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Lemmas.lean
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Lemmas.lean
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/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
! This file was ported from Lean 3 source module init.data.nat.lemmas
! leanprover-community/lean commit 855e5b74e3a52a40552e8f067169d747d48743fd
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
prelude
import Leanbin.Init.Data.Nat.Basic
import Leanbin.Init.Data.Nat.Div
import Leanbin.Init.Meta.Default
import Leanbin.Init.Algebra.Functions
universe u
namespace Nat
attribute [pre_smt] nat_zero_eq_zero
/-! addition -/
#print Nat.add_comm /-
protected theorem add_comm : ∀ n m : ℕ, n + m = m + n
| n, 0 => Eq.symm (Nat.zero_add n)
| n, m + 1 =>
suffices succ (n + m) = succ (m + n) from Eq.symm (succ_add m n) ▸ this
congr_arg succ (add_comm n m)
#align nat.add_comm Nat.add_comm
-/
#print Nat.add_assoc /-
protected theorem add_assoc : ∀ n m k : ℕ, n + m + k = n + (m + k)
| n, m, 0 => rfl
| n, m, succ k => by rw [add_succ, add_succ, add_assoc] <;> rfl
#align nat.add_assoc Nat.add_assoc
-/
#print Nat.add_left_comm /-
protected theorem add_left_comm : ∀ n m k : ℕ, n + (m + k) = m + (n + k) :=
left_comm Nat.add Nat.add_comm Nat.add_assoc
#align nat.add_left_comm Nat.add_left_comm
-/
#print Nat.add_left_cancel /-
protected theorem add_left_cancel : ∀ {n m k : ℕ}, n + m = n + k → m = k
| 0, m, k => by simp (config := { contextual := true }) [Nat.zero_add]
| succ n, m, k => fun h =>
have : n + m = n + k := by
simp [succ_add] at h
assumption
add_left_cancel this
#align nat.add_left_cancel Nat.add_left_cancel
-/
#print Nat.add_right_cancel /-
protected theorem add_right_cancel {n m k : ℕ} (h : n + m = k + m) : n = k :=
have : m + n = m + k := by rwa [Nat.add_comm n m, Nat.add_comm k m] at h
Nat.add_left_cancel this
#align nat.add_right_cancel Nat.add_right_cancel
-/
#print Nat.succ_ne_zero /-
theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 := fun h => Nat.noConfusion h
#align nat.succ_ne_zero Nat.succ_ne_zero
-/
#print Nat.succ_ne_self /-
theorem succ_ne_self : ∀ n : ℕ, succ n ≠ n
| 0, h => absurd h (Nat.succ_ne_zero 0)
| n + 1, h => succ_ne_self n (Nat.noConfusion h fun h => h)
#align nat.succ_ne_self Nat.succ_ne_self
-/
#print Nat.one_ne_zero /-
protected theorem one_ne_zero : 1 ≠ (0 : ℕ) := fun h => Nat.noConfusion h
#align nat.one_ne_zero Nat.one_ne_zero
-/
#print Nat.zero_ne_one /-
protected theorem zero_ne_one : 0 ≠ (1 : ℕ) := fun h => Nat.noConfusion h
#align nat.zero_ne_one Nat.zero_ne_one
-/
#print Nat.eq_zero_of_add_eq_zero_right /-
protected theorem eq_zero_of_add_eq_zero_right : ∀ {n m : ℕ}, n + m = 0 → n = 0
| 0, m => by simp [Nat.zero_add]
| n + 1, m => fun h => by
exfalso
rw [add_one, succ_add] at h
apply succ_ne_zero _ h
#align nat.eq_zero_of_add_eq_zero_right Nat.eq_zero_of_add_eq_zero_right
-/
#print Nat.eq_zero_of_add_eq_zero_left /-
protected theorem eq_zero_of_add_eq_zero_left {n m : ℕ} (h : n + m = 0) : m = 0 :=
@Nat.eq_zero_of_add_eq_zero_right m n (Nat.add_comm n m ▸ h)
#align nat.eq_zero_of_add_eq_zero_left Nat.eq_zero_of_add_eq_zero_left
-/
#print Nat.add_right_comm /-
protected theorem add_right_comm : ∀ n m k : ℕ, n + m + k = n + k + m :=
right_comm Nat.add Nat.add_comm Nat.add_assoc
#align nat.add_right_comm Nat.add_right_comm
-/
#print Nat.eq_zero_of_add_eq_zero /-
theorem eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
⟨Nat.eq_zero_of_add_eq_zero_right H, Nat.eq_zero_of_add_eq_zero_left H⟩
#align nat.eq_zero_of_add_eq_zero Nat.eq_zero_of_add_eq_zero
-/
/-! multiplication -/
#print Nat.mul_zero /-
protected theorem mul_zero (n : ℕ) : n * 0 = 0 :=
rfl
#align nat.mul_zero Nat.mul_zero
-/
#print Nat.mul_succ /-
theorem mul_succ (n m : ℕ) : n * succ m = n * m + n :=
rfl
#align nat.mul_succ Nat.mul_succ
