# Theory HOL-Library.Type_Length

```(*  Title:      HOL/Library/Type_Length.thy
Author:     John Matthews, Galois Connections, Inc., Copyright 2006
*)

section ‹Assigning lengths to types by type classes›

theory Type_Length
imports Numeral_Type
begin

text ‹
The aim of this is to allow any type as index type, but to provide a
default instantiation for numeral types. This independence requires
some duplication with the definitions in 🗏‹Numeral_Type.thy›.
›

class len0 =
fixes len_of :: "'a itself ⇒ nat"

syntax "_type_length" :: "type ⇒ nat" (‹(1LENGTH/(1'(_')))›)

translations "LENGTH('a)" ⇀
"CONST len_of (CONST Pure.type :: 'a itself)"

print_translation ‹
let
fun len_of_itself_tr' ctxt [Const (\<^const_syntax>‹Pure.type›, Type (_, [T]))] =
Syntax.const \<^syntax_const>‹_type_length› \$ Syntax_Phases.term_of_typ ctxt T
in [(\<^const_syntax>‹len_of›, len_of_itself_tr')] end
›

text ‹Some theorems are only true on words with length greater 0.›

class len = len0 +
assumes len_gt_0 [iff]: "0 < LENGTH('a)"
begin

lemma len_not_eq_0 [simp]:
"LENGTH('a) ≠ 0"
by simp

end

instantiation num0 and num1 :: len0
begin

definition len_num0: "len_of (_ :: num0 itself) = 0"
definition len_num1: "len_of (_ :: num1 itself) = 1"

instance ..

end

instantiation bit0 and bit1 :: (len0) len0
begin

definition len_bit0: "len_of (_ :: 'a::len0 bit0 itself) = 2 * LENGTH('a)"
definition len_bit1: "len_of (_ :: 'a::len0 bit1 itself) = 2 * LENGTH('a) + 1"

instance ..

end

lemmas len_of_numeral_defs [simp] = len_num0 len_num1 len_bit0 len_bit1

instance num1 :: len
by standard simp
instance bit0 :: (len) len
by standard simp
instance bit1 :: (len0) len
by standard simp

instantiation Enum.finite_1 :: len
begin

definition
"len_of_finite_1 (x :: Enum.finite_1 itself) ≡ (1 :: nat)"

instance
by standard (auto simp: len_of_finite_1_def)

end

instantiation Enum.finite_2 :: len
begin

definition
"len_of_finite_2 (x :: Enum.finite_2 itself) ≡ (2 :: nat)"

instance
by standard (auto simp: len_of_finite_2_def)

end

instantiation Enum.finite_3 :: len
begin

definition
"len_of_finite_3 (x :: Enum.finite_3 itself) ≡ (4 :: nat)"

instance
by standard (auto simp: len_of_finite_3_def)

end

lemma length_not_greater_eq_2_iff [simp]:
‹¬ 2 ≤ LENGTH('a::len) ⟷ LENGTH('a) = 1›
by (auto simp add: not_le dest: less_2_cases)

context linordered_idom
begin

lemma two_less_eq_exp_length [simp]:
‹2 ≤ 2 ^ LENGTH('b::len)›
using mult_left_mono [of 1 ‹2 ^ (LENGTH('b::len) - 1)› 2]
by (cases ‹LENGTH('b::len)›) simp_all

end

lemma less_eq_decr_length_iff [simp]:
‹n ≤ LENGTH('a::len) - Suc 0 ⟷ n < LENGTH('a)›
by (cases ‹LENGTH('a)›) (simp_all add: less_Suc_eq le_less)

lemma decr_length_less_iff [simp]:
‹LENGTH('a::len) - Suc 0 < n ⟷ LENGTH('a) ≤ n›
by (cases ‹LENGTH('a)›) auto

end
```