PROOF.AP.tree_ext: def of bend-osdev-array@0.1.0.1
Import: import bend-osdev-array@0.1.0.1/PROOF.bend as PROOF
Lines 1448-1469 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=1448&end_line=1469

import Base
import bend-osdev-array@0.1.0.1/array.bend as A

# a perfect tree is the tree of its own leaves
def AP.tree_ext(-T: Data, +d: Nat, +t: AP.Tree<T>, +f: T, +hp: {A.Arrays.is_perfect(T, d, AP.build(T, t)) == True{} : Bool}) -> {AP.build(T, t) == AP.build(T, AP.tof(T, d, A.Arrays.leaves(T, AP.build(T, t)), f)) : Array<T>}:
  match d t:
    case 0n APLeaf{x}:
      {==}
    case 0n APNode{xs, ys}:
      Empty.absurd({AP.build(T, APNode{xs, ys}) == AP.build(T, AP.tof(T, 0n, A.Arrays.leaves(T, AP.build(T, APNode{xs, ys})), f)) : Array<T>}, AP.ft(hp))
    case 1n+p APLeaf{x}:
      Empty.absurd({AP.build(T, APLeaf{x}) == AP.build(T, AP.tof(T, 1n+p, A.Arrays.leaves(T, AP.build(T, APLeaf{x})), f)) : Array<T>}, AP.ft(hp))
    case 1n+ +p APNode{xs, ys}:
      +s = A.Arrays.slots(p)
      +lx = A.Arrays.leaves(T, AP.build(T, xs))
      +ly = A.Arrays.leaves(T, AP.build(T, ys))
      +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)
      +hl = AP.len_perf(T, p, xs, hpx)
      %hl : {ANode{AP.build(T, xs), AP.build(T, ys)} == ANode{AP.build(T, AP.tof(T, p, List.take(&2, T, List.append(&2, T, lx, ly), _), f)), AP.build(T, AP.tof(T, p, List.drop(&2, T, List.append(&2, T, lx, ly), _), f))} : Array<T>}
      %Equal.sym(List<&2, T>, List.take(&2, T, List.append(&2, T, lx, ly), List.length(&2, T, lx)), lx, AP.take_app(T, lx, ly)) : {ANode{AP.build(T, xs), AP.build(T, ys)} == ANode{AP.build(T, AP.tof(T, p, _, f)), AP.build(T, AP.tof(T, p, List.drop(&2, T, List.append(&2, T, lx, ly), List.length(&2, T, lx)), f))} : Array<T>}
      %Equal.sym(List<&2, T>, List.drop(&2, T, List.append(&2, T, lx, ly), List.length(&2, T, lx)), ly, AP.drop_app(T, lx, ly)) : {ANode{AP.build(T, xs), AP.build(T, ys)} == ANode{AP.build(T, AP.tof(T, p, lx, f)), AP.build(T, AP.tof(T, p, _, f))} : Array<T>}
      %AP.tree_ext(T, p, xs, f, hpx) : {ANode{AP.build(T, xs), AP.build(T, ys)} == ANode{_, AP.build(T, AP.tof(T, p, ly, f))} : Array<T>}
      %AP.tree_ext(T, p, ys, f, hpy) : {ANode{AP.build(T, xs), AP.build(T, ys)} == ANode{AP.build(T, xs), _} : Array<T>}
      {==}

Uses (22):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Array
  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
  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.drop
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.length
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.take
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  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.drop_app
  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_perf
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.take_app
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.tof
  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