formal/lean/mathlib/order/basic.lean

/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import data.prod.basic
import data.subtype

/-!
# Basic definitions about `≤` and `<`

This file proves basic results about orders, provides extensive dot notation, defines useful order
classes and allows to transfer order instances.

## Type synonyms

* `order_dual α` : A type synonym reversing the meaning of all inequalities, with notation `αᵒᵈ`.
* `as_linear_order α`: A type synonym to promote `partial_order α` to `linear_order α` using
  `is_total α (≤)`.

### Transfering orders

- `order.preimage`, `preorder.lift`: Transfers a (pre)order on `β` to an order on `α`
  using a function `f : α → β`.
- `partial_order.lift`, `linear_order.lift`: Transfers a partial (resp., linear) order on `β` to a
  partial (resp., linear) order on `α` using an injective function `f`.

### Extra class

* `has_sup`: type class for the `⊔` notation
* `has_inf`: type class for the `⊓` notation
* `has_compl`: type class for the `ᶜ` notation
* `densely_ordered`: An order with no gap, i.e. for any two elements `a < b` there exists `c` such
  that `a < c < b`.

## Notes

`≤` and `<` are highly favored over `≥` and `>` in mathlib. The reason is that we can formulate all
lemmas using `≤`/`<`, and `rw` has trouble unifying `≤` and `≥`. Hence choosing one direction spares
us useless duplication. This is enforced by a linter. See Note [nolint_ge] for more infos.

Dot notation is particularly useful on `≤` (`has_le.le`) and `<` (`has_lt.lt`). To that end, we
provide many aliases to dot notation-less lemmas. For example, `le_trans` is aliased with
`has_le.le.trans` and can be used to construct `hab.trans hbc : a ≤ c` when `hab : a ≤ b`,
`hbc : b ≤ c`, `lt_of_le_of_lt` is aliased as `has_le.le.trans_lt` and can be used to construct
`hab.trans hbc : a < c` when `hab : a ≤ b`, `hbc : b < c`.

## TODO

- expand module docs
- automatic construction of dual definitions / theorems

## Tags

preorder, order, partial order, poset, linear order, chain
-/

open function

universes u v w
variables {α : Type u} {β : Type v} {γ : Type w} {r : α → α → Prop}

section preorder
variables [preorder α] {a b c : α}

lemma le_trans' : b ≤ c → a ≤ b → a ≤ c := flip le_trans
lemma lt_trans' : b < c → a < b → a < c := flip lt_trans
lemma lt_of_le_of_lt' : b ≤ c → a < b → a < c := flip lt_of_lt_of_le
lemma lt_of_lt_of_le' : b < c → a ≤ b → a < c := flip lt_of_le_of_lt

end preorder

section partial_order
variables [partial_order α] {a b : α}

lemma ge_antisymm : a ≤ b → b ≤ a → b = a := flip le_antisymm
lemma lt_of_le_of_ne' : a ≤ b → b ≠ a → a < b := λ h₁ h₂, lt_of_le_of_ne h₁ h₂.symm
lemma ne.lt_of_le : a ≠ b → a ≤ b → a < b := flip lt_of_le_of_ne
lemma ne.lt_of_le' : b ≠ a → a ≤ b → a < b := flip lt_of_le_of_ne'

end partial_order

attribute [simp] le_refl
attribute [ext] has_le

alias le_trans        ← has_le.le.trans
alias le_trans'       ← has_le.le.trans'
alias lt_of_le_of_lt  ← has_le.le.trans_lt
alias lt_of_le_of_lt' ← has_le.le.trans_lt'
alias le_antisymm     ← has_le.le.antisymm
alias ge_antisymm     ← has_le.le.antisymm'
alias lt_of_le_of_ne  ← has_le.le.lt_of_ne
alias lt_of_le_of_ne' ← has_le.le.lt_of_ne'
alias lt_of_le_not_le ← has_le.le.lt_of_not_le
alias lt_or_eq_of_le  ← has_le.le.lt_or_eq
alias decidable.lt_or_eq_of_le ← has_le.le.lt_or_eq_dec

alias le_of_lt        ← has_lt.lt.le
alias lt_trans        ← has_lt.lt.trans
alias lt_trans'       ← has_lt.lt.trans'
alias lt_of_lt_of_le  ← has_lt.lt.trans_le
alias lt_of_lt_of_le' ← has_lt.lt.trans_le'
alias ne_of_lt        ← has_lt.lt.ne
alias lt_asymm        ← has_lt.lt.asymm has_lt.lt.not_lt

alias le_of_eq        ← eq.le

attribute [nolint decidable_classical] has_le.le.lt_or_eq_dec

section
variables [preorder α] {a b c : α}

/-- A version of `le_refl` where the argument is implicit -/
lemma le_rfl : a ≤ a := le_refl a

