def rp_eq

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/u64mont.bend:109

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
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/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW

109# 2^64 r' == t + u m for r' = th + ph + carry
110def rp_eq(+one: Nat, +bp: Nat, +tl: WU.U64, +th: WU.U64, +mp: WU.U64, +pl: WU.U64, +ph: WU.U64, +T: Nat, +eT: {Nat.add(SW.value(tl), C.shift(64n, SW.value(th))) == T : Nat}, +eP: {Nat.add(SW.value(pl), C.shift(64n, SW.value(ph))) == Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp) : Nat}, +hinv: {1n+C.low(64n, Nat.mul(1n+bp, SW.value(mp))) == C.shift(64n, one) : Nat}) -> {C.shift(64n, Nat.add(Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl)))) == Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)) : Nat}:
111  Equal.trans(Nat, C.shift(64n, Nat.add(Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl)))), Nat.add(C.shift(64n, Nat.add(SW.value(th), SW.value(ph))), C.shift(64n, S.bit_value(X.add_over(tl, pl)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), WW.shift_add(64n, Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl))), Equal.trans(Nat, Nat.add(C.shift(64n, Nat.add(SW.value(th), SW.value(ph))), C.shift(64n, S.bit_value(X.add_over(tl, pl)))), Nat.add(Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph))), C.shift(64n, S.bit_value(X.add_over(tl, pl)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), Equal.cong(Nat, Nat, z => Nat.add(z, C.shift(64n, S.bit_value(X.add_over(tl, pl)))), C.shift(64n, Nat.add(SW.value(th), SW.value(ph))), Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph))), WW.shift_add(64n, SW.value(th), SW.value(ph))), Equal.trans(Nat, Nat.add(Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph))), C.shift(64n, S.bit_value(X.add_over(tl, pl)))), Nat.add(C.shift(64n, S.bit_value(X.add_over(tl, pl))), Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), NA.add_comm(Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph))), C.shift(64n, S.bit_value(X.add_over(tl, pl)))), Equal.trans(Nat, Nat.add(C.shift(64n, S.bit_value(X.add_over(tl, pl))), Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph)))), Nat.add(Nat.add(SW.value(tl), SW.value(pl)), Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), Equal.cong(Nat, Nat, z => Nat.add(z, Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph)))), C.shift(64n, S.bit_value(X.add_over(tl, pl))), Nat.add(SW.value(tl), SW.value(pl)), Equal.sym(Nat, Nat.add(SW.value(tl), SW.value(pl)), C.shift(64n, S.bit_value(X.add_over(tl, pl))), rp_low(one, bp, tl, mp, pl, ph, eP, hinv))), Equal.trans(Nat, Nat.add(Nat.add(SW.value(tl), SW.value(pl)), Nat.add(C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph)))), Nat.add(Nat.add(SW.value(tl), C.shift(64n, SW.value(th))), Nat.add(SW.value(pl), C.shift(64n, SW.value(ph)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), swap4(SW.value(tl), SW.value(pl), C.shift(64n, SW.value(th)), C.shift(64n, SW.value(ph))), Equal.trans(Nat, Nat.add(Nat.add(SW.value(tl), C.shift(64n, SW.value(th))), Nat.add(SW.value(pl), C.shift(64n, SW.value(ph)))), Nat.add(T, Nat.add(SW.value(pl), C.shift(64n, SW.value(ph)))), Nat.add(T, Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp)), Equal.cong(Nat, Nat, z => Nat.add(z, Nat.add(SW.value(pl), C.shift(64n, SW.value(ph)))), Nat.add(SW.value(tl), C.shift(64n, SW.value(th))), T, eT), Equal.cong(Nat, Nat, z => Nat.add(T, z), Nat.add(SW.value(pl), C.shift(64n, SW.value(ph))), Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp), eP)))))))