def lt_low

bend-collections-laws-crypto@1.0.0.0 · proofs/math/random/uint64n.bend:169

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend as SR
import bend-collections-laws-crypto@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/w64add.bend as WA

169# the implementation's rejection test is the specification's
170def lt_low(+x: WU.U64, +n: WU.U64, +m: Nat, +t: WU.U64, +hv: {SW.value(n) == 1n+m : Nat}, +ht: {SW.value(t) == SR.pow2mod(64n, 1n+m) : Nat}) -> {X.lt(X.pfst(X.mul128(x, n)), t) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), SR.pow2mod(64n, 1n+m)) : Bool}:
171  Equal.trans(Bool, X.lt(X.pfst(X.mul128(x, n)), t), Nat.is_lt(SW.value(X.pfst(X.mul128(x, n))), SW.value(t)), Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), SR.pow2mod(64n, 1n+m)),
172    WA.lt_value(X.pfst(X.mul128(x, n)), t),
173    %Equal.sym(Nat, SW.value(X.pfst(X.mul128(x, n))), C.low(64n, Nat.mul(SW.value(x), SW.value(n))), m128_lo(x, n)) : {Nat.is_lt(_, SW.value(t)) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), SR.pow2mod(64n, 1n+m)) : Bool}
174    %Equal.sym(Nat, SW.value(n), 1n+m, hv) : {Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), _)), SW.value(t)) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), SR.pow2mod(64n, 1n+m)) : Bool}
175    %Equal.sym(Nat, SW.value(t), SR.pow2mod(64n, 1n+m), ht) : {Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), _) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), SR.pow2mod(64n, 1n+m)) : Bool}
176    {==})