@[simp] lemma lt_self_iff_false (x : α) : x < x ↔ false := ⟨lt_irrefl x, false.elim⟩

lemma le_of_le_of_eq (hab : a ≤ b) (hbc : b = c) : a ≤ c := hab.trans hbc.le
lemma le_of_eq_of_le (hab : a = b) (hbc : b ≤ c) : a ≤ c := hab.le.trans hbc
lemma lt_of_lt_of_eq (hab : a < b) (hbc : b = c) : a < c := hab.trans_le hbc.le
lemma lt_of_eq_of_lt (hab : a = b) (hbc : b < c) : a < c := hab.le.trans_lt hbc
lemma le_of_le_of_eq' : b ≤ c → a = b → a ≤ c := flip le_of_eq_of_le
lemma le_of_eq_of_le' : b = c → a ≤ b → a ≤ c := flip le_of_le_of_eq
lemma lt_of_lt_of_eq' : b < c → a = b → a < c := flip lt_of_eq_of_lt
lemma lt_of_eq_of_lt' : b = c → a < b → a < c := flip lt_of_lt_of_eq

alias le_of_le_of_eq  ← has_le.le.trans_eq
alias le_of_le_of_eq' ← has_le.le.trans_eq'
alias lt_of_lt_of_eq  ← has_lt.lt.trans_eq
alias lt_of_lt_of_eq' ← has_lt.lt.trans_eq'
alias le_of_eq_of_le  ← eq.trans_le
alias le_of_eq_of_le' ← eq.trans_ge
alias lt_of_eq_of_lt  ← eq.trans_lt
alias lt_of_eq_of_lt' ← eq.trans_gt

end

namespace eq
variables [preorder α] {x y z : α}

/-- If `x = y` then `y ≤ x`. Note: this lemma uses `y ≤ x` instead of `x ≥ y`, because `le` is used
almost exclusively in mathlib. -/
protected lemma ge (h : x = y) : y ≤ x := h.symm.le

lemma not_lt (h : x = y) : ¬ x < y := λ h', h'.ne h
lemma not_gt (h : x = y) : ¬ y < x := h.symm.not_lt

end eq

namespace has_le.le

@[nolint ge_or_gt] -- see Note [nolint_ge]
protected lemma ge [has_le α] {x y : α} (h : x ≤ y) : y ≥ x := h

lemma lt_iff_ne [partial_order α] {x y : α} (h : x ≤ y) : x < y ↔ x ≠ y := ⟨λ h, h.ne, h.lt_of_ne⟩

lemma le_iff_eq [partial_order α] {x y : α} (h : x ≤ y) : y ≤ x ↔ y = x :=
⟨λ h', h'.antisymm h, eq.le⟩

lemma lt_or_le [linear_order α] {a b : α} (h : a ≤ b) (c : α) : a < c ∨ c ≤ b :=
(lt_or_ge a c).imp id $ λ hc, le_trans hc h

lemma le_or_lt [linear_order α] {a b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c < b :=
(le_or_gt a c).imp id $ λ hc, lt_of_lt_of_le hc h

lemma le_or_le [linear_order α] {a b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c ≤ b :=
(h.le_or_lt c).elim or.inl (λ h, or.inr $ le_of_lt h)

end has_le.le

namespace has_lt.lt

@[nolint ge_or_gt] -- see Note [nolint_ge]
protected lemma gt [has_lt α] {x y : α} (h : x < y) : y > x := h
protected lemma false [preorder α] {x : α} : x < x → false := lt_irrefl x

lemma ne' [preorder α] {x y : α} (h : x < y) : y ≠ x := h.ne.symm

lemma lt_or_lt [linear_order α] {x y : α} (h : x < y) (z : α) : x < z ∨ z < y :=
(lt_or_ge z y).elim or.inr (λ hz, or.inl $ h.trans_le hz)

end has_lt.lt

@[nolint ge_or_gt] -- see Note [nolint_ge]
protected lemma ge.le [has_le α] {x y : α} (h : x ≥ y) : y ≤ x := h

@[nolint ge_or_gt] -- see Note [nolint_ge]
protected lemma gt.lt [has_lt α] {x y : α} (h : x > y) : y < x := h

@[nolint ge_or_gt] -- see Note [nolint_ge]
theorem ge_of_eq [preorder α] {a b : α} (h : a = b) : a ≥ b := h.ge

@[simp, nolint ge_or_gt] -- see Note [nolint_ge]
lemma ge_iff_le [has_le α] {a b : α} : a ≥ b ↔ b ≤ a := iff.rfl
@[simp, nolint ge_or_gt] -- see Note [nolint_ge]
lemma gt_iff_lt [has_lt α] {a b : α} : a > b ↔ b < a := iff.rfl

lemma not_le_of_lt [preorder α] {a b : α} (h : a < b) : ¬ b ≤ a := (le_not_le_of_lt h).right

alias not_le_of_lt ← has_lt.lt.not_le

lemma not_lt_of_le [preorder α] {a b : α} (h : a ≤ b) : ¬ b < a := λ hba, hba.not_le h

alias not_lt_of_le ← has_le.le.not_lt

lemma ne_of_not_le [preorder α] {a b : α} (h : ¬ a ≤ b) : a ≠ b :=
λ hab, h (le_of_eq hab)

-- See Note [decidable namespace]
protected lemma decidable.le_iff_eq_or_lt [partial_order α] [@decidable_rel α (≤)]
  {a b : α} : a ≤ b ↔ a = b ∨ a < b := decidable.le_iff_lt_or_eq.trans or.comm

lemma le_iff_eq_or_lt [partial_order α] {a b : α} : a ≤ b ↔ a = b ∨ a < b :=
le_iff_lt_or_eq.trans or.comm

lemma lt_iff_le_and_ne [partial_order α] {a b : α} : a < b ↔ a ≤ b ∧ a ≠ b :=
⟨λ h, ⟨le_of_lt h, ne_of_lt h⟩, λ ⟨h1, h2⟩, h1.lt_of_ne h2⟩

