def and1

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/canon.bend:176

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
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/math/u64/u64div.bend as PD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend as M

176def and1(+one: Nat, +h1: {one == 1n : Nat}, +l: U32) -> {v(U32.and(l, 1)) == Nat.mod(v(l), 2n) : Nat}:
177  +a = PD.and32(1n, l, 1, {==})
178  +b = PD.and32_lt(1n, one, h1, l, 1, {==})
179  +hp = PD.hp32(1n, l)
180  +b2 = L.subst(Nat, o => {Nat.is_lt(v(U32.and(l, 1)), WD.sc(1n, o)) == True{} : Bool}, one, 1n, h1, b)
181  +e1 = Equal.trans(Nat, v(l), Nat.add(v(U32.and(l, 1)), WD.sc(1n, hp)), Nat.add(Nat.mul(hp, 2n), v(U32.and(l, 1))), a, Equal.trans(Nat, Nat.add(v(U32.and(l, 1)), WD.sc(1n, hp)), Nat.add(WD.sc(1n, hp), v(U32.and(l, 1))), Nat.add(Nat.mul(hp, 2n), v(U32.and(l, 1))), NA.add_comm(v(U32.and(l, 1)), WD.sc(1n, hp)), M.cong_l(WD.sc(1n, hp), Nat.mul(hp, 2n), v(U32.and(l, 1)), Equal.trans(Nat, Nat.double(hp), Nat.mul(2n, hp), Nat.mul(hp, 2n), NA.double_mul(hp), NA.mul_comm(2n, hp)))))
182  Equal.sym(Nat, Nat.mod(v(l), 2n), v(U32.and(l, 1)), Equal.trans(Nat, Nat.mod(v(l), 2n), Nat.mod(Nat.add(Nat.mul(hp, 2n), v(U32.and(l, 1))), 2n), v(U32.and(l, 1)), M.cong_mod(2n, v(l), Nat.add(Nat.mul(hp, 2n), v(U32.and(l, 1))), e1), NR.mod_of(hp, 1n, v(U32.and(l, 1)), b2)))