Fixed point

This module defines a generic fix operator for defining recursive computations that are not necessarily well-founded or productive. An instance is defined for Part.

Main definition

• class Fix
• Part.fix

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class Fix (α : Type u_1) : Type u_1

• fix f represents the computation of a fixed point for f.

  fix : (α → α) → α

Fix α provides a fix operator to define recursive computatiation via the fixed point of function of type α → α→ α.

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def Part.Fix.approx {α : Type u_1} {β : α → Type u_2} (f : ((a : α) → Part (β a)) → (a : α) → Part (β a)) : Stream' ((a : α) → Part (β a))

A series of successive, finite approximation of the fixed point of f, defined by approx f n = f^[n] ⊥⊥. The limit of this chain is the fixed point of f.

Equations

• Part.Fix.approx f 0 = ⊥
• Part.Fix.approx f (Nat.succ i) = f (Part.Fix.approx f i)

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def Part.fixAux {α : Type u_1} {β : α → Type u_2} (f : ((a : α) → Part (β a)) → (a : α) → Part (β a)) {p : ℕ → Prop} (i : Nat.Upto p) (g : (j : Nat.Upto p) → i < j → (a : α) → Part (β a)) (a : α) : Part (β a)

loop body for finding the fixed point of f

Equations

• Part.fixAux f i g = f fun x => Part.assert (¬p ↑i) fun h => g (Nat.Upto.succ i h) (_ : ↑i < Nat.succ ↑i) x

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noncomputable def Part.fix {α : Type u_1} {β : α → Type u_2} (f : ((a : α) → Part (β a)) → (a : α) → Part (β a)) (x : α) : Part (β x)

The least fixed point of f.

If f is a continuous function (according to complete partial orders), it satisfies the equations:

1. fix f = f (fix f) (is a fixed point)
2. ∀ X, f X ≤ X → fix f ≤ X∀ X, f X ≤ X → fix f ≤ X≤ X → fix f ≤ X→ fix f ≤ X≤ X (least fixed point)

Equations

• Part.fix f x = Part.assert (∃ i, (Part.Fix.approx f i x).Dom) fun h => WellFounded.fix (_ : WellFounded (Nat.Upto.GT fun x => (Part.Fix.approx f x x).Dom)) (Part.fixAux f) Nat.Upto.zero x

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theorem Part.fix_def {α : Type u_2} {β : α → Type u_1} (f : ((a : α) → Part (β a)) → (a : α) → Part (β a)) {x : α} (h' : ∃ i, (Part.Fix.approx f i x).Dom) : Part.fix f x = Part.Fix.approx f (Nat.succ (Nat.find h')) x

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theorem Part.fix_def' {α : Type u_2} {β : α → Type u_1} (f : ((a : α) → Part (β a)) → (a : α) → Part (β a)) {x : α} (h' : ¬∃ i, (Part.Fix.approx f i x).Dom) : Part.fix f x = Part.none

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noncomputable instance Part.instFixPart {α : Type u_1} : Fix (Part α)

Equations

• Part.instFixPart = { fix := fun f => Part.fix (fun x u => f (x u)) () }

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noncomputable instance Pi.Part.hasFix {α : Type u_1} {β : Type u_2} : Fix (α → Part β)

Equations

• Pi.Part.hasFix = { fix := Part.fix }