def mul_v

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldops.bend:109

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/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/crypto/secp256k1/limbs.bend as LV
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/fprot.bend as FR

109def mul_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.value(one, F.mul(a, b)) == FS.mmul(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}:
110  %Equal.sym(List<&2, Nat>, F.mul(a, b), F.mul_u(a, b), FR.f_mul(a, b)) : {FS.value(one, _) == FS.mmul(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}
111  %Equal.sym(Nat, FS.value(one, F.reduce(L.conv(a, b))), Nat.mod(LV.ev(L.conv(a, b)), FS.prime(one)), red_v(one, h1, L.conv(a, b), mul_hx(one, h1, a, b, ha, hb))) : {_ == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
112  %Equal.sym(Nat, LV.ev(L.conv(a, b)), Nat.mul(LV.ev(a), LV.ev(b)), LV.conv_v(a, b)) : {Nat.mod(_, FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
113  %Equal.sym(Nat, LV.ev(a), FS.value(one, a), Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a))) : {Nat.mod(Nat.mul(_, LV.ev(b)), FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
114  %Equal.sym(Nat, LV.ev(b), FS.value(one, b), Equal.sym(Nat, FS.value(one, b), LV.ev(b), LV.ev_value(one, h1, b))) : {Nat.mod(Nat.mul(FS.value(one, a), _), FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
115  {==}