-/
#print Nat.zero_mul /-
protected theorem zero_mul : ∀ n : ℕ, 0 * n = 0
| 0 => rfl
| succ n => by rw [mul_succ, zero_mul]
#align nat.zero_mul Nat.zero_mul
-/
/- ./././Mathport/Syntax/Translate/Expr.lean:333:4: warning: unsupported (TODO): `[tacs] -/
private unsafe def sort_add :=
sorry
#align nat.sort_add nat.sort_add
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.285555777.sort_add -/
#print Nat.succ_mul /-
theorem succ_mul : ∀ n m : ℕ, succ n * m = n * m + m
| n, 0 => rfl
| n, succ m => by
simp [mul_succ, add_succ, succ_mul n m]
run_tac
sort_add
#align nat.succ_mul Nat.succ_mul
-/
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.285555777.sort_add -/
#print Nat.right_distrib /-
protected theorem right_distrib : ∀ n m k : ℕ, (n + m) * k = n * k + m * k
| n, m, 0 => rfl
| n, m, succ k => by simp [mul_succ, right_distrib n m k];
run_tac
sort_add
#align nat.right_distrib Nat.right_distrib
-/
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.285555777.sort_add -/
#print Nat.left_distrib /-
protected theorem left_distrib : ∀ n m k : ℕ, n * (m + k) = n * m + n * k
| 0, m, k => by simp [Nat.zero_mul]
| succ n, m, k => by simp [succ_mul, left_distrib n m k];
run_tac
sort_add
#align nat.left_distrib Nat.left_distrib
-/
#print Nat.mul_comm /-
protected theorem mul_comm : ∀ n m : ℕ, n * m = m * n
| n, 0 => by rw [Nat.zero_mul, Nat.mul_zero]
| n, succ m => by simp [mul_succ, succ_mul, mul_comm n m]
#align nat.mul_comm Nat.mul_comm
-/
#print Nat.mul_assoc /-
protected theorem mul_assoc : ∀ n m k : ℕ, n * m * k = n * (m * k)
| n, m, 0 => rfl
| n, m, succ k => by simp [mul_succ, Nat.left_distrib, mul_assoc n m k]
#align nat.mul_assoc Nat.mul_assoc
-/
#print Nat.mul_one /-
protected theorem mul_one : ∀ n : ℕ, n * 1 = n :=
Nat.zero_add
#align nat.mul_one Nat.mul_one
-/
#print Nat.one_mul /-
protected theorem one_mul (n : ℕ) : 1 * n = n := by rw [Nat.mul_comm, Nat.mul_one]
#align nat.one_mul Nat.one_mul
-/
#print Nat.succ_add_eq_succ_add /-
theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m := by simp [succ_add, add_succ]
#align nat.succ_add_eq_succ_add Nat.succ_add_eq_succ_add
-/
#print Nat.eq_zero_of_mul_eq_zero /-
theorem eq_zero_of_mul_eq_zero : ∀ {n m : ℕ}, n * m = 0 → n = 0 ∨ m = 0
| 0, m => fun h => Or.inl rfl
| succ n, m => by
rw [succ_mul]; intro h
exact Or.inr (Nat.eq_zero_of_add_eq_zero_left h)
#align nat.eq_zero_of_mul_eq_zero Nat.eq_zero_of_mul_eq_zero
-/
/-! properties of inequality -/
#print Nat.le_of_eq /-
protected theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m :=
p ▸ less_than_or_equal.refl
#align nat.le_of_eq Nat.le_of_eq
-/
#print Nat.le_succ_of_le /-
theorem le_succ_of_le {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
Nat.le_trans h (le_succ m)
#align nat.le_succ_of_le Nat.le_succ_of_le
-/
#print Nat.le_of_succ_le /-
theorem le_of_succ_le {n m : ℕ} (h : succ n ≤ m) : n ≤ m :=
Nat.le_trans (le_succ n) h
#align nat.le_of_succ_le Nat.le_of_succ_le
-/
#print Nat.le_of_lt /-
protected theorem le_of_lt {n m : ℕ} (h : n < m) : n ≤ m :=
le_of_succ_le h
#align nat.le_of_lt Nat.le_of_lt
-/
#print Nat.lt.step /-
theorem lt.step {n m : ℕ} : n < m → n < succ m :=
less_than_or_equal.step
#align nat.lt.step Nat.lt.step
-/
#print Nat.eq_zero_or_pos /-
protected theorem eq_zero_or_pos (n : ℕ) : n = 0 ∨ 0 < n :=
by
cases n
exact Or.inl rfl
exact Or.inr (succ_pos _)
#align nat.eq_zero_or_pos Nat.eq_zero_or_pos
-/
#print Nat.pos_of_ne_zero /-
protected theorem pos_of_ne_zero {n : Nat} : n ≠ 0 → 0 < n :=
Or.resolve_left n.eq_zero_or_pos
#align nat.pos_of_ne_zero Nat.pos_of_ne_zero
-/
#print Nat.lt_trans /-
protected theorem lt_trans {n m k : ℕ} (h₁ : n < m) : m < k → n < k :=
Nat.le_trans (le.step h₁)
#align nat.lt_trans Nat.lt_trans
-/
#print Nat.lt_of_le_of_lt /-
protected theorem lt_of_le_of_lt {n m k : ℕ} (h₁ : n ≤ m) : m < k → n < k :=
