def reduce3_v

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/reducek.bend:102

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/src/crypto/secp256k1/limbs.bend as L
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

102# x mod m for x < 2^512, m = 2^256 - c, (1 + c) c + c <= 2^256, 2 c < 2^256
103def reduce3_v(+one: Nat, +h1: {one == 1n : Nat}, +mp: Nat, +c: List<&2, Nat>, +hm: {Nat.add(1n+mp, LV.ev(c)) == C.shift(256n, one) : Nat}, +hcc: {Nat.is_lt(Nat.add(LV.ev(c), LV.ev(c)), C.shift(256n, one)) == True{} : Bool}, +hb: {Nat.is_le(Nat.add(Nat.mul(1n+LV.ev(c), LV.ev(c)), LV.ev(c)), C.shift(256n, one)) == True{} : Bool}, +xs: List<&2, Nat>, +hx: {Nat.is_lt(LV.ev(xs), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}) -> {LV.ev(L.reduce(c, 3n, xs)) == Nat.mod(LV.ev(xs), 1n+mp) : Nat}:
104  %Equal.sym(List<&2, Nat>, L.reduce(c, 3n, xs), L.reduce_go(c, 3n, xs), FR.lred(c, 3n, xs)) : {LV.ev(_) == Nat.mod(LV.ev(xs), 1n+mp) : Nat}
105  +r = L.rounds(3n, c, L.carry(32n, xs, 0n))
106  %Equal.sym(Nat, LV.ev(L.canon(c, L.take(16n, L.rounds(3n, c, L.carry(32n, xs, 0n))))), Nat.mod(LV.ev(L.rounds(3n, c, L.carry(32n, xs, 0n))), 1n+mp), red_fin_v(one, h1, mp, c, hm, hcc, L.rounds(3n, c, L.carry(32n, xs, 0n)), r3_lo(one, h1, c, xs), red3_x3(one, h1, c, xs, hb, hx))) : {_ == Nat.mod(LV.ev(xs), 1n+mp) : Nat}
107  %Equal.sym(Nat, Nat.mod(LV.ev(L.rounds(3n, c, L.carry(32n, xs, 0n))), 1n+mp), Nat.mod(LV.ev(xs), 1n+mp), red3_mod(one, h1, mp, c, hm, xs)) : {_ == Nat.mod(LV.ev(xs), 1n+mp) : Nat}
108  {==}