow.bend:174

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/src/crypto/secp256k1/field.bend as F
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/crypto/secp256k1/semiring.bend as SR
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as LV
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

174# hv(bits of p - 2) + 2 = p
175def inv_h2(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(FS.cp(one), 1n+K.pp(one)) : Nat}:
176  +bs = L.bits(L.norm16([978n, 0n, 1n]))
177  +hcb = BV.cb(one, bs, 0n, 0n, BV.bits01(L.norm16([978n, 0n, 1n])))
178  %Equal.sym(Nat, Nat.add(FS.cp(one), 1n+K.pp(one)), Nat.add(1n+K.pp(one), FS.cp(one)), NA.add_comm(FS.cp(one), 1n+K.pp(one))) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == _ : Nat}
179  %Equal.sym(Nat, Nat.add(1n+K.pp(one), FS.cp(one)), C.shift(256n, one), K.hm_p(one, h1)) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == _ : Nat}
180  %Equal.sym(Nat, C.shift(256n, one), Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), BV.hv(L.bits(L.norm16([978n, 0n, 1n])), one, 0n)), one), Equal.sym(Nat, Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), BV.hv(L.bits(L.norm16([978n, 0n, 1n])), one, 0n)), one), C.shift(256n, one), hcb)) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == _ : Nat}
181  %Equal.sym(Nat, BV.hv(L.bits(L.norm16([978n, 0n, 1n])), one, 0n), Nat.add(LV.pos(16n, 0n), FS.digits(one, L.norm16([978n, 0n, 1n]))), BV.bits_v(one, h1, 16n, L.norm16([978n, 0n, 1n]), 0n, xp_limbs(one, h1))) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), _), one) : Nat}
182  %Equal.sym(Nat, LV.pos(16n, 0n), 0n, LV.pos_zero(16n)) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(_, FS.digits(one, L.norm16([978n, 0n, 1n])))), one) : Nat}
183  %Equal.sym(Nat, FS.digits(one, L.norm16([978n, 0n, 1n])), Nat.add(FS.cp(one), one), xp_digits(one, h1)) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(0n, _)), one) : Nat}
184  %Equal.sym(Nat, one, 1n, h1) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(0n, Nat.add(FS.cp(one), _))), one) : Nat}
185  %Equal.sym(Nat, one, 1n, h1) : {Nat.add(FS.cp(one), Nat.add(BV.hv(F.inv_bits(), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(0n, Nat.add(FS.cp(one), 1n))), _) : Nat}
186  %Equal.sym(Nat, Nat.add(FS.cp(one), Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), 2n)), Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 2n)), NA.add_swap(FS.cp(one), BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), 2n)) : {_ == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(0n, Nat.add(FS.cp(one), 1n))), 1n) : Nat}
187  %Equal.sym(Nat, Nat.add(0n, Nat.add(FS.cp(one), 1n)), Nat.add(FS.cp(one), 1n), SR.zero_add(Nat.add(FS.cp(one), 1n))) : {Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), _), 1n) : Nat}
188  %Equal.sym(Nat, Nat.add(Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 1n)), 1n), Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(Nat.add(FS.cp(one), 1n), 1n)), NA.add_assoc(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 1n), 1n)) : {Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 2n)) == _ : Nat}
189  %Equal.sym(Nat, Nat.add(Nat.add(FS.cp(one), 1n), 1n), Nat.add(FS.cp(one), Nat.add(1n, 1n)), NA.add_assoc(FS.cp(one), 1n, 1n)) : {Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 2n)) == Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), _) : Nat}
190  %Equal.sym(Nat, Nat.add(1n, 1n), 2n, {==}) : {Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), 2n)) == Nat.add(BV.hv(L.cbits(L.bits(L.norm16([978n, 0n, 1n]))), one, 0n), Nat.add(FS.cp(one), _)) : Nat}
191  {==}