-- See Note [decidable namespace]
protected lemma decidable.eq_iff_le_not_lt [partial_order α] [@decidable_rel α (≤)]
  {a b : α} : a = b ↔ a ≤ b ∧ ¬ a < b :=
⟨λ h, ⟨h.le, h ▸ lt_irrefl _⟩, λ ⟨h₁, h₂⟩, h₁.antisymm $
  decidable.by_contradiction $ λ h₃, h₂ (h₁.lt_of_not_le h₃)⟩

lemma eq_iff_le_not_lt [partial_order α] {a b : α} : a = b ↔ a ≤ b ∧ ¬ a < b :=
by haveI := classical.dec; exact decidable.eq_iff_le_not_lt

lemma eq_or_lt_of_le [partial_order α] {a b : α} (h : a ≤ b) : a = b ∨ a < b := h.lt_or_eq.symm
lemma eq_or_gt_of_le [partial_order α] {a b : α} (h : a ≤ b) : b = a ∨ a < b :=
h.lt_or_eq.symm.imp eq.symm id

alias decidable.eq_or_lt_of_le ← has_le.le.eq_or_lt_dec
alias eq_or_lt_of_le ← has_le.le.eq_or_lt
alias eq_or_gt_of_le ← has_le.le.eq_or_gt

attribute [nolint decidable_classical] has_le.le.eq_or_lt_dec

lemma eq_of_le_of_not_lt [partial_order α] {a b : α} (hab : a ≤ b) (hba : ¬ a < b) : a = b :=
hab.eq_or_lt.resolve_right hba

lemma eq_of_ge_of_not_gt [partial_order α] {a b : α} (hab : a ≤ b) (hba : ¬ a < b) : b = a :=
(hab.eq_or_lt.resolve_right hba).symm

alias eq_of_le_of_not_lt ← has_le.le.eq_of_not_lt
alias eq_of_ge_of_not_gt ← has_le.le.eq_of_not_gt

lemma ne.le_iff_lt [partial_order α] {a b : α} (h : a ≠ b) : a ≤ b ↔ a < b :=
⟨λ h', lt_of_le_of_ne h' h, λ h, h.le⟩

lemma ne.not_le_or_not_le [partial_order α] {a b : α} (h : a ≠ b) : ¬ a ≤ b ∨ ¬ b ≤ a :=
not_and_distrib.1 $ le_antisymm_iff.not.1 h

-- See Note [decidable namespace]
protected lemma decidable.ne_iff_lt_iff_le [partial_order α] [decidable_eq α] {a b : α} :
  (a ≠ b ↔ a < b) ↔ a ≤ b :=
⟨λ h, decidable.by_cases le_of_eq (le_of_lt ∘ h.mp), λ h, ⟨lt_of_le_of_ne h, ne_of_lt⟩⟩

@[simp] lemma ne_iff_lt_iff_le [partial_order α] {a b : α} : (a ≠ b ↔ a < b) ↔ a ≤ b :=
by haveI := classical.dec; exact decidable.ne_iff_lt_iff_le

lemma lt_of_not_le [linear_order α] {a b : α} (h : ¬ b ≤ a) : a < b :=
((le_total _ _).resolve_right h).lt_of_not_le h

lemma lt_iff_not_le [linear_order α] {x y : α} : x < y ↔ ¬ y ≤ x := ⟨not_le_of_lt, lt_of_not_le⟩

lemma ne.lt_or_lt [linear_order α] {x y : α} (h : x ≠ y) : x < y ∨ y < x := lt_or_gt_of_ne h

/-- A version of `ne_iff_lt_or_gt` with LHS and RHS reversed. -/
@[simp] lemma lt_or_lt_iff_ne [linear_order α] {x y : α} : x < y ∨ y < x ↔ x ≠ y :=
ne_iff_lt_or_gt.symm

lemma not_lt_iff_eq_or_lt [linear_order α] {a b : α} : ¬ a < b ↔ a = b ∨ b < a :=
not_lt.trans $ decidable.le_iff_eq_or_lt.trans $ or_congr eq_comm iff.rfl

lemma exists_ge_of_linear [linear_order α] (a b : α) : ∃ c, a ≤ c ∧ b ≤ c :=
match le_total a b with
| or.inl h := ⟨_, h, le_rfl⟩
| or.inr h := ⟨_, le_rfl, h⟩
end

lemma lt_imp_lt_of_le_imp_le {β} [linear_order α] [preorder β] {a b : α} {c d : β}
  (H : a ≤ b → c ≤ d) (h : d < c) : b < a :=
lt_of_not_le $ λ h', (H h').not_lt h

lemma le_imp_le_iff_lt_imp_lt {β} [linear_order α] [linear_order β] {a b : α} {c d : β} :
  (a ≤ b → c ≤ d) ↔ (d < c → b < a) :=
⟨lt_imp_lt_of_le_imp_le, le_imp_le_of_lt_imp_lt⟩

lemma lt_iff_lt_of_le_iff_le' {β} [preorder α] [preorder β] {a b : α} {c d : β}
  (H : a ≤ b ↔ c ≤ d) (H' : b ≤ a ↔ d ≤ c) : b < a ↔ d < c :=
lt_iff_le_not_le.trans $ (and_congr H' (not_congr H)).trans lt_iff_le_not_le.symm