Nat.le_trans (succ_le_succ h₁)
#align nat.lt_of_le_of_lt Nat.lt_of_le_of_lt
-/
#print Nat.lt.base /-
theorem lt.base (n : ℕ) : n < succ n :=
Nat.le_refl (succ n)
#align nat.lt.base Nat.lt.base
-/
#print Nat.lt_succ_self /-
theorem lt_succ_self (n : ℕ) : n < succ n :=
lt.base n
#align nat.lt_succ_self Nat.lt_succ_self
-/
#print Nat.le_antisymm /-
protected theorem le_antisymm {n m : ℕ} (h₁ : n ≤ m) : m ≤ n → n = m :=
le.cases_on h₁ (fun a => rfl) fun a b c => absurd (Nat.lt_of_le_of_lt b c) (Nat.lt_irrefl n)
#align nat.le_antisymm Nat.le_antisymm
-/
#print Nat.lt_or_ge /-
protected theorem lt_or_ge : ∀ a b : ℕ, a < b ∨ b ≤ a
| a, 0 => Or.inr a.zero_le
| a, b + 1 =>
match lt_or_ge a b with
| Or.inl h => Or.inl (le_succ_of_le h)
| Or.inr h =>
match Nat.eq_or_lt_of_le h with
| Or.inl h1 => Or.inl (h1 ▸ lt_succ_self b)
| Or.inr h1 => Or.inr h1
#align nat.lt_or_ge Nat.lt_or_ge
-/
#print Nat.le_total /-
protected theorem le_total {m n : ℕ} : m ≤ n ∨ n ≤ m :=
Or.imp_left Nat.le_of_lt (Nat.lt_or_ge m n)
#align nat.le_total Nat.le_total
-/
protected theorem lt_of_le_and_ne {m n : ℕ} (h1 : m ≤ n) : m ≠ n → m < n :=
Or.resolve_right (Or.symm (Nat.eq_or_lt_of_le h1))
#align nat.lt_of_le_and_ne Nat.lt_of_le_and_ne
#print Nat.lt_iff_le_not_le /-
protected theorem lt_iff_le_not_le {m n : ℕ} : m < n ↔ m ≤ n ∧ ¬n ≤ m :=
⟨fun hmn => ⟨Nat.le_of_lt hmn, fun hnm => Nat.lt_irrefl _ (Nat.lt_of_le_of_lt hnm hmn)⟩,
fun ⟨hmn, hnm⟩ => Nat.lt_of_le_and_ne hmn fun heq => hnm (HEq ▸ Nat.le_refl _)⟩
#align nat.lt_iff_le_not_le Nat.lt_iff_le_not_le
-/
instance : LinearOrder ℕ where
le := Nat.le
le_refl := @Nat.le_refl
le_trans := @Nat.le_trans
le_antisymm := @Nat.le_antisymm
le_total := @Nat.le_total
lt := Nat.lt
lt_iff_le_not_le := @Nat.lt_iff_le_not_le
decidableLt := Nat.decidableLt
decidableLe := Nat.decidableLe
DecidableEq := Nat.decidableEq
#print Nat.eq_zero_of_le_zero /-
protected theorem eq_zero_of_le_zero {n : Nat} (h : n ≤ 0) : n = 0 :=
le_antisymm h n.zero_le
#align nat.eq_zero_of_le_zero Nat.eq_zero_of_le_zero
-/
#print Nat.succ_lt_succ /-
theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
succ_le_succ
#align nat.succ_lt_succ Nat.succ_lt_succ
-/
#print Nat.lt_of_succ_lt /-
theorem lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
le_of_succ_le
#align nat.lt_of_succ_lt Nat.lt_of_succ_lt
-/
#print Nat.lt_of_succ_lt_succ /-
theorem lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
le_of_succ_le_succ
#align nat.lt_of_succ_lt_succ Nat.lt_of_succ_lt_succ
-/
#print Nat.pred_lt_pred /-
theorem pred_lt_pred : ∀ {n m : ℕ}, n ≠ 0 → n < m → pred n < pred m
| 0, _, h₁, h => absurd rfl h₁
| n, 0, h₁, h => absurd h n.not_lt_zero
| succ n, succ m, _, h => lt_of_succ_lt_succ h
#align nat.pred_lt_pred Nat.pred_lt_pred
-/
#print Nat.lt_of_succ_le /-
theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b :=
h
#align nat.lt_of_succ_le Nat.lt_of_succ_le
-/
#print Nat.succ_le_of_lt /-
theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b :=
h
#align nat.succ_le_of_lt Nat.succ_le_of_lt
-/
#print Nat.le_add_right /-
protected theorem le_add_right : ∀ n k : ℕ, n ≤ n + k
| n, 0 => Nat.le_refl n
| n, k + 1 => le_succ_of_le (le_add_right n k)
#align nat.le_add_right Nat.le_add_right
-/
#print Nat.le_add_left /-
protected theorem le_add_left (n m : ℕ) : n ≤ m + n :=
Nat.add_comm n m ▸ n.le_add_right m
#align nat.le_add_left Nat.le_add_left
-/
#print Nat.le.dest /-
theorem le.dest : ∀ {n m : ℕ}, n ≤ m → ∃ k, n + k = m
| n, _, less_than_or_equal.refl => ⟨0, rfl⟩
| n, _, less_than_or_equal.step h =>
match le.dest h with
| ⟨w, hw⟩ => ⟨succ w, hw ▸ add_succ n w⟩
#align nat.le.dest Nat.le.dest
-/
#print Nat.le.intro /-
protected theorem le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
h ▸ n.le_add_right k
#align nat.le.intro Nat.le.intro
-/
#print Nat.add_le_add_left /-
