Uint64n.lt_low_n: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/uint64n.bend as Uint64n
Lines 220-225 of bend-collections-laws-math@1.0.0.0/proofs/math/random/uint64n.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/uint64n.bend&start_line=220&end_line=225

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

def 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}:
  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),
    WA.lt_value(X.pfst(X.mul128(x, n)), n),
    %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}
    %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}
    {==})

Uses (14):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/uint64n.bend&name=m128_lo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend&name=lt_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/w64.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/u64.bend&name=U64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=mul128
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=pfst