Uint64n.loop: 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 207-218 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=207&end_line=218

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/spec/math/random/source.bend as SRC
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/src/math/random/rand.bend as R
import bend-collections-laws-crypto@1.0.0.0/spec/math/random.bend as SRM

# THEOREM (the rejection loop): f retries from the draw p follow the
# specification's draws
def loop(~S: Data, ~next: S -> WU.U64 & S, +f: Nat, +n: WU.U64, +m: Nat, +t: WU.U64, p: WU.U64 & S, +hv: {SW.value(n) == 1n+m : Nat}, +hp: {Nat.is_eq(SR.and_bits(64n, 1n+m, m), 0n) == False{} : Bool}, +htn: {Nat.is_lt(SR.pow2mod(64n, 1n+m), 1n+m) == True{} : Bool}, +ht: {SW.value(t) == SR.pow2mod(64n, 1n+m) : Nat}) -> {SRM.val_pair(S, R.result(S, R.retry(~S, ~next, f, n, t, R.draw(S, n, p)))) == cont(~S, ~next, f, 1n+m, SW.value(SRC.fst64(S, p)), SRC.snd64(S, p)) : Nat & S}:
  match f p:
    case 0n Tuple{+x, +s1}:
      +H = C.high(64n, Nat.mul(SW.value(x), 1n+m))
      +c0 = X.lt(X.pfst(X.mul128(x, n)), t)
      %Equal.sym(Maybe<&2, Nat>, SR.draw(64n, SW.value(x), 1n+m), SR.accept(H, Bool.not(c0)), hd_of(x, n, m, t, hv, hp, htn, ht, c0, {==})) : {(SW.value(X.psnd(X.mul128(x, n))), s1) == SR.below_go(~S, ~next, 0n, 1n+m, _, SW.value(x), s1) : Nat & S}
      %Equal.sym(Nat & S, SR.below_go(~S, ~next, 0n, 1n+m, SR.accept(H, Bool.not(c0)), SW.value(x), s1), (H, s1), last_acc(~S, ~next, H, Bool.not(c0), SW.value(x), 1n+m, s1, {==})) : {(SW.value(X.psnd(X.mul128(x, n))), s1) == _ : Nat & S}
      Equal.cong(Nat, Nat & S, z => (z, s1), SW.value(X.psnd(X.mul128(x, n))), H, hi_value(x, n, m, hv))
    case 1n+g Tuple{+x, +s1}:
      step_case(~S, ~next, g, n, m, t, x, s1, hv, hp, htn, ht, X.lt(X.pfst(X.mul128(x, n)), t), {==}, loop(~S, ~next, g, n, m, t, next(s1), hv, hp, htn, ht))

Uses (34):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Maybe
  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_eq
  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=Base/base.bend&name=Pair
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Sigma
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=cont
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=hd_of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=hi_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=last_acc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/uint64n.bend&name=step_case
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=high
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=accept
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=and_bits
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=below_go
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=draw
  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/random/source.bend&name=fst64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/source.bend&name=snd64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random.bend&name=val_pair
  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/random/rand.bend&name=draw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/random/rand.bend&name=result
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/random/rand.bend&name=retry
  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
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend&name=psnd