PROOF.AP.len_perf: def of bend-osdev-array@0.1.0.1
Import: import bend-osdev-array@0.1.0.1/PROOF.bend as PROOF
Lines 1051-1068 of bend-osdev-array@0.1.0.1/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&start_line=1051&end_line=1068

import Base
import bend-osdev-array@0.1.0.1/array.bend as A
import bend-osdev-u32@0.1.0.0/LAWS.bend as UL

def AP.len_perf(-T: Data, +d: Nat, +t: AP.Tree<T>, +hp: {A.Arrays.is_perfect(T, d, AP.build(T, t)) == True{} : Bool}) -> {List.length(&2, T, A.Arrays.leaves(T, AP.build(T, t))) == A.Arrays.slots(d) : Nat}:
  match d t:
    case 0n APLeaf{x}:
      {==}
    case 0n APNode{xs, ys}:
      Empty.absurd({List.length(&2, T, A.Arrays.leaves(T, AP.build(T, APNode{xs, ys}))) == A.Arrays.slots(0n) : Nat}, AP.ft(hp))
    case 1n+p APLeaf{x}:
      Empty.absurd({List.length(&2, T, A.Arrays.leaves(T, AP.build(T, APLeaf{x}))) == A.Arrays.slots(1n+p) : Nat}, AP.ft(hp))
    case 1n+ +p APNode{xs, ys}:
      +lx = A.Arrays.leaves(T, AP.build(T, xs))
      +ly = A.Arrays.leaves(T, AP.build(T, ys))
      +s = A.Arrays.slots(p)
      +hpx = AP.and_l(A.Arrays.is_perfect(T, p, AP.build(T, xs)), A.Arrays.is_perfect(T, p, AP.build(T, ys)), hp)
      +hpy = AP.and_r(A.Arrays.is_perfect(T, p, AP.build(T, xs)), A.Arrays.is_perfect(T, p, AP.build(T, ys)), hp)
      %Equal.sym(Nat, List.length(&2, T, List.append(&2, T, lx, ly)), Nat.add(List.length(&2, T, lx), List.length(&2, T, ly)), AP.len_app(T, lx, ly)) : {_ == Nat.double(s) : Nat}
      %Equal.sym(Nat, List.length(&2, T, lx), s, AP.len_perf(T, p, xs, hpx)) : {Nat.add(_, List.length(&2, T, ly)) == Nat.double(s) : Nat}
      %Equal.sym(Nat, List.length(&2, T, ly), s, AP.len_perf(T, p, ys, hpy)) : {Nat.add(s, _) == Nat.double(s) : Nat}
      Equal.sym(Nat, Nat.double(s), Nat.add(s, s), UL.nat_double(s))

Uses (18):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.append
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.length
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.Tree
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.and_l
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.and_r
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.build
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.ft
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.len_app
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.is_perfect
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.leaves
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.slots
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_double