proofs/math/random/bounded.bend source

proofs/math/random/bounded.bend on the hub · documented module

import Baseimport ../../../src/math/u64.bend as Wimport ../../../src/math/w64.bend as Ximport ../../../src/math/random/rand.bend as Rimport ../../../spec/math/w64.bend as SWimport ../../../spec/math/random.bend as SRMimport ../../../spec/math/random/rand.bend as SRimport ../../lib/logic.bend as Limport ../typed/w64add.bend as WAimport ./uint64n.bend as UNimport ./below.bend as BL# Uint64n.lt: uint64n(n) is the specification's bounded draw# (Uint64n.value), which is below n.def fst_val(-S: Data, r: W.U64 & S) -> {BL.pfst(S, SRM.val_pair(S, r)) == SW.value(R.fst(W.U64, S, r)) : Nat}:  match r:    case Tuple{x, s}:      {==}def nz_pos(+v: Nat, +h: {Nat.is_eq(v, 0n) == False{} : Bool}) -> {Nat.is_lt(0n, v) == True{} : Bool}:  match v:    case 0n:      Empty.absurd({Nat.is_lt(0n, 0n) == True{} : Bool}, L.true_false(h))    case 1n+p:      {==}def uint64n_lt(~S: Data, ~next: S -> W.U64 & S, +s: S, +n: W.U64, +hn: {X.is_zero(n) == False{} : Bool}) -> SRM.Uint64n.lt(~S, ~next, s, n, hn):  +vn = SW.value(n)  +hv = Equal.trans(Bool, Nat.is_eq(vn, 0n), X.is_zero(n), False{}, Equal.sym(Bool, X.is_zero(n), Nat.is_eq(vn, 0n), WA.is_zero_value(n)), hn)  +e = Equal.trans(Nat, SW.value(R.fst(W.U64, S, R.uint64n(~S, ~next, s, n))), BL.pfst(S, SRM.val_pair(S, R.uint64n(~S, ~next, s, n))), BL.pfst(S, SR.below(~S, ~next, 127n, vn, s)),    Equal.sym(Nat, BL.pfst(S, SRM.val_pair(S, R.uint64n(~S, ~next, s, n))), SW.value(R.fst(W.U64, S, R.uint64n(~S, ~next, s, n))), fst_val(S, R.uint64n(~S, ~next, s, n))),    Equal.cong(Nat & S, Nat, p => BL.pfst(S, p), SRM.val_pair(S, R.uint64n(~S, ~next, s, n)), SR.below(~S, ~next, 127n, vn, s), UN.uint64n_value(~S, ~next, s, n)))  Equal.trans(Bool, Nat.is_lt(SW.value(R.fst(W.U64, S, R.uint64n(~S, ~next, s, n))), vn), Nat.is_lt(BL.pfst(S, SR.below(~S, ~next, 127n, vn, s)), vn), True{},    Equal.cong(Nat, Bool, z => Nat.is_lt(z, vn), SW.value(R.fst(W.U64, S, R.uint64n(~S, ~next, s, n))), BL.pfst(S, SR.below(~S, ~next, 127n, vn, s)), e),    BL.below_lt(~S, ~next, 127n, vn, s, nz_pos(vn, hv)))