PROOF.nat.nlt_le: def of bend-osdev-algo@0.1.0.0
Import: import bend-osdev-algo@0.1.0.0/PROOF.bend as PROOF
Lines 958-970 of bend-osdev-algo@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&start_line=958&end_line=970

import Base

# not a < c, a <= b: not b < c
def nat.nlt_le(a: Nat, b: Nat, c: Nat, e: {Nat.is_lt(a, c) == False{} : Bool}, f: {Nat.is_le(a, b) == True{} : Bool}) -> {Nat.is_lt(b, c) == False{} : Bool}:
  match a b c:
    case 0n _ 0n:
      nat.lt_zero(b)
    case 0n _ 1n+cp:
      Empty.absurd({Nat.is_lt(b, 1n+cp) == False{} : Bool}, absurd.tf(e))
    case 1n+ap 0n _:
      Empty.absurd({Nat.is_lt(0n, c) == False{} : Bool}, absurd.ft(f))
    case 1n+ap 1n+bp 0n:
      {==}
    case 1n+ap 1n+bp 1n+cp:
      nat.nlt_le(ap, bp, cp, e, f)

Uses (8):
  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.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=absurd.ft
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=absurd.tf
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=nat.lt_zero