protected theorem add_le_add_left {n m : ℕ} (h : n ≤ m) (k : ℕ) : k + n ≤ k + m :=
match le.dest h with
| ⟨w, hw⟩ => @le.intro _ _ w (by rw [Nat.add_assoc, hw])
#align nat.add_le_add_left Nat.add_le_add_left
-/
#print Nat.add_le_add_right /-
protected theorem add_le_add_right {n m : ℕ} (h : n ≤ m) (k : ℕ) : n + k ≤ m + k := by
rw [Nat.add_comm n k, Nat.add_comm m k]; apply Nat.add_le_add_left h
#align nat.add_le_add_right Nat.add_le_add_right
-/
#print Nat.le_of_add_le_add_left /-
protected theorem le_of_add_le_add_left {k n m : ℕ} (h : k + n ≤ k + m) : n ≤ m :=
match le.dest h with
| ⟨w, hw⟩ =>
@le.intro _ _ w
(by
rw [Nat.add_assoc] at hw
apply Nat.add_left_cancel hw)
#align nat.le_of_add_le_add_left Nat.le_of_add_le_add_left
-/
protected theorem le_of_add_le_add_right {k n m : ℕ} : n + k ≤ m + k → n ≤ m :=
by
rw [Nat.add_comm _ k, Nat.add_comm _ k]
apply Nat.le_of_add_le_add_left
#align nat.le_of_add_le_add_right Nat.le_of_add_le_add_rightₓ
protected theorem add_le_add_iff_right {k n m : ℕ} : n + k ≤ m + k ↔ n ≤ m :=
⟨Nat.le_of_add_le_add_right, fun h => Nat.add_le_add_right h _⟩
#align nat.add_le_add_iff_right Nat.add_le_add_iff_right
#print Nat.lt_of_add_lt_add_left /-
protected theorem lt_of_add_lt_add_left {k n m : ℕ} (h : k + n < k + m) : n < m :=
let h' := Nat.le_of_lt h
Nat.lt_of_le_and_ne (Nat.le_of_add_le_add_left h') fun heq =>
Nat.lt_irrefl (k + m) (by rw [HEq] at h; assumption)
#align nat.lt_of_add_lt_add_left Nat.lt_of_add_lt_add_left
-/
#print Nat.lt_of_add_lt_add_right /-
protected theorem lt_of_add_lt_add_right {a b c : ℕ} (h : a + b < c + b) : a < c :=
Nat.lt_of_add_lt_add_left <| show b + a < b + c by rwa [Nat.add_comm b a, Nat.add_comm b c]
#align nat.lt_of_add_lt_add_right Nat.lt_of_add_lt_add_right
-/
#print Nat.add_lt_add_left /-
protected theorem add_lt_add_left {n m : ℕ} (h : n < m) (k : ℕ) : k + n < k + m :=
lt_of_succ_le (add_succ k n ▸ Nat.add_le_add_left (succ_le_of_lt h) k)
#align nat.add_lt_add_left Nat.add_lt_add_left
-/
#print Nat.add_lt_add_right /-
protected theorem add_lt_add_right {n m : ℕ} (h : n < m) (k : ℕ) : n + k < m + k :=
Nat.add_comm k m ▸ Nat.add_comm k n ▸ Nat.add_lt_add_left h k
#align nat.add_lt_add_right Nat.add_lt_add_right
-/
#print Nat.lt_add_of_pos_right /-
protected theorem lt_add_of_pos_right {n k : ℕ} (h : 0 < k) : n < n + k :=
Nat.add_lt_add_left h n
#align nat.lt_add_of_pos_right Nat.lt_add_of_pos_right
-/
#print Nat.lt_add_of_pos_left /-
protected theorem lt_add_of_pos_left {n k : ℕ} (h : 0 < k) : n < k + n := by
rw [Nat.add_comm] <;> exact Nat.lt_add_of_pos_right h
#align nat.lt_add_of_pos_left Nat.lt_add_of_pos_left
-/
#print Nat.add_lt_add /-
protected theorem add_lt_add {a b c d : ℕ} (h₁ : a < b) (h₂ : c < d) : a + c < b + d :=
lt_trans (Nat.add_lt_add_right h₁ c) (Nat.add_lt_add_left h₂ b)
#align nat.add_lt_add Nat.add_lt_add
-/
#print Nat.add_le_add /-
protected theorem add_le_add {a b c d : ℕ} (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d :=
le_trans (Nat.add_le_add_right h₁ c) (Nat.add_le_add_left h₂ b)
#align nat.add_le_add Nat.add_le_add
-/
#print Nat.zero_lt_one /-
protected theorem zero_lt_one : 0 < (1 : Nat) :=
zero_lt_succ 0
#align nat.zero_lt_one Nat.zero_lt_one
-/
#print Nat.mul_le_mul_left /-
protected theorem mul_le_mul_left {n m : ℕ} (k : ℕ) (h : n ≤ m) : k * n ≤ k * m :=
match le.dest h with
| ⟨l, hl⟩ =>
have : k * n + k * l = k * m := by rw [← Nat.left_distrib, hl]
le.intro this
#align nat.mul_le_mul_left Nat.mul_le_mul_left
-/
#print Nat.mul_le_mul_right /-
protected theorem mul_le_mul_right {n m : ℕ} (k : ℕ) (h : n ≤ m) : n * k ≤ m * k :=
Nat.mul_comm k m ▸ Nat.mul_comm k n ▸ k.mul_le_mul_left h
#align nat.mul_le_mul_right Nat.mul_le_mul_right
-/
#print Nat.mul_lt_mul_of_pos_left /-
protected theorem mul_lt_mul_of_pos_left {n m k : ℕ} (h : n < m) (hk : 0 < k) : k * n < k * m :=
