mathlib documentation

data.W

W types #

Given α : Type and β : α → Type, the W type determined by this data, W_type β, is the inductively defined type of trees where the nodes are labeled by elements of α and the children of a node labeled a are indexed by elements of β a.

This file is currently a stub, awaiting a full development of the theory. Currently, the main result is that if α is an encodable fintype and β a is encodable for every a : α, then W_type β is encodable. This can be used to show the encodability of other inductive types, such as those that are commonly used to formalize syntax, e.g. terms and expressions in a given language. The strategy is illustrated in the example found in the file prop_encodable in the archive/examples folder of mathlib.

Implementation details #

While the name W_type is somewhat verbose, it is preferable to putting a single character identifier W in the root namespace.

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inductive W_type {α : Type u_1} (β : α → Type u_2) :
Type (max u_1 u_2)
  • mk : Π {α : Type u_1} (β : α → Type u_2) (a : α), (β a → W_type β) → W_type β

Given β : α → Type*, W_type β is the type of finitely branching trees where nodes are labeled by elements of α and the children of a node labeled a are indexed by elements of β a.

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@[protected, instance]
def W_type.inhabited  :
inhabited (W_type (λ (_x : unit), empty))
Equations
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def W_type.depth {α : Type u_1} {β : α → Type u_2} [Π (a : α), fintype (β a)] :
W_type β → ℕ

The depth of a finitely branching tree.

Equations
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theorem W_type.depth_pos {α : Type u_1} {β : α → Type u_2} [Π (a : α), fintype (β a)] (t : W_type β) :
0 < t.depth
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theorem W_type.depth_lt_depth_mk {α : Type u_1} {β : α → Type u_2} [Π (a : α), fintype (β a)] (a : α) (f : β a → W_type β) (i : β a) :
(f i).depth < (W_type.mk a f).depth
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@[protected, instance]
def encodable.W_type.encodable {α : Type u_1} {β : α → Type u_2} [Π (a : α), fintype (β a)] [Π (a : α), encodable (β a)] [encodable α] :
encodable (W_type β)

W_type is encodable when α is an encodable fintype and for every a : α, β a is encodable.

Equations