def lt_add_r2

bend-collections-laws-math@1.0.0.0 · proofs/lib/nat.bend:566

import Base

566# a < b  ==>  a + c < b + c
567def 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}:
568  match a b:
569    case 0n 0n:
570      Empty.absurd({Nat.is_lt(Nat.add(0n, c), Nat.add(0n, c)) == True{} : Bool}, lt_zero_absurd(0n, h))
571    case 0n 1n+q:
572      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))))
573    case 1n+p 0n:
574      Empty.absurd({Nat.is_lt(Nat.add(1n+p, c), Nat.add(0n, c)) == True{} : Bool}, lt_zero_absurd(1n+p, h))
575    case 1n+p 1n+q:
576      lt_add_r2(p, q, c, h)