Uint64n.lt_low: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend as Uint64n
Lines 169-176 of bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&start_line=169&end_line=176

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

# the implementation's rejection test is the specification's
def 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}:
  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)),
    WA.lt_value(X.pfst(X.mul128(x, n)), t),
    %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}
    %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}
    %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}
    {==})

Uses (15):
  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-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=m128_lo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/w64add.bend&name=lt_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=pow2mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/w64.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/u64.bend&name=U64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend&name=lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend&name=mul128
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend&name=pfst