lemma lt_iff_lt_of_le_iff_le {β} [linear_order α] [linear_order β] {a b : α} {c d : β}
  (H : a ≤ b ↔ c ≤ d) : b < a ↔ d < c :=
not_le.symm.trans $ (not_congr H).trans $ not_le

lemma le_iff_le_iff_lt_iff_lt {β} [linear_order α] [linear_order β] {a b : α} {c d : β} :
  (a ≤ b ↔ c ≤ d) ↔ (b < a ↔ d < c) :=
⟨lt_iff_lt_of_le_iff_le, λ H, not_lt.symm.trans $ (not_congr H).trans $ not_lt⟩

lemma eq_of_forall_le_iff [partial_order α] {a b : α}
  (H : ∀ c, c ≤ a ↔ c ≤ b) : a = b :=
((H _).1 le_rfl).antisymm ((H _).2 le_rfl)

lemma le_of_forall_le [preorder α] {a b : α}
  (H : ∀ c, c ≤ a → c ≤ b) : a ≤ b :=
H _ le_rfl

lemma le_of_forall_le' [preorder α] {a b : α}
  (H : ∀ c, a ≤ c → b ≤ c) : b ≤ a :=
H _ le_rfl

lemma le_of_forall_lt [linear_order α] {a b : α}
  (H : ∀ c, c < a → c < b) : a ≤ b :=
le_of_not_lt $ λ h, lt_irrefl _ (H _ h)

lemma forall_lt_iff_le [linear_order α] {a b : α} :
  (∀ ⦃c⦄, c < a → c < b) ↔ a ≤ b :=
⟨le_of_forall_lt, λ h c hca, lt_of_lt_of_le hca h⟩

lemma le_of_forall_lt' [linear_order α] {a b : α}
  (H : ∀ c, a < c → b < c) : b ≤ a :=
le_of_not_lt $ λ h, lt_irrefl _ (H _ h)

lemma forall_lt_iff_le' [linear_order α] {a b : α} :
  (∀ ⦃c⦄, a < c → b < c) ↔ b ≤ a :=
⟨le_of_forall_lt', λ h c hac, lt_of_le_of_lt h hac⟩

lemma eq_of_forall_ge_iff [partial_order α] {a b : α}
  (H : ∀ c, a ≤ c ↔ b ≤ c) : a = b :=
((H _).2 le_rfl).antisymm ((H _).1 le_rfl)

/-- A symmetric relation implies two values are equal, when it implies they're less-equal.  -/
lemma rel_imp_eq_of_rel_imp_le [partial_order β] (r : α → α → Prop) [is_symm α r] {f : α → β}
  (h : ∀ a b, r a b → f a ≤ f b) {a b : α} : r a b → f a = f b :=
λ hab, le_antisymm (h a b hab) (h b a $ symm hab)

/-- monotonicity of `≤` with respect to `→` -/
lemma le_implies_le_of_le_of_le {a b c d : α} [preorder α] (hca : c ≤ a) (hbd : b ≤ d) :
  a ≤ b → c ≤ d :=
λ hab, (hca.trans hab).trans hbd

@[ext]
lemma preorder.to_has_le_injective {α : Type*} :
  function.injective (@preorder.to_has_le α) :=
λ A B h, begin
  cases A, cases B,
  injection h with h_le,
  have : A_lt = B_lt,
  { funext a b,
    dsimp [(≤)] at A_lt_iff_le_not_le B_lt_iff_le_not_le h_le,
    simp [A_lt_iff_le_not_le, B_lt_iff_le_not_le, h_le], },
  congr',
end

@[ext]
lemma partial_order.to_preorder_injective {α : Type*} :
  function.injective (@partial_order.to_preorder α) :=
λ A B h, by { cases A, cases B, injection h, congr' }

@[ext]
lemma linear_order.to_partial_order_injective {α : Type*} :
  function.injective (@linear_order.to_partial_order α) :=
begin
  intros A B h,
  cases A, cases B, injection h,
  obtain rfl : A_le = B_le := ‹_›, obtain rfl : A_lt = B_lt := ‹_›,
  obtain rfl : A_decidable_le = B_decidable_le := subsingleton.elim _ _,
  obtain rfl : A_max = B_max := A_max_def.trans B_max_def.symm,
  obtain rfl : A_min = B_min := A_min_def.trans B_min_def.symm,
  congr
end

theorem preorder.ext {α} {A B : preorder α}
  (H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { ext x y, exact H x y }

theorem partial_order.ext {α} {A B : partial_order α}
  (H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { ext x y, exact H x y }

theorem linear_order.ext {α} {A B : linear_order α}
  (H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { ext x y, exact H x y }

/-- Given a relation `R` on `β` and a function `f : α → β`, the preimage relation on `α` is defined
by `x ≤ y ↔ f x ≤ f y`. It is the unique relation on `α` making `f` a `rel_embedding` (assuming `f`
is injective). -/
@[simp] def order.preimage {α β} (f : α → β) (s : β → β → Prop) (x y : α) : Prop := s (f x) (f y)

infix ` ⁻¹'o `:80 := order.preimage

/-- The preimage of a decidable order is decidable. -/
instance order.preimage.decidable {α β} (f : α → β) (s : β → β → Prop) [H : decidable_rel s] :
  decidable_rel (f ⁻¹'o s) :=
λ x y, H _ _

