ReduceFiberProof.quotient_below: 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 147-164 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=147&end_line=164

import Base
import bendtorch-core@0.1.0.0/src/shape/core.bend as Shape
import bendtorch-core@0.1.0.0/src/storage/arith.bend as Arith
import bendtorch-core@0.1.0.0/src/storage/api.bend as Api

# Below a * size, the quotient by size is below a.
def quotient_below(+t: Nat, +a: Nat, +size: Nat, below: {Nat.is_lt(t, Nat.mul(a, size)) == True{} : Bool},
  decision: Shape.BoolCase<Nat.is_lt(Nat.div(t, size), a)>, given: Decomposed<t, size>) ->
  {Nat.is_lt(Nat.div(t, size), a) == True{} : Bool}:
  match decision:
    case Shape.BTrue{equal}:
      Equal.sym(Bool, True{}, Nat.is_lt(Nat.div(t, size), a), equal)
    case Shape.BFalse{equal}:
      Decomposed{rest, sum} = given
      +q = Nat.div(t, size)
      +r = Nat.mod(t, size)
      +grown = le_trans(Nat.mul(a, size), Nat.mul(q, size), t,
        mul_le_mono(a, q, size, Arith.not_lt_le(q, a, Equal.sym(Bool, False{}, Nat.is_lt(q, a), equal))),
        %sum : {Nat.is_le(Nat.mul(q, size), _) == True{} : Bool}
        Api.le_add_right(Nat.mul(q, size), r))
      Empty.absurd({Nat.is_lt(q, a) == True{} : Bool},
        Arith.clash(Equal.trans(Bool, False{}, Nat.is_lt(t, t), True{}, Equal.sym(Bool, Nat.is_lt(t, t), False{}, lt_irrefl(t)),
          Arith.lt_le_trans(t, Nat.mul(a, size), t, below, grown))))

Uses (19):
  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=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.div
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&name=Decomposed
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&name=lt_irrefl
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/autograd/reduce_fiber_proof.bend&name=mul_le_mono
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/shape/core.bend&name=BoolCase
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/storage/api.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/storage/arith.bend&name=clash
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/storage/arith.bend&name=lt_le_trans
  https://jend.leeeet.dev/definition.txt?path=bendtorch-core@0.1.0.0/src/storage/arith.bend&name=not_lt_le