Documentation

Mathlib.Order.WellFounded

Well-founded relations #

A relation is well-founded if it can be used for induction: for each x, (∀ y, r y x → P y) → P x implies P x. Well-founded relations can be used for induction and recursion, including construction of fixed points in the space of dependent functions Π x : α , β x.

The predicate WellFounded is defined in the core library. In this file we prove some extra lemmas and provide a few new definitions: WellFounded.min, WellFounded.sup, and WellFounded.succ, and an induction principle WellFounded.induction_bot.

theorem WellFounded.isAsymm {α : Type u_1} {r : ααProp} (h : WellFounded r) :
IsAsymm α r
theorem WellFounded.isIrrefl {α : Type u_1} {r : ααProp} (h : WellFounded r) :
theorem WellFounded.mono {α : Type u_1} {r : ααProp} {r' : ααProp} (hr : WellFounded r) (h : (a b : α) → r' a br a b) :
theorem WellFounded.onFun {α : Sort u_4} {β : Sort u_5} {r : ββProp} {f : αβ} :
theorem WellFounded.has_min {α : Type u_4} {r : ααProp} (H : WellFounded r) (s : Set α) :
Set.Nonempty sa, a s ∀ (x : α), x s¬r x a

If r is a well-founded relation, then any nonempty set has a minimal element with respect to r.

noncomputable def WellFounded.min {α : Type u_1} {r : ααProp} (H : WellFounded r) (s : Set α) (h : Set.Nonempty s) :
α

A minimal element of a nonempty set in a well-founded order.

If you're working with a nonempty linear order, consider defining a ConditionallyCompleteLinearOrderBot instance via WellFounded.conditionallyCompleteLinearOrderWithBot and using Inf instead.

Equations
Instances For
    theorem WellFounded.min_mem {α : Type u_1} {r : ααProp} (H : WellFounded r) (s : Set α) (h : Set.Nonempty s) :
    theorem WellFounded.not_lt_min {α : Type u_1} {r : ααProp} (H : WellFounded r) (s : Set α) (h : Set.Nonempty s) {x : α} (hx : x s) :
    ¬r x (WellFounded.min H s h)
    theorem WellFounded.wellFounded_iff_has_min {α : Type u_1} {r : ααProp} :
    WellFounded r ∀ (s : Set α), Set.Nonempty sm, m s ∀ (x : α), x s¬r x m
    noncomputable def WellFounded.sup {α : Type u_1} {r : ααProp} (wf : WellFounded r) (s : Set α) (h : Set.Bounded r s) :
    α

    The supremum of a bounded, well-founded order

    Equations
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      theorem WellFounded.lt_sup {α : Type u_1} {r : ααProp} (wf : WellFounded r) {s : Set α} (h : Set.Bounded r s) {x : α} (hx : x s) :
      r x (WellFounded.sup wf s h)
      noncomputable def WellFounded.succ {α : Type u_1} {r : ααProp} (wf : WellFounded r) (x : α) :
      α

      A successor of an element x in a well-founded order is a minimal element y such that x < y if one exists. Otherwise it is x itself.

      Equations
      Instances For
        theorem WellFounded.lt_succ {α : Type u_1} {r : ααProp} (wf : WellFounded r) {x : α} (h : y, r x y) :
        r x (WellFounded.succ wf x)
        theorem WellFounded.lt_succ_iff {α : Type u_1} {r : ααProp} [wo : IsWellOrder α r] {x : α} (h : y, r x y) (y : α) :
        r y (WellFounded.succ (_ : WellFounded r) x) r y x y = x
        theorem WellFounded.min_le {β : Type u_2} [LinearOrder β] (h : WellFounded fun x x_1 => x < x_1) {x : β} {s : Set β} (hx : x s) (hne : optParam (Set.Nonempty s) (_ : x, x s)) :
        theorem WellFounded.eq_strictMono_iff_eq_range {β : Type u_2} {γ : Type u_3} [LinearOrder β] (h : WellFounded fun x x_1 => x < x_1) [PartialOrder γ] {f : βγ} {g : βγ} (hf : StrictMono f) (hg : StrictMono g) :
        theorem WellFounded.self_le_of_strictMono {β : Type u_2} [LinearOrder β] (h : WellFounded fun x x_1 => x < x_1) {f : ββ} (hf : StrictMono f) (n : β) :
        n f n
        noncomputable def Function.argmin {α : Type u_1} {β : Type u_2} (f : αβ) [LT β] (h : WellFounded fun x x_1 => x < x_1) [Nonempty α] :
        α

        Given a function f : α → β where β carries a well-founded <, this is an element of α whose image under f is minimal in the sense of Function.not_lt_argmin.

