ProofNat.div_lt.go: def of lawful-words@0.1.0.0
Import: import lawful-words@0.1.0.0/proof_nat.bend as ProofNat
Lines 1345-1359 of lawful-words@0.1.0.0/proof_nat.bend: https://jend.leeeet.dev/read.txt?path=lawful-words@0.1.0.0/proof_nat.bend&start_line=1345&end_line=1359

import Base

# a < b * c with b > 0 gives a / b < c.
def div_lt.go(+a: Nat, +b: Nat, +c: Nat, +h: {Nat.is_lt(a, Nat.mul(b, c)) == True{} : Bool},
  t: Bool, et: {Nat.is_lt(Nat.div(a, b), c) == t : Bool}) -> {Nat.is_lt(Nat.div(a, b), c) == True{} : Bool}:
  match t:
    case True{}:
      et
    case False{}:
      +q = Nat.div(a, b)
      +hcq = not_lt_le(q, c, et)
      +hle = le_trans(Nat.mul(c, b), Nat.mul(q, b), a, le_mul_r(c, q, b, hcq),
        %div_mod(a, b) : {Nat.is_le(Nat.mul(q, b), _) == True{} : Bool}
        le_add_r(Nat.mul(q, b), Nat.mod(a, b)))
      +hlt = lt_le_trans(a, Nat.mul(b, c), a, h, %mul_comm(c, b) : {Nat.is_le(_, a) == True{} : Bool}; hle)
      Empty.absurd({Nat.is_lt(q, c) == True{} : Bool}, false_ne_true(
        Equal.trans(Bool, False{}, Nat.is_lt(a, a), True{}, Equal.sym(Bool, Nat.is_lt(a, a), False{}, lt_irrefl(a)), hlt)))

Uses (19):
  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=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.div
  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.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=div_mod
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=false_ne_true
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=le_add_r
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=le_mul_r
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=lt_irrefl
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=lt_le_trans
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_nat.bend&name=not_lt_le