Nat.lt_of_lt_of_le (Nat.lt_add_of_pos_right hk)
(mul_succ k n ▸ Nat.mul_le_mul_left k (succ_le_of_lt h))
#align nat.mul_lt_mul_of_pos_left Nat.mul_lt_mul_of_pos_left
-/
#print Nat.mul_lt_mul_of_pos_right /-
protected theorem mul_lt_mul_of_pos_right {n m k : ℕ} (h : n < m) (hk : 0 < k) : n * k < m * k :=
Nat.mul_comm k m ▸ Nat.mul_comm k n ▸ Nat.mul_lt_mul_of_pos_left h hk
#align nat.mul_lt_mul_of_pos_right Nat.mul_lt_mul_of_pos_right
-/
#print Nat.le_of_mul_le_mul_left /-
protected theorem le_of_mul_le_mul_left {a b c : ℕ} (h : c * a ≤ c * b) (hc : 0 < c) : a ≤ b :=
not_lt.1 fun h1 : b < a =>
have h2 : c * b < c * a := Nat.mul_lt_mul_of_pos_left h1 hc
not_le_of_gt h2 h
#align nat.le_of_mul_le_mul_left Nat.le_of_mul_le_mul_left
-/
#print Nat.le_of_lt_succ /-
theorem le_of_lt_succ {m n : Nat} : m < succ n → m ≤ n :=
le_of_succ_le_succ
#align nat.le_of_lt_succ Nat.le_of_lt_succ
-/
#print Nat.eq_of_mul_eq_mul_left /-
protected theorem eq_of_mul_eq_mul_left {m k n : ℕ} (Hn : 0 < n) (H : n * m = n * k) : m = k :=
le_antisymm (Nat.le_of_mul_le_mul_left (le_of_eq H) Hn)
(Nat.le_of_mul_le_mul_left (le_of_eq H.symm) Hn)
#align nat.eq_of_mul_eq_mul_left Nat.eq_of_mul_eq_mul_left
-/
#print Nat.mul_pos /-
protected theorem mul_pos {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : 0 < a * b :=
by
have h : 0 * b < a * b := Nat.mul_lt_mul_of_pos_right ha hb
rwa [Nat.zero_mul] at h
#align nat.mul_pos Nat.mul_pos
-/
#print Nat.le_succ_of_pred_le /-
theorem le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
Nat.casesOn n le.step fun a => succ_le_succ
#align nat.le_succ_of_pred_le Nat.le_succ_of_pred_le
-/
#print Nat.le_lt_antisymm /-
theorem le_lt_antisymm {n m : ℕ} (h₁ : n ≤ m) (h₂ : m < n) : False :=
Nat.lt_irrefl n (Nat.lt_of_le_of_lt h₁ h₂)
#align nat.le_lt_antisymm Nat.le_lt_antisymm
-/
#print Nat.lt_le_antisymm /-
theorem lt_le_antisymm {n m : ℕ} (h₁ : n < m) (h₂ : m ≤ n) : False :=
le_lt_antisymm h₂ h₁
#align nat.lt_le_antisymm Nat.lt_le_antisymm
-/
#print Nat.lt_asymm /-
protected theorem lt_asymm {n m : ℕ} (h₁ : n < m) : ¬m < n :=
le_lt_antisymm (Nat.le_of_lt h₁)
#align nat.lt_asymm Nat.lt_asymm
-/
protected def ltGeByCases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C :=
Decidable.byCases h₁ fun h => h₂ (Or.elim (Nat.lt_or_ge a b) (fun a => absurd a h) fun a => a)
#align nat.lt_ge_by_cases Nat.ltGeByCases
protected def ltByCases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : a = b → C) (h₃ : b < a → C) :
C :=
Nat.ltGeByCases h₁ fun h₁ => Nat.ltGeByCases h₃ fun h => h₂ (Nat.le_antisymm h h₁)
#align nat.lt_by_cases Nat.ltByCases
#print Nat.lt_trichotomy /-
protected theorem lt_trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
Nat.ltByCases (fun h => Or.inl h) (fun h => Or.inr (Or.inl h)) fun h => Or.inr (Or.inr h)
#align nat.lt_trichotomy Nat.lt_trichotomy
-/
#print Nat.eq_or_lt_of_not_lt /-
protected theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬a < b) : a = b ∨ b < a :=
(Nat.lt_trichotomy a b).resolve_left hnlt
#align nat.eq_or_lt_of_not_lt Nat.eq_or_lt_of_not_lt
-/
#print Nat.lt_succ_of_lt /-
theorem lt_succ_of_lt {a b : Nat} (h : a < b) : a < succ b :=
le_succ_of_le h
#align nat.lt_succ_of_lt Nat.lt_succ_of_lt
-/
#print Nat.one_pos /-
theorem one_pos : 0 < 1 :=
Nat.zero_lt_one
#align nat.one_pos Nat.one_pos
-/
#print Nat.mul_le_mul_of_nonneg_left /-
protected theorem mul_le_mul_of_nonneg_left {a b c : ℕ} (h₁ : a ≤ b) : c * a ≤ c * b :=
by
by_cases hba : b ≤ a; · simp [le_antisymm hba h₁]
by_cases hc0 : c ≤ 0; · simp [le_antisymm hc0 c.zero_le, Nat.zero_mul]
exact
(le_not_le_of_lt
(Nat.mul_lt_mul_of_pos_left (lt_of_le_not_le h₁ hba) (lt_of_le_not_le c.zero_le hc0))).left
#align nat.mul_le_mul_of_nonneg_left Nat.mul_le_mul_of_nonneg_left
-/
#print Nat.mul_le_mul_of_nonneg_right /-
protected theorem mul_le_mul_of_nonneg_right {a b c : ℕ} (h₁ : a ≤ b) : a * c ≤ b * c :=