/-! ### Order dual -/

/-- Type synonym to equip a type with the dual order: `≤` means `≥` and `<` means `>`. `αᵒᵈ` is
notation for `order_dual α`. -/
def order_dual (α : Type*) : Type* := α

notation α `ᵒᵈ`:std.prec.max_plus := order_dual α

namespace order_dual

instance (α : Type*) [h : nonempty α] : nonempty αᵒᵈ := h
instance (α : Type*) [h : subsingleton α] : subsingleton αᵒᵈ := h
instance (α : Type*) [has_le α] : has_le αᵒᵈ := ⟨λ x y : α, y ≤ x⟩
instance (α : Type*) [has_lt α] : has_lt αᵒᵈ := ⟨λ x y : α, y < x⟩
instance (α : Type*) [has_zero α] : has_zero αᵒᵈ := ⟨(0 : α)⟩

instance (α : Type*) [preorder α] : preorder αᵒᵈ :=
{ le_refl          := le_refl,
  le_trans         := λ a b c hab hbc, hbc.trans hab,
  lt_iff_le_not_le := λ _ _, lt_iff_le_not_le,
  .. order_dual.has_le α,
  .. order_dual.has_lt α }

instance (α : Type*) [partial_order α] : partial_order αᵒᵈ :=
{ le_antisymm := λ a b hab hba, @le_antisymm α _ a b hba hab, .. order_dual.preorder α }

instance (α : Type*) [linear_order α] : linear_order αᵒᵈ :=
{ le_total     := λ a b : α, le_total b a,
  decidable_le := (infer_instance : decidable_rel (λ a b : α, b ≤ a)),
  decidable_lt := (infer_instance : decidable_rel (λ a b : α, b < a)),
  min := @max α _,
  max := @min α _,
  min_def := @linear_order.max_def α _,
  max_def := @linear_order.min_def α _,
  .. order_dual.partial_order α }

instance : Π [inhabited α], inhabited αᵒᵈ := id

theorem preorder.dual_dual (α : Type*) [H : preorder α] :
  order_dual.preorder αᵒᵈ = H :=
preorder.ext $ λ _ _, iff.rfl

theorem partial_order.dual_dual (α : Type*) [H : partial_order α] :
  order_dual.partial_order αᵒᵈ = H :=
partial_order.ext $ λ _ _, iff.rfl

theorem linear_order.dual_dual (α : Type*) [H : linear_order α] :
  order_dual.linear_order αᵒᵈ = H :=
linear_order.ext $ λ _ _, iff.rfl

end order_dual

/-! ### `has_compl` -/

/-- Set / lattice complement -/
@[notation_class] class has_compl (α : Type*) := (compl : α → α)

export has_compl (compl)

postfix `ᶜ`:(max+1) := compl

instance Prop.has_compl : has_compl Prop := ⟨not⟩

instance pi.has_compl {ι : Type u} {α : ι → Type v} [∀ i, has_compl (α i)] :
  has_compl (Π i, α i) :=
⟨λ x i, (x i)ᶜ⟩

lemma pi.compl_def {ι : Type u} {α : ι → Type v} [∀ i, has_compl (α i)] (x : Π i, α i) :
  xᶜ = λ i, (x i)ᶜ := rfl

@[simp]
lemma pi.compl_apply {ι : Type u} {α : ι → Type v} [∀ i, has_compl (α i)] (x : Π i, α i) (i : ι)  :
  xᶜ i = (x i)ᶜ := rfl

instance is_irrefl.compl (r) [is_irrefl α r] : is_refl α rᶜ := ⟨@irrefl α r _⟩
instance is_refl.compl (r) [is_refl α r] : is_irrefl α rᶜ := ⟨λ a, not_not_intro (refl a)⟩

/-! ### Order instances on the function space -/

instance pi.has_le {ι : Type u} {α : ι → Type v} [∀ i, has_le (α i)] : has_le (Π i, α i) :=
{ le       := λ x y, ∀ i, x i ≤ y i }

lemma pi.le_def {ι : Type u} {α : ι → Type v} [∀ i, has_le (α i)] {x y : Π i, α i} :
  x ≤ y ↔ ∀ i, x i ≤ y i :=
iff.rfl

instance pi.preorder {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] : preorder (Π i, α i) :=
{ le_refl  := λ a i, le_refl (a i),
  le_trans := λ a b c h₁ h₂ i, le_trans (h₁ i) (h₂ i),
  ..pi.has_le }

lemma pi.lt_def {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] {x y : Π i, α i} :
  x < y ↔ x ≤ y ∧ ∃ i, x i < y i :=
by simp [lt_iff_le_not_le, pi.le_def] {contextual := tt}

lemma le_update_iff {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] [decidable_eq ι]
  {x y : Π i, α i} {i : ι} {a : α i} :
  x ≤ function.update y i a ↔ x i ≤ a ∧ ∀ j ≠ i, x j ≤ y j :=
function.forall_update_iff _ (λ j z, x j ≤ z)

lemma update_le_iff {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] [decidable_eq ι]
  {x y : Π i, α i} {i : ι} {a : α i} :
  function.update x i a ≤ y ↔ a ≤ y i ∧ ∀ j ≠ i, x j ≤ y j :=
function.forall_update_iff _ (λ j z, z ≤ y j)

