/- Copyright 2025 The Formal Conjectures Authors. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at https://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ module public import Mathlib.Data.Nat.Init@[expose] public sectionnamespace Natvariable {a b c : ℕ} @[simp] protected lemma dvd_add_mul_self : a ∣ b + c * a ↔ a ∣ b := bya:ℕb:ℕc:ℕ⊢ a ∣ b + c * a ↔ a ∣ b rw [Nat.dvd_add_left (Nat.dvd_mul_left _ _)a:ℕb:ℕc:ℕ⊢ a ∣ b ↔ a ∣ bAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_add_self_mul : a ∣ b + a * c ↔ a ∣ b := bya:ℕb:ℕc:ℕ⊢ a ∣ b + a * c ↔ a ∣ b rw [Nat.mul_comm,a:ℕb:ℕc:ℕ⊢ a ∣ b + c * a ↔ a ∣ bAll goals completed! 🐙 Nat.dvd_add_mul_selfa:ℕb:ℕc:ℕ⊢ a ∣ b ↔ a ∣ bAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_mul_self_add : a ∣ b * a + c ↔ a ∣ c := bya:ℕb:ℕc:ℕ⊢ a ∣ b * a + c ↔ a ∣ c rw [Nat.add_comm,a:ℕb:ℕc:ℕ⊢ a ∣ c + b * a ↔ a ∣ cAll goals completed! 🐙 Nat.dvd_add_mul_selfa:ℕb:ℕc:ℕ⊢ a ∣ c ↔ a ∣ cAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_self_mul_add : a ∣ a * b + c ↔ a ∣ c := bya:ℕb:ℕc:ℕ⊢ a ∣ a * b + c ↔ a ∣ c rw [Nat.mul_comm,a:ℕb:ℕc:ℕ⊢ a ∣ b * a + c ↔ a ∣ cAll goals completed! 🐙 Nat.dvd_mul_self_adda:ℕb:ℕc:ℕ⊢ a ∣ c ↔ a ∣ cAll goals completed! 🐙]All goals completed! 🐙@[simp] lemma dvd_sub_iff_right' (hab : a ∣ b) : a ∣ b - c ↔ a ∣ c ∨ b ≤ c := bya:ℕb:ℕc:ℕhab:a ∣ b⊢ a ∣ b - c ↔ a ∣ c ∨ b ≤ c obtain hbc | hcb := b.le_total cinla:ℕb:ℕc:ℕhab:a ∣ bhbc:b ≤ c⊢ a ∣ b - c ↔ a ∣ c ∨ b ≤ cinra:ℕb:ℕc:ℕhab:a ∣ bhcb:c ≤ b⊢ a ∣ b - c ↔ a ∣ c ∨ b ≤ c ·inla:ℕb:ℕc:ℕhab:a ∣ bhbc:b ≤ c⊢ a ∣ b - c ↔ a ∣ c ∨ b ≤ c simp [*]All goals completed! 🐙 ·inra:ℕb:ℕc:ℕhab:a ∣ bhcb:c ≤ b⊢ a ∣ b - c ↔ a ∣ c ∨ b ≤ c simp +contextual [Nat.dvd_sub_iff_right, Nat.le_antisymm hcb, *]All goals completed! 🐙@[simp] lemma dvd_sub_iff_left' (hac : a ∣ c) : a ∣ b - c ↔ a ∣ b ∨ b ≤ c := bya:ℕb:ℕc:ℕhac:a ∣ c⊢ a ∣ b - c ↔ a ∣ b ∨ b ≤ c obtain hbc | hcb := b.le_total cinla:ℕb:ℕc:ℕhac:a ∣ chbc:b ≤ c⊢ a ∣ b - c ↔ a ∣ b ∨ b ≤ cinra:ℕb:ℕc:ℕhac:a ∣ chcb:c ≤ b⊢ a ∣ b - c ↔ a ∣ b ∨ b ≤ c ·inla:ℕb:ℕc:ℕhac:a ∣ chbc:b ≤ c⊢ a ∣ b - c ↔ a ∣ b ∨ b ≤ c simp [*]All goals completed! 🐙 ·inra:ℕb:ℕc:ℕhac:a ∣ chcb:c ≤ b⊢ a ∣ b - c ↔ a ∣ b ∨ b ≤ c simp +contextual [Nat.dvd_sub_iff_left, ← Nat.le_antisymm hcb, *]All goals completed! 🐙 @[simp] protected lemma dvd_sub_mul_self : a ∣ b - c * a ↔ a ∣ b ∨ b ≤ c * a := bya:ℕb:ℕc:ℕ⊢ a ∣ b - c * a ↔ a ∣ b ∨ b ≤ c * a rw [Nat.dvd_sub_iff_left' (Nat.dvd_mul_left _ _)a:ℕb:ℕc:ℕ⊢ a ∣ b ∨ b ≤ c * a ↔ a ∣ b ∨ b ≤ c * aAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_sub_self_mul : a ∣ b - a * c ↔ a ∣ b ∨ b ≤ c * a := bya:ℕb:ℕc:ℕ⊢ a ∣ b - a * c ↔ a ∣ b ∨ b ≤ c * a rw [Nat.mul_comm,a:ℕb:ℕc:ℕ⊢ a ∣ b - c * a ↔ a ∣ b ∨ b ≤ c * aAll goals completed! 🐙 Nat.dvd_sub_mul_selfa:ℕb:ℕc:ℕ⊢ a ∣ b ∨ b ≤ c * a ↔ a ∣ b ∨ b ≤ c * aAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_mul_self_sub : a ∣ b * a - c ↔ a ∣ c ∨ b * a ≤ c := bya:ℕb:ℕc:ℕ⊢ a ∣ b * a - c ↔ a ∣ c ∨ b * a ≤ c rw [Nat.dvd_sub_iff_right' (Nat.dvd_mul_left _ _)a:ℕb:ℕc:ℕ⊢ a ∣ c ∨ b * a ≤ c ↔ a ∣ c ∨ b * a ≤ cAll goals completed! 🐙]All goals completed! 🐙 @[simp] protected lemma dvd_self_mul_sub : a ∣ a * b - c ↔ a ∣ c ∨ b * a ≤ c := bya:ℕb:ℕc:ℕ⊢ a ∣ a * b - c ↔ a ∣ c ∨ b * a ≤ c rw [Nat.mul_comm,a:ℕb:ℕc:ℕ⊢ a ∣ b * a - c ↔ a ∣ c ∨ b * a ≤ cAll goals completed! 🐙 Nat.dvd_mul_self_suba:ℕb:ℕc:ℕ⊢ a ∣ c ∨ b * a ≤ c ↔ a ∣ c ∨ b * a ≤ cAll goals completed! 🐙]All goals completed! 🐙end Nat