def loop

bend-collections-laws-math@1.0.0.0 · proofs/math/random/uint64n.bend:207

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/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/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/random/rand.bend as R
import bend-collections-laws-math@1.0.0.0/spec/math/random.bend as SRM

207# THEOREM (the rejection loop): f retries from the draw p follow the
208# specification's draws
209def 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}:
210  match f p:
211    case 0n Tuple{+x, +s1}:
212      +H = C.high(64n, Nat.mul(SW.value(x), 1n+m))
213      +c0 = X.lt(X.pfst(X.mul128(x, n)), t)
214      %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}
215      %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}
216      Equal.cong(Nat, Nat & S, z => (z, s1), SW.value(X.psnd(X.mul128(x, n))), H, hi_value(x, n, m, hv))
217    case 1n+g Tuple{+x, +s1}:
218      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))