lemma update_le_update_iff {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] [decidable_eq ι]
  {x y : Π i, α i} {i : ι} {a b : α i} :
  function.update x i a ≤ function.update y i b ↔ a ≤ b ∧ ∀ j ≠ i, x j ≤ y j :=
by simp [update_le_iff] {contextual := tt}

instance pi.partial_order {ι : Type u} {α : ι → Type v} [∀ i, partial_order (α i)] :
  partial_order (Π i, α i) :=
{ le_antisymm := λ f g h1 h2, funext (λ b, (h1 b).antisymm (h2 b)),
  ..pi.preorder }

instance pi.has_sdiff {ι : Type u} {α : ι → Type v} [∀ i, has_sdiff (α i)] :
  has_sdiff (Π i, α i) :=
⟨λ x y i, x i \ y i⟩

lemma pi.sdiff_def {ι : Type u} {α : ι → Type v} [∀ i, has_sdiff (α i)] (x y : Π i, α i) :
  (x \ y) = λ i, x i \ y i := rfl

@[simp]
lemma pi.sdiff_apply {ι : Type u} {α : ι → Type v} [∀ i, has_sdiff (α i)] (x y : Π i, α i) (i : ι) :
  (x \ y) i = x i \ y i := rfl

/-! ### `min`/`max` recursors -/

section min_max_rec

variables [linear_order α] {p : α → Prop} {x y : α}

lemma min_rec (hx : x ≤ y → p x) (hy : y ≤ x → p y) : p (min x y) :=
(le_total x y).rec (λ h, (min_eq_left h).symm.subst (hx h))
  (λ h, (min_eq_right h).symm.subst (hy h))

lemma max_rec (hx : y ≤ x → p x) (hy : x ≤ y → p y) : p (max x y) := @min_rec αᵒᵈ _ _ _ _ hx hy
lemma min_rec' (p : α → Prop) (hx : p x) (hy : p y) : p (min x y) := min_rec (λ _, hx) (λ _, hy)
lemma max_rec' (p : α → Prop) (hx : p x) (hy : p y) : p (max x y) := max_rec (λ _, hx) (λ _, hy)

end min_max_rec

/-! ### `has_sup` and `has_inf` -/

/-- Typeclass for the `⊔` (`\lub`) notation -/
@[notation_class] class has_sup (α : Type u) := (sup : α → α → α)
/-- Typeclass for the `⊓` (`\glb`) notation -/
@[notation_class] class has_inf (α : Type u) := (inf : α → α → α)

infix ⊔ := has_sup.sup
infix ⊓ := has_inf.inf

/-! ### Lifts of order instances -/

/-- Transfer a `preorder` on `β` to a `preorder` on `α` using a function `f : α → β`.
See note [reducible non-instances]. -/
@[reducible] def preorder.lift {α β} [preorder β] (f : α → β) : preorder α :=
{ le               := λ x y, f x ≤ f y,
  le_refl          := λ a, le_rfl,
  le_trans         := λ a b c, le_trans,
  lt               := λ x y, f x < f y,
  lt_iff_le_not_le := λ a b, lt_iff_le_not_le }

/-- Transfer a `partial_order` on `β` to a `partial_order` on `α` using an injective
function `f : α → β`. See note [reducible non-instances]. -/
@[reducible] def partial_order.lift {α β} [partial_order β] (f : α → β) (inj : injective f) :
  partial_order α :=
{ le_antisymm := λ a b h₁ h₂, inj (h₁.antisymm h₂), .. preorder.lift f }

/-- Transfer a `linear_order` on `β` to a `linear_order` on `α` using an injective
function `f : α → β`. This version takes `[has_sup α]` and `[has_inf α]` as arguments, then uses
them for `max` and `min` fields. See `linear_order.lift'` for a version that autogenerates `min` and
`max` fields. See note [reducible non-instances]. -/
@[reducible] def linear_order.lift {α β} [linear_order β] [has_sup α] [has_inf α] (f : α → β)
  (inj : injective f) (hsup : ∀ x y, f (x ⊔ y) = max (f x) (f y))
  (hinf : ∀ x y, f (x ⊓ y) = min (f x) (f y)) :
  linear_order α :=
{ le_total     := λ x y, le_total (f x) (f y),
  decidable_le := λ x y, (infer_instance : decidable (f x ≤ f y)),
  decidable_lt := λ x y, (infer_instance : decidable (f x < f y)),
  decidable_eq := λ x y, decidable_of_iff (f x = f y) inj.eq_iff,
  min := (⊓),
  max := (⊔),
  min_def := by { ext x y, apply inj, rw [hinf, min_def, min_default, apply_ite f], refl },
  max_def := by { ext x y, apply inj, rw [hsup, max_def, max_default, apply_ite f], refl },
  .. partial_order.lift f inj }

/-- Transfer a `linear_order` on `β` to a `linear_order` on `α` using an injective
function `f : α → β`. This version autogenerates `min` and `max` fields. See `linear_order.lift`
for a version that takes `[has_sup α]` and `[has_inf α]`, then uses them as `max` and `min`.
See note [reducible non-instances]. -/
@[reducible] def linear_order.lift' {α β} [linear_order β] (f : α → β) (inj : injective f) :
  linear_order α :=
@linear_order.lift α β _ ⟨λ x y, if f y ≤ f x then x else y⟩ ⟨λ x y, if f x ≤ f y then x else y⟩
  f inj (λ x y, (apply_ite f _ _ _).trans (max_def _ _).symm)
  (λ x y, (apply_ite f _ _ _).trans (min_def _ _).symm)

