ReduceFiberProof.shift_shift: def of bendtorch-core@0.1.0.0
Import: import bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend as ReduceFiberProof
Lines 574-581 of bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend: https://jend.leeeet.dev/read.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&start_line=574&end_line=581

import Base
import bendtorch-core@0.1.0.0/src/shape/index_proofs.bend as Index

def shift_shift(+a: Nat, +b: Nat, +xs: List<&2, Nat>) -> {shift(a, shift(b, xs)) == shift(Nat.add(a, b), xs) : List<&2, Nat>}:
  match xs:
    case Nil{}:
      {==}
    case +x <> +rest:
      %Equal.sym(Nat, Nat.add(a, Nat.add(b, x)), Nat.add(Nat.add(a, b), x), Index.add_assoc(a, b, x)) : {_ <> shift(a, shift(b, rest)) == Nat.add(Nat.add(a, b), x) <> shift(Nat.add(a, b), rest) : List<&2, Nat>}
      Equal.cong(List<&2, Nat>, List<&2, Nat>, tail => Nat.add(Nat.add(a, b), x) <> tail,
        shift(a, shift(b, rest)), shift(Nat.add(a, b), rest), shift_shift(a, b, rest))

Uses (7):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&name=shift
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/shape/index_proofs.bend&name=add_assoc