def pm_mont

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/u64int.bend:477

import Base
import bend-collections-laws-math@1.0.0.0/src/math/num.bend as NM
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/src/math/instances.bend as I
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64laws.bend as LW
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend as NF
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64mont.bend as MO
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X

477# Montgomery multiplication (mont_ok(m) True)
478def pm_mont(+k: Nat, +hk: {k == 64n : Nat}, +b: WU.U64, +e: WU.U64, +m: WU.U64, +mp: Nat, +hnm: {v(m) == 1n+mp : Nat}, +hok: {X.mont_ok(m) == True{} : Bool}) -> {v(I.u64_op(NM.PowMod{I.u64_op(NM.Rem{b, m}), e, m})) == M.pow_mod_go(v(e), 1n+mp, v(e), Nat.mod(v(b), 1n+mp), Nat.mod(1n, 1n+mp)) : Nat}:
479  +hj = L.subst(Nat, z => {Nat.is_le(z, 140n) == True{} : Bool}, 64n, k, Equal.sym(Nat, k, 64n, hk), {==})
480  Equal.trans(Nat, v(X.mpow(X.rem(b, m), e, m)), M.pow_mod_go(140n, 1n+mp, v(e), Nat.mod(v(b), 1n+mp), Nat.mod(1n, 1n+mp)), M.pow_mod_go(v(e), 1n+mp, v(e), Nat.mod(v(b), 1n+mp), Nat.mod(1n, 1n+mp)), MO.mpow_top(1n, {==}, mp, m, hnm, hok, b, e), NF.pm_140(k, 1n+mp, v(e), Nat.mod(v(b), 1n+mp), Nat.mod(1n, 1n+mp), hj, LW.val_lt(k, hk, e)))