Below.below_lt: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend as Below
Lines 125-127 of bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend&start_line=125&end_line=127

import Base
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend as SR
import bend-collections-laws-math@1.0.0.0/spec/math/random/source.bend as SRC
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/math/typed/w64m128.bend as M128

# THEOREM: the bounded draw from any source is below n > 0
def below_lt(~S: Data, ~next: S -> WU.U64 & S, +f: Nat, +n: Nat, +s: S, +hn: {Nat.is_lt(0n, n) == True{} : Bool}) -> {Nat.is_lt(pfst(S, SR.below(~S, ~next, f, n, s)), n) == True{} : Bool}:
  go_lt(~S, ~next, f, n, SR.draw(64n, SW.value(SRC.fst64(S, next(s))), n), SW.value(SRC.fst64(S, next(s))), SRC.snd64(S, next(s)), M128.fit64(SRC.fst64(S, next(s))), hn, {==})

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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=Pair
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend&name=go_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend&name=pfst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64m128.bend&name=fit64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend&name=below
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend&name=draw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/source.bend&name=fst64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/source.bend&name=snd64
  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