proofs/math/number/bitcount.bend source

proofs/math/number/bitcount.bend on the hub · documented module

import Baseimport ../../../spec/lib/common.bend as Cimport ../../lib/nat.bend as Nimport ../../lib/lemmas/proofs/nat_algebra.bend as Aimport ../../../src/math/number.bend as NBimport ../../../spec/math/number.bend as SNimport ../typed/width.bend as WW# bit_count counts the 1 bits (Python int.bit_count; Lean 4 Mathlib# Nat.bits and List.count true): each step adds n mod 2 == bit(n) and# continues with n div 2 == half(n), the definition of ones.def half_div2(+n: Nat) -> {C.half(n) == Nat.div(n, 2n) : Nat}:  match n:    case 0n:      {==}    case 1n:      {==}    case 2n+ +q:      Equal.trans(Nat, 1n+C.half(q), 1n+Nat.div(q, 2n), Nat.div(2n+q, 2n), Equal.cong(Nat, Nat, t => 1n+t, C.half(q), Nat.div(q, 2n), half_div2(q)), Equal.sym(Nat, Nat.div(2n+q, 2n), 1n+Nat.div(q, 2n), N.div2_step(q)))def ones_zero(+k: Nat) -> {SN.ones(k, 0n) == 0n : Nat}:  match k:    case 0n:      {==}    case 1n+ +p:      ones_zero(p)# the loop adds the ones of its fuel's worth of bitsdef go(+f: Nat, +n: Nat, +acc: Nat) -> {NB.bit_count_go(f, n, acc) == Nat.add(acc, SN.ones(f, n)) : Nat}:  match f n:    case 0n _:      Equal.sym(Nat, Nat.add(acc, 0n), acc, A.add_zero(acc))    case 1n+ +g 0n:      %Equal.sym(Nat, SN.ones(g, 0n), 0n, ones_zero(g)) : {acc == Nat.add(acc, _) : Nat}      Equal.sym(Nat, Nat.add(acc, 0n), acc, A.add_zero(acc))    case 1n+ +g 1n+ +np:      %Equal.sym(Nat, C.bit(1n+np), Nat.mod(1n+np, 2n), WW.bit_mod(1n+np)) : {NB.bit_count_go(g, Nat.div(1n+np, 2n), Nat.add(acc, Nat.mod(1n+np, 2n))) == Nat.add(acc, Nat.add(_, SN.ones(g, C.half(1n+np)))) : Nat}      %Equal.sym(Nat, C.half(1n+np), Nat.div(1n+np, 2n), half_div2(1n+np)) : {NB.bit_count_go(g, Nat.div(1n+np, 2n), Nat.add(acc, Nat.mod(1n+np, 2n))) == Nat.add(acc, Nat.add(Nat.mod(1n+np, 2n), SN.ones(g, _))) : Nat}      %A.add_assoc(acc, Nat.mod(1n+np, 2n), SN.ones(g, Nat.div(1n+np, 2n))) : {NB.bit_count_go(g, Nat.div(1n+np, 2n), Nat.add(acc, Nat.mod(1n+np, 2n))) == _ : Nat}      go(g, Nat.div(1n+np, 2n), Nat.add(acc, Nat.mod(1n+np, 2n)))def bit_count_value(+n: Nat) -> SN.BitCount.value(n):  go(n, n, 0n)