def mok_inv_o

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/u64mont.bend:352

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/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

352# over an open all-ones word o: a literal 2^64 - 1 under SW.value would be
353# expanded in unary by any conversion that unfolds it
354def mok_inv_o(+one: Nat, +h1: {one == 1n : Nat}, +bp: Nat, +ml: U32, +mh: U32, +hM: {SW.value(WU.U64{ml, mh}) == 1n+bp : Nat}, +o: U32, +po: {o == U32{WD.mask(32n, 32n)} : U32}, +heq: {X.eq(X.mul(WU.U64{ml, mh}, X.minv(WU.U64{ml, mh})), WU.U64{o, o}) == True{} : Bool}) -> {1n+C.low(64n, Nat.mul(1n+bp, SW.value(X.minv(WU.U64{ml, mh})))) == C.shift(64n, one) : Nat}:
355  +mp = X.minv(WU.U64{ml, mh})
356  +w = X.mul(WU.U64{ml, mh}, mp)
357  +h = Equal.trans(Bool, Nat.is_eq(SW.value(w), SW.value(WU.U64{o, o})), X.eq(w, WU.U64{o, o}), True{}, Equal.sym(Bool, X.eq(w, WU.U64{o, o}), Nat.is_eq(SW.value(w), SW.value(WU.U64{o, o})), WA.eq_value(w, WU.U64{o, o})), heq)
358  +e1 = N.eq_from_is_eq(SW.value(w), SW.value(WU.U64{o, o}), h)
359  +e2 = Equal.trans(Nat, C.low(64n, Nat.mul(1n+bp, SW.value(mp))), C.low(64n, Nat.mul(SW.value(WU.U64{ml, mh}), SW.value(mp))), SW.value(w), Equal.cong(Nat, Nat, z => C.low(64n, Nat.mul(z, SW.value(mp))), 1n+bp, SW.value(WU.U64{ml, mh}), Equal.sym(Nat, SW.value(WU.U64{ml, mh}), 1n+bp, hM)), Equal.sym(Nat, SW.value(w), C.low(64n, Nat.mul(SW.value(WU.U64{ml, mh}), SW.value(mp))), WA.mul_value(WU.U64{ml, mh}, mp)))
360  Equal.trans(Nat, 1n+C.low(64n, Nat.mul(1n+bp, SW.value(mp))), 1n+SW.value(WU.U64{o, o}), C.shift(64n, one), Equal.cong(Nat, Nat, z => 1n+z, C.low(64n, Nat.mul(1n+bp, SW.value(mp))), SW.value(WU.U64{o, o}), Equal.trans(Nat, C.low(64n, Nat.mul(1n+bp, SW.value(mp))), SW.value(w), SW.value(WU.U64{o, o}), e2, e1)), ones_v(one, h1, o, po))