Perm.internal_apply_invert: def of bend-mathlib@0.8.1.0
Import: import bend-mathlib@0.8.1.0/perm.bend as Perm
Lines 286-293 of bend-mathlib@0.8.1.0/perm.bend: https://jend.leeeet.dev/read.txt?path=bend-mathlib@0.8.1.0/perm.bend&start_line=286&end_line=293

import Base
import bendlib-kernel-list@1.0.0.0/list.bend as K

def internal_apply_invert(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>) -> {K.apply(A, internal_invert(s), K.apply(A, s, xs)) == xs : List<&2, A>}:
  match s:
    case Nil{}:
      {==}
    case +i <> rest:
      %Equal.sym(List<&2, A>, K.apply(A, List.append(&2, Nat, internal_invert(rest), i <> Nil{}), K.apply(A, rest, K.swap_at(A, i, xs))), K.apply(A, i <> Nil{}, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs)))), internal_apply_app(A, internal_invert(rest), i <> Nil{}, K.apply(A, rest, K.swap_at(A, i, xs)))) : {_ == xs : List<&2, A>}
      %Equal.sym(List<&2, A>, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs))), K.swap_at(A, i, xs), internal_apply_invert(A, rest, K.swap_at(A, i, xs))) : {K.apply(A, i <> Nil{}, _) == xs : List<&2, A>}
      internal_swap_at_invol(A, i, xs)

Uses (9):
  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=Nat
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/perm.bend&name=internal_apply_app
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/perm.bend&name=internal_invert
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/perm.bend&name=internal_swap_at_invol
  https://jend.leeeet.dev/definition.txt?path=bendlib-kernel-list@1.0.0.0/list.bend&name=apply
  https://jend.leeeet.dev/definition.txt?path=bendlib-kernel-list@1.0.0.0/list.bend&name=swap_at