by
by_cases hba : b ≤ a; · simp [le_antisymm hba h₁]
by_cases hc0 : c ≤ 0; · simp [le_antisymm hc0 c.zero_le, Nat.mul_zero]
exact
(le_not_le_of_lt
(Nat.mul_lt_mul_of_pos_right (lt_of_le_not_le h₁ hba) (lt_of_le_not_le c.zero_le hc0))).left
#align nat.mul_le_mul_of_nonneg_right Nat.mul_le_mul_of_nonneg_right
-/
protected theorem mul_lt_mul {a b c d : ℕ} (hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) :
a * b < c * d :=
calc
a * b < c * b := Nat.mul_lt_mul_of_pos_right hac pos_b
_ ≤ c * d := Nat.mul_le_mul_of_nonneg_left hbd
#align nat.mul_lt_mul Nat.mul_lt_mulₓ
protected theorem mul_lt_mul' {a b c d : ℕ} (h1 : a ≤ c) (h2 : b < d) (h3 : 0 < c) :
a * b < c * d :=
calc
a * b ≤ c * b := Nat.mul_le_mul_of_nonneg_right h1
_ < c * d := Nat.mul_lt_mul_of_pos_left h2 h3
#align nat.mul_lt_mul' Nat.mul_lt_mul'ₓ
#print Nat.mul_le_mul /-
-- TODO: there are four variations, depending on which variables we assume to be nonneg
protected theorem mul_le_mul {a b c d : ℕ} (hac : a ≤ c) (hbd : b ≤ d) : a * b ≤ c * d :=
calc
a * b ≤ c * b := Nat.mul_le_mul_of_nonneg_right hac
_ ≤ c * d := Nat.mul_le_mul_of_nonneg_left hbd
#align nat.mul_le_mul Nat.mul_le_mul
-/
/-! bit0/bit1 properties -/
#print Nat.bit1_eq_succ_bit0 /-
protected theorem bit1_eq_succ_bit0 (n : ℕ) : bit1 n = succ (bit0 n) :=
rfl
#align nat.bit1_eq_succ_bit0 Nat.bit1_eq_succ_bit0
-/
#print Nat.bit1_succ_eq /-
protected theorem bit1_succ_eq (n : ℕ) : bit1 (succ n) = succ (succ (bit1 n)) :=
Eq.trans (Nat.bit1_eq_succ_bit0 (succ n)) (congr_arg succ (Nat.bit0_succ_eq n))
#align nat.bit1_succ_eq Nat.bit1_succ_eq
-/
#print Nat.bit1_ne_one /-
protected theorem bit1_ne_one : ∀ {n : ℕ}, n ≠ 0 → bit1 n ≠ 1
| 0, h, h1 => absurd rfl h
| n + 1, h, h1 => Nat.noConfusion h1 fun h2 => absurd h2 (succ_ne_zero _)
#align nat.bit1_ne_one Nat.bit1_ne_one
-/
#print Nat.bit0_ne_one /-
protected theorem bit0_ne_one : ∀ n : ℕ, bit0 n ≠ 1
| 0, h => absurd h (Ne.symm Nat.one_ne_zero)
| n + 1, h =>
have h1 : succ (succ (n + n)) = 1 := succ_add n n ▸ h
Nat.noConfusion h1 fun h2 => absurd h2 (succ_ne_zero (n + n))
#align nat.bit0_ne_one Nat.bit0_ne_one
-/
#print Nat.add_self_ne_one /-
protected theorem add_self_ne_one : ∀ n : ℕ, n + n ≠ 1
| 0, h => Nat.noConfusion h
| n + 1, h =>
have h1 : succ (succ (n + n)) = 1 := succ_add n n ▸ h
Nat.noConfusion h1 fun h2 => absurd h2 (Nat.succ_ne_zero (n + n))
#align nat.add_self_ne_one Nat.add_self_ne_one
-/
#print Nat.bit1_ne_bit0 /-
protected theorem bit1_ne_bit0 : ∀ n m : ℕ, bit1 n ≠ bit0 m
| 0, m, h => absurd h (Ne.symm (Nat.add_self_ne_one m))
| n + 1, 0, h =>
have h1 : succ (bit0 (succ n)) = 0 := h
absurd h1 (Nat.succ_ne_zero _)
| n + 1, m + 1, h =>
have h1 : succ (succ (bit1 n)) = succ (succ (bit0 m)) :=
Nat.bit0_succ_eq m ▸ Nat.bit1_succ_eq n ▸ h
have h2 : bit1 n = bit0 m := Nat.noConfusion h1 fun h2' => Nat.noConfusion h2' fun h2'' => h2''
absurd h2 (bit1_ne_bit0 n m)
#align nat.bit1_ne_bit0 Nat.bit1_ne_bit0
-/
#print Nat.bit0_ne_bit1 /-
protected theorem bit0_ne_bit1 : ∀ n m : ℕ, bit0 n ≠ bit1 m := fun n m : Nat =>
Ne.symm (Nat.bit1_ne_bit0 m n)
#align nat.bit0_ne_bit1 Nat.bit0_ne_bit1
-/
#print Nat.bit0_inj /-
protected theorem bit0_inj : ∀ {n m : ℕ}, bit0 n = bit0 m → n = m
| 0, 0, h => rfl
| 0, m + 1, h => by contradiction
| n + 1, 0, h => by contradiction
| n + 1, m + 1, h =>
by
have : succ (succ (n + n)) = succ (succ (m + m)) :=
by
unfold bit0 at h
simp [add_one, add_succ, succ_add] at h
have aux : n + n = m + m := h
rw [aux]
have : n + n = m + m := by repeat injection this with this
have : n = m := bit0_inj this
rw [this]
#align nat.bit0_inj Nat.bit0_inj
-/
#print Nat.bit1_inj /-
protected theorem bit1_inj : ∀ {n m : ℕ}, bit1 n = bit1 m → n = m := fun n m h =>
have : succ (bit0 n) = succ (bit0 m) := by simp [Nat.bit1_eq_succ_bit0] at h; rw [h]
have : bit0 n = bit0 m := by injection this
