# Theory Type_Length

theory Type_Length
imports Numeral_Type
```(*  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)"

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

end
```