def lt_low_n

bend-collections-laws-math@1.0.0.0 · proofs/math/random/uint64n.bend:220

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/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

220def lt_low_n(+x: WU.U64, +n: WU.U64, +m: Nat, +hv: {SW.value(n) == 1n+m : Nat}) -> {X.lt(X.pfst(X.mul128(x, n)), n) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), 1n+m) : Bool}:
221  Equal.trans(Bool, X.lt(X.pfst(X.mul128(x, n)), n), Nat.is_lt(SW.value(X.pfst(X.mul128(x, n))), SW.value(n)), Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), 1n+m),
222    WA.lt_value(X.pfst(X.mul128(x, n)), n),
223    %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(n)) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), 1n+m) : Bool}
224    %Equal.sym(Nat, SW.value(n), 1n+m, hv) : {Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), _)), _) == Nat.is_lt(C.low(64n, Nat.mul(SW.value(x), 1n+m)), 1n+m) : Bool}
225    {==})