Arith.quot_lt_g: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend as Arith
Lines 81-90 of bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&start_line=81&end_line=90

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 quot_lt_g(+k: Nat, +q: Nat, +d: Nat, +r: Nat, +c: Bool, +hc: {Nat.is_lt(q, sc(k, 1n)) == c : Bool}, +h: {Nat.is_lt(Nat.add(Nat.mul(q, d), r), sc(k, d)) == True{} : Bool}) -> {Nat.is_lt(q, sc(k, 1n)) == True{} : Bool}:
  match c:
    case True{}:
      hc
    case False{}:
      +l1 = mul_le(sc(k, 1n), q, d, N.not_lt_le(q, sc(k, 1n), hc))
      +e1 = Equal.trans(Nat, Nat.mul(sc(k, 1n), d), sc(k, Nat.mul(1n, d)), sc(k, d), Equal.sym(Nat, sc(k, Nat.mul(1n, d)), Nat.mul(sc(k, 1n), d), sc_mul(k, 1n, d)), Equal.cong(Nat, Nat, z => sc(k, z), Nat.mul(1n, d), d, one_mul(d)))
      +l2 = L.subst(Nat, z => {Nat.is_le(z, Nat.mul(q, d)) == True{} : Bool}, Nat.mul(sc(k, 1n), d), sc(k, d), e1, l1)
      +l3 = N.le_trans(sc(k, d), Nat.mul(q, d), Nat.add(Nat.mul(q, d), r), l2, N.le_add_right(Nat.mul(q, d), r))
      Empty.absurd({Nat.is_lt(q, sc(k, 1n)) == True{} : Bool}, L.true_false(Equal.trans(Bool, True{}, Nat.is_le(sc(k, d), Nat.add(Nat.mul(q, d), r)), False{}, Equal.sym(Bool, Nat.is_le(sc(k, d), Nat.add(Nat.mul(q, d), r)), True{}, l3), N.lt_not_le(Nat.add(Nat.mul(q, d), r), sc(k, d), h))))

Uses (20):
  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.cong
  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.add
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&name=mul_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&name=one_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&name=sc_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=true_false
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_not_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=not_lt_le