Misc.max_eq_l: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/misc.bend as Misc
Lines 36-45 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/misc.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/misc.bend&start_line=36&end_line=45

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

def max_eq_l(+x: Nat, +lo: Nat, +h: {Nat.is_le(lo, x) == True{} : Bool}) -> {Nat.max(x, lo) == x : Nat}:
  match x lo:
    case 0n 0n:
      {==}
    case 0n 1n+lp:
      Empty.absurd({Nat.max(0n, 1n+lp) == 0n : Nat}, L.false_true(h))
    case 1n+xp 0n:
      {==}
    case 1n+ +xp 1n+ +lp:
      N.succ_cong(Nat.max(xp, lp), xp, max_eq_l(xp, lp, h))

Uses (7):
  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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.max
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=succ_cong