/-! ### Subtype of an order -/

namespace subtype

instance [has_le α] {p : α → Prop} : has_le (subtype p) := ⟨λ x y, (x : α) ≤ y⟩
instance [has_lt α] {p : α → Prop} : has_lt (subtype p) := ⟨λ x y, (x : α) < y⟩

@[simp] lemma mk_le_mk [has_le α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :
  (⟨x, hx⟩ : subtype p) ≤ ⟨y, hy⟩ ↔ x ≤ y :=
iff.rfl

@[simp] lemma mk_lt_mk [has_lt α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :
  (⟨x, hx⟩ : subtype p) < ⟨y, hy⟩ ↔ x < y :=
iff.rfl

@[simp, norm_cast]
lemma coe_le_coe [has_le α] {p : α → Prop} {x y : subtype p} : (x : α) ≤ y ↔ x ≤ y := iff.rfl

@[simp, norm_cast]
lemma coe_lt_coe [has_lt α] {p : α → Prop} {x y : subtype p} : (x : α) < y ↔ x < y := iff.rfl

instance [preorder α] (p : α → Prop) : preorder (subtype p) := preorder.lift (coe : subtype p → α)

instance partial_order [partial_order α] (p : α → Prop) :
  partial_order (subtype p) :=
partial_order.lift coe subtype.coe_injective

instance decidable_le [preorder α] [h : @decidable_rel α (≤)] {p : α → Prop} :
  @decidable_rel (subtype p) (≤) :=
λ a b, h a b

instance decidable_lt [preorder α] [h : @decidable_rel α (<)] {p : α → Prop} :
  @decidable_rel (subtype p) (<) :=
λ a b, h a b

/-- A subtype of a linear order is a linear order. We explicitly give the proofs of decidable
equality and decidable order in order to ensure the decidability instances are all definitionally
equal. -/
instance [linear_order α] (p : α → Prop) : linear_order (subtype p) :=
@linear_order.lift (subtype p) _ _ ⟨λ x y, ⟨max x y, max_rec' _ x.2 y.2⟩⟩
  ⟨λ x y, ⟨min x y, min_rec' _ x.2 y.2⟩⟩ coe subtype.coe_injective (λ _ _, rfl) (λ _ _, rfl)

end subtype

/-!
### Pointwise order on `α × β`

The lexicographic order is defined in `data.prod.lex`, and the instances are available via the
type synonym `α ×ₗ β = α × β`.
-/

namespace prod

instance (α : Type u) (β : Type v) [has_le α] [has_le β] : has_le (α × β) :=
⟨λ p q, p.1 ≤ q.1 ∧ p.2 ≤ q.2⟩

lemma le_def [has_le α] [has_le β] {x y : α × β} : x ≤ y ↔ x.1 ≤ y.1 ∧ x.2 ≤ y.2 := iff.rfl

@[simp] lemma mk_le_mk [has_le α] [has_le β] {x₁ x₂ : α} {y₁ y₂ : β} :
  (x₁, y₁) ≤ (x₂, y₂) ↔ x₁ ≤ x₂ ∧ y₁ ≤ y₂ :=
iff.rfl

@[simp] lemma swap_le_swap [has_le α] [has_le β] {x y : α × β} : x.swap ≤ y.swap ↔ x ≤ y :=
and_comm _ _

section preorder
variables [preorder α] [preorder β] {a a₁ a₂ : α} {b b₁ b₂ : β} {x y : α × β}

instance (α : Type u) (β : Type v) [preorder α] [preorder β] : preorder (α × β) :=
{ le_refl  := λ ⟨a, b⟩, ⟨le_refl a, le_refl b⟩,
  le_trans := λ ⟨a, b⟩ ⟨c, d⟩ ⟨e, f⟩ ⟨hac, hbd⟩ ⟨hce, hdf⟩,
    ⟨le_trans hac hce, le_trans hbd hdf⟩,
  .. prod.has_le α β }

@[simp] lemma swap_lt_swap : x.swap < y.swap ↔ x < y :=
and_congr swap_le_swap (not_congr swap_le_swap)

lemma mk_le_mk_iff_left : (a₁, b) ≤ (a₂, b) ↔ a₁ ≤ a₂ := and_iff_left le_rfl
lemma mk_le_mk_iff_right : (a, b₁) ≤ (a, b₂) ↔ b₁ ≤ b₂ := and_iff_right le_rfl

lemma mk_lt_mk_iff_left : (a₁, b) < (a₂, b) ↔ a₁ < a₂ :=
lt_iff_lt_of_le_iff_le' mk_le_mk_iff_left mk_le_mk_iff_left

lemma mk_lt_mk_iff_right : (a, b₁) < (a, b₂) ↔ b₁ < b₂ :=
lt_iff_lt_of_le_iff_le' mk_le_mk_iff_right mk_le_mk_iff_right

lemma lt_iff : x < y ↔ x.1 < y.1 ∧ x.2 ≤ y.2 ∨ x.1 ≤ y.1 ∧ x.2 < y.2 :=
begin
  refine ⟨λ h, _, _⟩,
  { by_cases h₁ : y.1 ≤ x.1,
    { exact or.inr ⟨h.1.1, h.1.2.lt_of_not_le $ λ h₂, h.2 ⟨h₁, h₂⟩⟩ },
    { exact or.inl ⟨h.1.1.lt_of_not_le h₁, h.1.2⟩ } },
  { rintro (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩),
    { exact ⟨⟨h₁.le, h₂⟩, λ h, h₁.not_le h.1⟩ },
    { exact ⟨⟨h₁, h₂.le⟩, λ h, h₂.not_le h.2⟩ } }
end