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          theorem Function.not_lt_argmin {α : Type u_1} {β : Type u_2} (f : αβ) [LT β] (h : WellFounded fun x x_1 => x < x_1) [Nonempty α] (a : α) :
          ¬f a < f (Function.argmin f h)
          noncomputable def Function.argminOn {α : Type u_1} {β : Type u_2} (f : αβ) [LT β] (h : WellFounded fun x x_1 => x < x_1) (s : Set α) (hs : Set.Nonempty s) :
          α

          Given a function f : α → β where β carries a well-founded <, and a non-empty subset s of α, this is an element of s whose image under f is minimal in the sense of Function.not_lt_argminOn.

          Equations
          Instances For
            @[simp]
            theorem Function.argminOn_mem {α : Type u_1} {β : Type u_2} (f : αβ) [LT β] (h : WellFounded fun x x_1 => x < x_1) (s : Set α) (hs : Set.Nonempty s) :
            theorem Function.not_lt_argminOn {α : Type u_1} {β : Type u_2} (f : αβ) [LT β] (h : WellFounded fun x x_1 => x < x_1) (s : Set α) {a : α} (ha : a s) (hs : optParam (Set.Nonempty s) (_ : Set.Nonempty s)) :
            ¬f a < f (Function.argminOn f h s hs)
            theorem Function.argmin_le {α : Type u_1} {β : Type u_2} (f : αβ) [LinearOrder β] (h : WellFounded fun x x_1 => x < x_1) (a : α) [Nonempty α] :
            f (Function.argmin f h) f a
            theorem Function.argminOn_le {α : Type u_1} {β : Type u_2} (f : αβ) [LinearOrder β] (h : WellFounded fun x x_1 => x < x_1) (s : Set α) {a : α} (ha : a s) (hs : optParam (Set.Nonempty s) (_ : Set.Nonempty s)) :
            f (Function.argminOn f h s hs) f a
            theorem Acc.induction_bot' {α : Sort u_4} {β : Sort u_5} {r : ααProp} {a : α} {bot : α} (ha : Acc r a) {C : βProp} {f : αβ} (ih : ∀ (b : α), f b f botC (f b)c, r c b C (f c)) :
            C (f a)C (f bot)

            Let r be a relation on α, let f : α → β be a function, let C : β → Prop, and let bot : α. This induction principle shows that C (f bot) holds, given that

            • some a that is accessible by r satisfies C (f a), and
            • for each b such that f b ≠ f bot and C (f b) holds, there is c satisfying r c b and C (f c).
            theorem Acc.induction_bot {α : Sort u_4} {r : ααProp} {a : α} {bot : α} (ha : Acc r a) {C : αProp} (ih : ∀ (b : α), b botC bc, r c b C c) :
            C aC bot

            Let r be a relation on α, let C : α → Prop and let bot : α. This induction principle shows that C bot holds, given that

            • some a that is accessible by r satisfies C a, and
            • for each b ≠ bot such that C b holds, there is c satisfying r c b and C c.
            theorem WellFounded.induction_bot' {α : Sort u_4} {β : Sort u_5} {r : ααProp} (hwf : WellFounded r) {a : α} {bot : α} {C : βProp} {f : αβ} (ih : ∀ (b : α), f b f botC (f b)c, r c b C (f c)) :
            C (f a)C (f bot)

            Let r be a well-founded relation on α, let f : α → β be a function, let C : β → Prop, and let bot : α. This induction principle shows that C (f bot) holds, given that

            • some a satisfies C (f a), and
            • for each b such that f b ≠ f bot and C (f b) holds, there is c satisfying r c b and C (f c).
            theorem WellFounded.induction_bot {α : Sort u_4} {r : ααProp} (hwf : WellFounded r) {a : α} {bot : α} {C : αProp} (ih : ∀ (b : α), b botC bc, r c b C c) :
            C aC bot

            Let r be a well-founded relation on α, let C : α → Prop, and let bot : α. This induction principle shows that C bot holds, given that

            • some a satisfies C a, and
            • for each b that satisfies C b, there is c satisfying r c b and C c.

            The naming is inspired by the fact that when r is transitive, it follows that bot is the smallest element w.r.t. r that satisfies C.