NatOrder.add_le: def of stelliferous@0.0.2.0
Import: import stelliferous@0.0.2.0/proofs/nat_order.bend as NatOrder
Lines 317-329 of stelliferous@0.0.2.0/proofs/nat_order.bend: https://jend.leeeet.dev/read.txt?path=stelliferous@0.0.2.0/proofs/nat_order.bend&start_line=317&end_line=329

import Base
import stelliferous@0.0.2.0/proofs/word_encoding.bend as WordEncoding

# Order structure shared by padding, allocation and mixed-radix indexing.
def add_le(+a: Nat,+b: Nat,+c: Nat,h: {Nat.is_le(a,b) == True{} : Bool}) -> {Nat.is_le(Nat.add(a,c),Nat.add(b,c)) == True{} : Bool}:
  match a b:
    case 0n 0n:
      reflexive(c)
    case 0n 1n+ +b:
      successor(c,Nat.add(b,c),
        %commutative(c,b) : {Nat.is_le(c,_) == True{} : Bool}
        le_add(c,b))
    case 1n+a 0n:
      Empty.absurd({Nat.is_le(Nat.add(1n+a,c),c) == True{} : Bool},WordEncoding.false_true(h))
    case 1n+a 1n+b:
      add_le(a,b,c,h)

Uses (10):
  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=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=stelliferous@0.0.2.0/proofs/nat_order.bend&name=commutative
  https://jend.leeeet.dev/definition.txt?path=stelliferous@0.0.2.0/proofs/nat_order.bend&name=le_add
  https://jend.leeeet.dev/definition.txt?path=stelliferous@0.0.2.0/proofs/nat_order.bend&name=reflexive
  https://jend.leeeet.dev/definition.txt?path=stelliferous@0.0.2.0/proofs/nat_order.bend&name=successor
  https://jend.leeeet.dev/definition.txt?path=stelliferous@0.0.2.0/proofs/word_encoding.bend&name=false_true