Nat.bit0_inj this
#align nat.bit1_inj Nat.bit1_inj
-/
#print Nat.bit0_ne /-
protected theorem bit0_ne {n m : ℕ} : n ≠ m → bit0 n ≠ bit0 m := fun h₁ h₂ =>
absurd (Nat.bit0_inj h₂) h₁
#align nat.bit0_ne Nat.bit0_ne
-/
#print Nat.bit1_ne /-
protected theorem bit1_ne {n m : ℕ} : n ≠ m → bit1 n ≠ bit1 m := fun h₁ h₂ =>
absurd (Nat.bit1_inj h₂) h₁
#align nat.bit1_ne Nat.bit1_ne
-/
#print Nat.zero_ne_bit0 /-
protected theorem zero_ne_bit0 {n : ℕ} : n ≠ 0 → 0 ≠ bit0 n := fun h => Ne.symm (Nat.bit0_ne_zero h)
#align nat.zero_ne_bit0 Nat.zero_ne_bit0
-/
#print Nat.zero_ne_bit1 /-
protected theorem zero_ne_bit1 (n : ℕ) : 0 ≠ bit1 n :=
Ne.symm (Nat.bit1_ne_zero n)
#align nat.zero_ne_bit1 Nat.zero_ne_bit1
-/
#print Nat.one_ne_bit0 /-
protected theorem one_ne_bit0 (n : ℕ) : 1 ≠ bit0 n :=
Ne.symm (Nat.bit0_ne_one n)
#align nat.one_ne_bit0 Nat.one_ne_bit0
-/
#print Nat.one_ne_bit1 /-
protected theorem one_ne_bit1 {n : ℕ} : n ≠ 0 → 1 ≠ bit1 n := fun h => Ne.symm (Nat.bit1_ne_one h)
#align nat.one_ne_bit1 Nat.one_ne_bit1
-/
#print Nat.one_lt_bit1 /-
protected theorem one_lt_bit1 : ∀ {n : Nat}, n ≠ 0 → 1 < bit1 n
| 0, h => by contradiction
| succ n, h => by
rw [Nat.bit1_succ_eq]
apply succ_lt_succ
apply zero_lt_succ
#align nat.one_lt_bit1 Nat.one_lt_bit1
-/
#print Nat.one_lt_bit0 /-
protected theorem one_lt_bit0 : ∀ {n : Nat}, n ≠ 0 → 1 < bit0 n
| 0, h => by contradiction
| succ n, h => by
rw [Nat.bit0_succ_eq]
apply succ_lt_succ
apply zero_lt_succ
#align nat.one_lt_bit0 Nat.one_lt_bit0
-/
#print Nat.bit0_lt /-
protected theorem bit0_lt {n m : Nat} (h : n < m) : bit0 n < bit0 m :=
Nat.add_lt_add h h
#align nat.bit0_lt Nat.bit0_lt
-/
#print Nat.bit1_lt /-
protected theorem bit1_lt {n m : Nat} (h : n < m) : bit1 n < bit1 m :=
succ_lt_succ (Nat.add_lt_add h h)
#align nat.bit1_lt Nat.bit1_lt
-/
#print Nat.bit0_lt_bit1 /-
protected theorem bit0_lt_bit1 {n m : Nat} (h : n ≤ m) : bit0 n < bit1 m :=
lt_succ_of_le (Nat.add_le_add h h)
#align nat.bit0_lt_bit1 Nat.bit0_lt_bit1
-/
#print Nat.bit1_lt_bit0 /-
protected theorem bit1_lt_bit0 : ∀ {n m : Nat}, n < m → bit1 n < bit0 m
| n, 0, h => absurd h n.not_lt_zero
| n, succ m, h =>
have : n ≤ m := le_of_lt_succ h
have : succ (n + n) ≤ succ (m + m) := succ_le_succ (Nat.add_le_add this this)
have : succ (n + n) ≤ succ m + m := by rw [succ_add]; assumption
show succ (n + n) < succ (succ m + m) from lt_succ_of_le this
#align nat.bit1_lt_bit0 Nat.bit1_lt_bit0
-/
#print Nat.one_le_bit1 /-
protected theorem one_le_bit1 (n : ℕ) : 1 ≤ bit1 n :=
show 1 ≤ succ (bit0 n) from succ_le_succ (bit0 n).zero_le
#align nat.one_le_bit1 Nat.one_le_bit1
-/
#print Nat.one_le_bit0 /-
protected theorem one_le_bit0 : ∀ n : ℕ, n ≠ 0 → 1 ≤ bit0 n
| 0, h => absurd rfl h
| n + 1, h =>
suffices 1 ≤ succ (succ (bit0 n)) from Eq.symm (Nat.bit0_succ_eq n) ▸ this
succ_le_succ (bit0 n).succ.zero_le
#align nat.one_le_bit0 Nat.one_le_bit0
-/
/-! successor and predecessor -/
#print Nat.pred_zero /-
@[simp]
theorem pred_zero : pred 0 = 0 :=
rfl
#align nat.pred_zero Nat.pred_zero
-/
#print Nat.pred_succ /-
@[simp]
theorem pred_succ (n : ℕ) : pred (succ n) = n :=
rfl
#align nat.pred_succ Nat.pred_succ
-/
#print Nat.add_one_ne_zero /-
theorem add_one_ne_zero (n : ℕ) : n + 1 ≠ 0 :=
succ_ne_zero _
#align nat.add_one_ne_zero Nat.add_one_ne_zero
-/
#print Nat.eq_zero_or_eq_succ_pred /-
theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) := by cases n <;> simp
#align nat.eq_zero_or_eq_succ_pred Nat.eq_zero_or_eq_succ_pred
-/
#print Nat.exists_eq_succ_of_ne_zero /-
theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃ k : ℕ, n = succ k :=
⟨_, (eq_zero_or_eq_succ_pred _).resolve_left H⟩
#align nat.exists_eq_succ_of_ne_zero Nat.exists_eq_succ_of_ne_zero
-/
/- warning: nat.discriminate clashes with [anonymous] -> [anonymous]
warning: nat.discriminate -> [anonymous] is a dubious translation:
