Roots.lt_sub: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend as Roots
Lines 88-106 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend&start_line=88&end_line=106

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
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend as F

# b < c <= a gives a - c < a - b
def lt_sub(+a: Nat, +b: Nat, +c: Nat, +h1: {Nat.is_lt(b, c) == True{} : Bool}, +h2: {Nat.is_le(c, a) == True{} : Bool}) -> {Nat.is_lt(Nat.sub(a, c), Nat.sub(a, b)) == True{} : Bool}:
  match a b c:
    case 0n 0n 0n:
      Empty.absurd({Nat.is_lt(Nat.sub(0n, 0n), Nat.sub(0n, 0n)) == True{} : Bool}, L.false_true(h1))
    case 0n 0n 1n+cp:
      Empty.absurd({Nat.is_lt(Nat.sub(0n, 1n+cp), Nat.sub(0n, 0n)) == True{} : Bool}, L.false_true(h2))
    case 0n 1n+bp 0n:
      Empty.absurd({Nat.is_lt(Nat.sub(0n, 0n), Nat.sub(0n, 1n+bp)) == True{} : Bool}, L.false_true(h1))
    case 0n 1n+bp 1n+cp:
      Empty.absurd({Nat.is_lt(Nat.sub(0n, 1n+cp), Nat.sub(0n, 1n+bp)) == True{} : Bool}, L.false_true(h2))
    case 1n+ap 0n 0n:
      Empty.absurd({Nat.is_lt(Nat.sub(1n+ap, 0n), Nat.sub(1n+ap, 0n)) == True{} : Bool}, L.false_true(h1))
    case 1n+ +ap 0n 1n+ +cp:
      N.le_lt_succ(Nat.sub(ap, cp), ap, F.sub_le_self(ap, cp))
    case 1n+ap 1n+bp 0n:
      Empty.absurd({Nat.is_lt(Nat.sub(1n+ap, 0n), Nat.sub(1n+ap, 1n+bp)) == True{} : Bool}, L.false_true(h1))
    case 1n+ +ap 1n+ +bp 1n+ +cp:
      lt_sub(ap, bp, cp, h1, h2)

Uses (9):
  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=Base/base.bend&name=Nat.sub
  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=le_lt_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=sub_le_self