def min_succ

bend-collections-laws-math@1.0.0.0 · proofs/math/random/lemire.bend:213

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

213# min(N + 1, C) = min(N, C) + [N < C]
214def min_succ(+n: Nat, +c: Nat) -> {Nat.min(1n+n, c) == Nat.add(Nat.min(n, c), b2n(Nat.is_lt(n, c))) : Nat}:
215  match n c:
216    case _ 0n:
217      %Equal.sym(Nat, Nat.min(n, 0n), 0n, min_zero(n)) : {0n == Nat.add(_, b2n(Nat.is_lt(n, 0n))) : Nat}
218      %Equal.sym(Bool, Nat.is_lt(n, 0n), False{}, N.not_lt_zero(n)) : {0n == Nat.add(0n, b2n(_)) : Nat}
219      {==}
220    case 0n 1n+q:
221      {==}
222    case 1n+p 1n+q:
223      Equal.cong(Nat, Nat, z => 1n+z, Nat.min(1n+p, q), Nat.add(Nat.min(p, q), b2n(Nat.is_lt(p, q))), min_succ(p, q))