Nat.lt_add_r2: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as Nat
Lines 566-576 of bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&start_line=566&end_line=576

import Base

# a < b  ==>  a + c < b + c
def lt_add_r2(+a: Nat, +b: Nat, +c: Nat, +h: {Nat.is_lt(a, b) == True{} : Bool}) -> {Nat.is_lt(Nat.add(a, c), Nat.add(b, c)) == True{} : Bool}:
  match a b:
    case 0n 0n:
      Empty.absurd({Nat.is_lt(Nat.add(0n, c), Nat.add(0n, c)) == True{} : Bool}, lt_zero_absurd(0n, h))
    case 0n 1n+q:
      le_lt_succ(c, Nat.add(q, c), le_trans(c, Nat.add(c, q), Nat.add(q, c), le_add_right(c, q), eq_le(Nat.add(c, q), Nat.add(q, c), add_comm(c, q))))
    case 1n+p 0n:
      Empty.absurd({Nat.is_lt(Nat.add(1n+p, c), Nat.add(0n, c)) == True{} : Bool}, lt_zero_absurd(1n+p, h))
    case 1n+p 1n+q:
      lt_add_r2(p, q, c, h)

Uses (11):
  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_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=eq_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_lt_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_zero_absurd