@[simp] lemma mk_lt_mk : (a₁, b₁) < (a₂, b₂) ↔ a₁ < a₂ ∧ b₁ ≤ b₂ ∨ a₁ ≤ a₂ ∧ b₁ < b₂ := lt_iff

end preorder

/-- The pointwise partial order on a product.
    (The lexicographic ordering is defined in order/lexicographic.lean, and the instances are
    available via the type synonym `α ×ₗ β = α × β`.) -/
instance (α : Type u) (β : Type v) [partial_order α] [partial_order β] :
  partial_order (α × β) :=
{ le_antisymm := λ ⟨a, b⟩ ⟨c, d⟩ ⟨hac, hbd⟩ ⟨hca, hdb⟩,
    prod.ext (hac.antisymm hca) (hbd.antisymm hdb),
  .. prod.preorder α β }

end prod

/-! ### Additional order classes -/

/-- An order is dense if there is an element between any pair of distinct elements. -/
class densely_ordered (α : Type u) [has_lt α] : Prop :=
(dense : ∀ a₁ a₂ : α, a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂)

lemma exists_between [has_lt α] [densely_ordered α] :
  ∀ {a₁ a₂ : α}, a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂ :=
densely_ordered.dense

instance order_dual.densely_ordered (α : Type u) [has_lt α] [densely_ordered α] :
  densely_ordered αᵒᵈ :=
⟨λ a₁ a₂ ha, (@exists_between α _ _ _ _ ha).imp $ λ a, and.symm⟩

lemma le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
  (h : ∀ a, a₂ < a → a₁ ≤ a) :
  a₁ ≤ a₂ :=
le_of_not_gt $ λ ha,
  let ⟨a, ha₁, ha₂⟩ := exists_between ha in
  lt_irrefl a $ lt_of_lt_of_le ‹a < a₁› (h _ ‹a₂ < a›)

lemma eq_of_le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
  (h₁ : a₂ ≤ a₁) (h₂ : ∀ a, a₂ < a → a₁ ≤ a) : a₁ = a₂ :=
le_antisymm (le_of_forall_le_of_dense h₂) h₁

lemma le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
  (h : ∀ a₃ < a₁, a₃ ≤ a₂) :
  a₁ ≤ a₂ :=
le_of_not_gt $ λ ha,
  let ⟨a, ha₁, ha₂⟩ := exists_between ha in
  lt_irrefl a $ lt_of_le_of_lt (h _ ‹a < a₁›) ‹a₂ < a›

lemma eq_of_le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
  (h₁ : a₂ ≤ a₁) (h₂ : ∀ a₃ < a₁, a₃ ≤ a₂) : a₁ = a₂ :=
(le_of_forall_ge_of_dense h₂).antisymm h₁

lemma dense_or_discrete [linear_order α] (a₁ a₂ : α) :
  (∃ a, a₁ < a ∧ a < a₂) ∨ ((∀ a, a₁ < a → a₂ ≤ a) ∧ (∀ a < a₂, a ≤ a₁)) :=
or_iff_not_imp_left.2 $ λ h,
  ⟨λ a ha₁, le_of_not_gt $ λ ha₂, h ⟨a, ha₁, ha₂⟩,
    λ a ha₂, le_of_not_gt $ λ ha₁, h ⟨a, ha₁, ha₂⟩⟩

namespace punit
variables (a b : punit.{u+1})

instance : linear_order punit :=
by refine_struct
{ le := λ _ _, true,
  lt := λ _ _, false,
  max := λ _ _, star,
  min := λ _ _, star,
  decidable_eq := punit.decidable_eq,
  decidable_le := λ _ _, decidable.true,
  decidable_lt := λ _ _, decidable.false };
    intros; trivial <|> simp only [eq_iff_true_of_subsingleton, not_true, and_false] <|>
      exact or.inl trivial

lemma max_eq : max a b = star := rfl
lemma min_eq : min a b = star := rfl
@[simp] protected lemma le : a ≤ b := trivial
@[simp] lemma not_lt : ¬ a < b := not_false

instance : densely_ordered punit := ⟨λ _ _, false.elim⟩

end punit

variables {s : β → β → Prop} {t : γ → γ → Prop}

/-! ### Linear order from a total partial order -/

/-- Type synonym to create an instance of `linear_order` from a `partial_order` and
`is_total α (≤)` -/
def as_linear_order (α : Type u) := α

instance {α} [inhabited α] : inhabited (as_linear_order α) :=
⟨ (default : α) ⟩

noncomputable instance as_linear_order.linear_order {α} [partial_order α] [is_total α (≤)] :
  linear_order (as_linear_order α) :=
{ le_total     := @total_of α (≤) _,
  decidable_le := classical.dec_rel _,
  .. (_ : partial_order α) }