def rp_lt_m

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/u64mont.bend:167

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/logic.bend as L
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/math/natural/arith.bend as NR

167# redc_p is below m and is t / 2^64 (mod m)
168def rp_lt_m(+one: Nat, +h1: {one == 1n : Nat}, +bp: Nat, +m: WU.U64, +mp: WU.U64, +hM: {SW.value(m) == 1n+bp : Nat}, +hinv: {1n+C.low(64n, Nat.mul(1n+bp, SW.value(mp))) == C.shift(64n, one) : Nat}, +h63: {Nat.is_lt(1n+bp, C.shift(63n, one)) == True{} : Bool}, +tl: WU.U64, +th: WU.U64, +pl: WU.U64, +ph: WU.U64, +T: Nat, +eT: {Nat.add(SW.value(tl), C.shift(64n, SW.value(th))) == T : Nat}, +hT: {Nat.is_lt(T, Nat.mul(1n+bp, 1n+bp)) == True{} : Bool}, +eP: {Nat.add(SW.value(pl), C.shift(64n, SW.value(ph))) == Nat.mul(SW.value(X.mul(tl, mp)), 1n+bp) : Nat}) -> {Nat.is_lt(SW.value(X.redc_p(m, tl, th, (pl, ph))), 1n+bp) == True{} : Bool}:
169  L.subst(Nat, z => {Nat.is_lt(z, 1n+bp) == True{} : Bool}, Nat.mod(Nat.add(Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl))), 1n+bp), SW.value(X.redc_p(m, tl, th, (pl, ph))), Equal.sym(Nat, SW.value(X.redc_p(m, tl, th, (pl, ph))), Nat.mod(Nat.add(Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl))), 1n+bp), rp_mod(one, h1, bp, m, mp, hM, hinv, h63, tl, th, pl, ph, T, eT, hT, eP)), NR.dm_lt(bp, Nat.add(Nat.add(SW.value(th), SW.value(ph)), S.bit_value(X.add_over(tl, pl)))))