BackwardPROOF.length_append: def of bendtorch-core@0.1.0.0
Import: import bendtorch-core@0.1.0.0/src/autograd/backward_PROOF.bend as BackwardPROOF
Lines 39-46 of bendtorch-core@0.1.0.0/src/autograd/backward_PROOF.bend: https://jend.leeeet.dev/read.txt?path=bendtorch-core@0.1.0.0/src/autograd/backward_PROOF.bend&start_line=39&end_line=46

import Base

def length_append(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>) ->
  {List.length(&2, A, List.append(&2, A, xs, ys)) == Nat.add(List.length(&2, A, xs), List.length(&2, A, ys)) : Nat}:
  match xs:
    case Nil{}:
      {==}
    case Con{head, tail}:
      Equal.cong(Nat, Nat, number => 1n+number, List.length(&2, A, List.append(&2, A, tail, ys)),
        Nat.add(List.length(&2, A, tail), List.length(&2, A, ys)), length_append(A, tail, ys))

Uses (6):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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.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