lean 3 declaration is
forall {B : Sort.{u}} {n : Nat}, ((Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> B) -> (forall (m : Nat), (Eq.{1} Nat n (Nat.succ m)) -> B) -> B
but is expected to have type
forall {B : Type.{u}} {n : Type.{v}}, (Nat -> B -> n) -> Nat -> (List.{u} B) -> (List.{v} n)
Case conversion may be inaccurate. Consider using '#align nat.discriminate [anonymous]ₓ'. -/
def [anonymous] {B : Sort u} {n : ℕ} (H1 : n = 0 → B) (H2 : ∀ m, n = succ m → B) : B := by
induction' h : n with <;> [exact H1 h, exact H2 _ h]
#align nat.discriminate [anonymous]
theorem one_succ_zero : 1 = succ 0 :=
rfl
#align nat.one_succ_zero Nat.one_succ_zero
#print Nat.pred_inj /-
theorem pred_inj : ∀ {a b : Nat}, 0 < a → 0 < b → Nat.pred a = Nat.pred b → a = b
| succ a, succ b, ha, hb, h => by
have : a = b := h
rw [this]
| succ a, 0, ha, hb, h => absurd hb (lt_irrefl _)
| 0, succ b, ha, hb, h => absurd ha (lt_irrefl _)
| 0, 0, ha, hb, h => rfl
#align nat.pred_inj Nat.pred_inj
-/
/-! subtraction
Many lemmas are proven more generally in mathlib `algebra/order/sub` -/
#print Nat.zero_sub /-
@[simp]
protected theorem zero_sub : ∀ a : ℕ, 0 - a = 0
| 0 => rfl
| a + 1 => congr_arg pred (zero_sub a)
#align nat.zero_sub Nat.zero_sub
-/
#print Nat.sub_lt_succ /-
theorem sub_lt_succ (a b : ℕ) : a - b < succ a :=
lt_succ_of_le (a.sub_le b)
#align nat.sub_lt_succ Nat.sub_lt_succ
-/
#print Nat.sub_le_sub_right /-
protected theorem sub_le_sub_right {n m : ℕ} (h : n ≤ m) : ∀ k, n - k ≤ m - k
| 0 => h
| succ z => pred_le_pred (sub_le_sub_right z)
#align nat.sub_le_sub_right Nat.sub_le_sub_right
-/
#print Nat.sub_zero /-
@[simp]
protected theorem sub_zero (n : ℕ) : n - 0 = n :=
rfl
#align nat.sub_zero Nat.sub_zero
-/
#print Nat.sub_succ /-
theorem sub_succ (n m : ℕ) : n - succ m = pred (n - m) :=
rfl
#align nat.sub_succ Nat.sub_succ
-/
#print Nat.succ_sub_succ /-
theorem succ_sub_succ (n m : ℕ) : succ n - succ m = n - m :=
succ_sub_succ_eq_sub n m
#align nat.succ_sub_succ Nat.succ_sub_succ
-/
#print Nat.sub_self /-
protected theorem sub_self : ∀ n : ℕ, n - n = 0
| 0 => by rw [Nat.sub_zero]
| succ n => by rw [succ_sub_succ, sub_self n]
#align nat.sub_self Nat.sub_self
-/
#print Nat.add_sub_add_right /-
/- TODO(Leo): remove the following ematch annotations as soon as we have
arithmetic theory in the smt_stactic -/
@[ematch_lhs]
protected theorem add_sub_add_right : ∀ n k m : ℕ, n + k - (m + k) = n - m
| n, 0, m => by rw [Nat.add_zero, Nat.add_zero]
| n, succ k, m => by rw [add_succ, add_succ, succ_sub_succ, add_sub_add_right n k m]
#align nat.add_sub_add_right Nat.add_sub_add_right
-/
#print Nat.add_sub_add_left /-
@[ematch_lhs]
protected theorem add_sub_add_left (k n m : ℕ) : k + n - (k + m) = n - m := by
rw [Nat.add_comm k n, Nat.add_comm k m, Nat.add_sub_add_right]
#align nat.add_sub_add_left Nat.add_sub_add_left
-/
#print Nat.add_sub_cancel /-
@[ematch_lhs]
protected theorem add_sub_cancel (n m : ℕ) : n + m - m = n :=
by
suffices n + m - (0 + m) = n by rwa [Nat.zero_add] at this
rw [Nat.add_sub_add_right, Nat.sub_zero]
#align nat.add_sub_cancel Nat.add_sub_cancel
-/
#print Nat.add_sub_cancel_left /-
@[ematch_lhs]
protected theorem add_sub_cancel_left (n m : ℕ) : n + m - n = m :=
show n + m - (n + 0) = m by rw [Nat.add_sub_add_left, Nat.sub_zero]
#align nat.add_sub_cancel_left Nat.add_sub_cancel_left
-/
#print Nat.sub_sub /-
protected theorem sub_sub : ∀ n m k : ℕ, n - m - k = n - (m + k)
| n, m, 0 => by rw [Nat.add_zero, Nat.sub_zero]
| n, m, succ k => by rw [add_succ, Nat.sub_succ, Nat.sub_succ, sub_sub n m k]
#align nat.sub_sub Nat.sub_sub
-/
#print Nat.le_of_le_of_sub_le_sub_right /-
protected theorem le_of_le_of_sub_le_sub_right {n m k : ℕ} (h₀ : k ≤ m) (h₁ : n - k ≤ m - k) :
n ≤ m := by
revert k m
induction' n with n <;> intro k m h₀ h₁
· exact m.zero_le
· cases' k with k