ReducePROOF.neither_below: def of bendtorch-core@0.1.0.0
Import: import bendtorch-core@0.1.0.0/src/autograd/reduce_PROOF.bend as ReducePROOF
Lines 513-523 of bendtorch-core@0.1.0.0/src/autograd/reduce_PROOF.bend: https://jend.leeeet.dev/read.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_PROOF.bend&start_line=513&end_line=523

import Base
import bendtorch-core@0.1.0.0/src/storage/arith.bend as Arith

def neither_below(+left: Nat, +right: Nat, first: {Nat.is_lt(left, right) == False{} : Bool},
  second: {Nat.is_lt(right, left) == False{} : Bool}) -> {left == right : Nat}:
  match left right:
    case 0n 0n:
      {==}
    case 0n 1n+r:
      Empty.absurd({0n == 1n+r : Nat}, Arith.clash_true(first))
    case 1n+l 0n:
      Empty.absurd({1n+l == 0n : Nat}, Arith.clash_true(second))
    case 1n+l 1n+r:
      Equal.cong(Nat, Nat, n => 1n+n, l, r, neither_below(l, r, first, second))

Uses (6):
  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.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/storage/arith.bend&name=clash_true