mathlib3 documentation

data.W.cardinal

Cardinality of W-types #

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This file proves some theorems about the cardinality of W-types. The main result is cardinal_mk_le_max_aleph_0_of_fintype which says that if for any a : α, β a is finite, then the cardinality of W_type β is at most the maximum of the cardinality of α and ℵ₀. This can be used to prove theorems about the cardinality of algebraic constructions such as polynomials. There is a surjection from a W_type to mv_polynomial for example, and this surjection can be used to put an upper bound on the cardinality of mv_polynomial.

Tags #

W, W type, cardinal, first order

theorem W_type.cardinal_mk_eq_sum {α : Type u} {β : α Type u} :
theorem W_type.cardinal_mk_le_of_le {α : Type u} {β : α Type u} {κ : cardinal} (hκ : cardinal.sum (λ (a : α), κ ^ cardinal.mk (β a)) κ) :

#(W_type β) is the least cardinal κ such that sum (λ a : α, κ ^ #(β a)) ≤ κ

If, for any a : α, β a is finite, then the cardinality of W_type β is at most the maximum of the cardinality of α and ℵ₀