src/HOL/Library/Cardinality.thy
 author wenzelm Wed, 06 Apr 2011 13:33:46 +0200 changeset 42247 12fe41a92cd5 parent 42245 29e3967550d5 child 44142 8e27e0177518 permissions -rw-r--r--
typed_print_translation: discontinued show_sorts argument;
```
(*  Title:      HOL/Library/Cardinality.thy
Author:     Brian Huffman
*)

header {* Cardinality of types *}

theory Cardinality
imports Main
begin

subsection {* Preliminary lemmas *}
(* These should be moved elsewhere *)

lemma (in type_definition) univ:
"UNIV = Abs ` A"
proof
show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
show "UNIV \<subseteq> Abs ` A"
proof
fix x :: 'b
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x \<in> A" by (rule Rep)
ultimately show "x \<in> Abs ` A" by (rule image_eqI)
qed
qed

lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
by (simp add: univ card_image inj_on_def Abs_inject)

subsection {* Cardinalities of types *}

syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")

translations "CARD('t)" => "CONST card (CONST UNIV \<Colon> 't set)"

let
fun card_univ_tr' ctxt _ [Const (@{const_syntax UNIV}, Type (_, [T, _]))] =
Syntax.const @{syntax_const "_type_card"} \$ Syntax_Phases.term_of_typ ctxt T;
in [(@{const_syntax card}, card_univ_tr')] end
*}

lemma card_unit [simp]: "CARD(unit) = 1"
unfolding UNIV_unit by simp

lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a::finite) * CARD('b::finite)"
unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)

lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)

lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
unfolding UNIV_option_conv
apply (subgoal_tac "(None::'a option) \<notin> range Some")
apply fast
done

lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
unfolding Pow_UNIV [symmetric]
by (simp only: card_Pow finite numeral_2_eq_2)

lemma card_nat [simp]: "CARD(nat) = 0"

subsection {* Classes with at least 1 and 2  *}

text {* Class finite already captures "at least 1" *}

lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
unfolding neq0_conv [symmetric] by simp

lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"

text {* Class for cardinality "at least 2" *}

class card2 = finite +
assumes two_le_card: "2 \<le> CARD('a)"

lemma one_less_card: "Suc 0 < CARD('a::card2)"
using two_le_card [where 'a='a] by simp

lemma one_less_int_card: "1 < int CARD('a::card2)"
using one_less_card [where 'a='a] by simp

end
```