Fieldpow.sqrt_v: 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 336-343 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=336&end_line=343

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/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

def sqrt_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}) -> {FS.value(one, F.sqrt(a)) == FS.fsqrt(FS.prime(one), FS.value(one, a)) : Nat}:
  %Equal.sym(Nat, FS.value(one, F.pow(a, F.sqrt_bits(), F.one())), LV.ev(F.pow(a, F.sqrt_bits(), F.one())), LV.ev_value(one, h1, F.pow(a, F.sqrt_bits(), F.one()))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(F.pow(a, F.sqrt_bits(), F.one())), Nat.mod(Nat.pow(LV.ev(a), BV.hv(F.sqrt_bits(), one, 0n)), 1n+K.pp(one)), pow_ev(one, h1, a, ha, F.sqrt_bits(), F.one(), one_r(one, h1), 0n, one_e(one, h1, LV.ev(a)), sqrt_bits01(one))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, BV.hv(F.sqrt_bits(), one, 0n), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n), exp_sqrt(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), _), 1n+K.pp(one)) == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(_, Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(_, 1n), 4n)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), _) : Nat}
  {==}

Uses (26):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.div
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.pow
  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/consts.bend&name=prime_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=exp_sqrt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=one_e
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=one_r
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=pow_ev
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fieldpow.bend&name=sqrt_bits01
  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=ev_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend&name=fsqrt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend&name=prime
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend&name=reduced
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend&name=one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend&name=pow
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend&name=sqrt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend&name=sqrt_bits