The Coq Proof Assistant

# Library Coq.Strings.Ascii

Contributed by Laurent Théry (INRIA); Adapted to Coq V8 by the Coq Development Team
Require Import Bool BinPos BinNat PeanoNat Nnat Coq.Strings.Byte.
Import IfNotations.

# Definition of ascii characters

Definition of ascii character as a 8 bits constructor
Inductive ascii : Set := Ascii (_ _ _ _ _ _ _ _ : bool).
Declare Scope char_scope.
Delimit Scope char_scope with char.
Bind Scope char_scope with ascii.
Register ascii as core.ascii.type.
Register Ascii as core.ascii.ascii.
Definition zero := Ascii false false false false false false false false.
Definition one := Ascii true false false false false false false false.
Definition shift (c : bool) (a : ascii) :=
match a with
| Ascii a1 a2 a3 a4 a5 a6 a7 a8 => Ascii c a1 a2 a3 a4 a5 a6 a7
end.
Definition of a decidable function that is effective
Definition ascii_dec : forall a b : ascii, {a = b} + {a <> b}.
Local Open Scope lazy_bool_scope.
Definition eqb (a b : ascii) : bool :=
match a, b with
| Ascii a0 a1 a2 a3 a4 a5 a6 a7,
Ascii b0 b1 b2 b3 b4 b5 b6 b7 =>
Bool.eqb a0 b0 &&& Bool.eqb a1 b1 &&& Bool.eqb a2 b2 &&& Bool.eqb a3 b3
&&& Bool.eqb a4 b4 &&& Bool.eqb a5 b5 &&& Bool.eqb a6 b6 &&& Bool.eqb a7 b7
end.
Infix "=?" := eqb : char_scope.
Lemma eqb_spec (a b : ascii) : reflect (a = b) (a =? b)%char.
Local Ltac t_eqb :=
repeat first [ congruence
| progress subst
| apply conj
| match goal with
| [ |- context[eqb ?x ?y] ] => destruct (eqb_spec x y)
end
| intro ].
Lemma eqb_refl x : (x =? x)%char = true.
Lemma eqb_sym x y : (x =? y)%char = (y =? x)%char.
Lemma eqb_eq n m : (n =? m)%char = true <-> n = m.
Lemma eqb_neq x y : (x =? y)%char = false <-> x <> y.
Lemma eqb_compat: Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) eqb.

# Conversion between natural numbers modulo 256 and ascii characters

Auxiliary function that turns a positive into an ascii by looking at the last 8 bits, ie z mod 2^8
Definition ascii_of_pos : positive -> ascii :=
let loop := fix loop n p :=
match n with
| O => zero
| S n' =>
match p with
| xH => one
| xI p' => shift true (loop n' p')
| xO p' => shift false (loop n' p')
end
end
in loop 8.
Conversion from N to ascii
Definition ascii_of_N (n : N) :=
match n with
| N0 => zero
| Npos p => ascii_of_pos p
end.
Same for nat
Definition ascii_of_nat (a : nat) := ascii_of_N (N.of_nat a).
The opposite functions
Local Open Scope list_scope.
Fixpoint N_of_digits (l:list bool) : N :=
match l with
| nil => 0
| b :: l' => (if b then 1 else 0) + 2*(N_of_digits l')
end%N.
Definition N_of_ascii (a : ascii) : N :=
let (a0,a1,a2,a3,a4,a5,a6,a7) := a in
N_of_digits (a0::a1::a2::a3::a4::a5::a6::a7::nil).
Definition nat_of_ascii (a : ascii) : nat := N.to_nat (N_of_ascii a).
Proofs that we have indeed opposite function (below 256)
Theorem ascii_N_embedding :
forall a : ascii, ascii_of_N (N_of_ascii a) = a.
Theorem N_ascii_embedding :
forall n:N, (n < 256)%N -> N_of_ascii (ascii_of_N n) = n.
Theorem N_ascii_bounded :
forall a : ascii, (N_of_ascii a < 256)%N.
Theorem ascii_nat_embedding :
forall a : ascii, ascii_of_nat (nat_of_ascii a) = a.
Theorem nat_ascii_embedding :
forall n : nat, n < 256 -> nat_of_ascii (ascii_of_nat n) = n.
Theorem nat_ascii_bounded :
forall a : ascii, nat_of_ascii a < 256.
Definition compare (a b : ascii) : comparison :=
N.compare (N_of_ascii a) (N_of_ascii b).
Lemma compare_antisym (a b : ascii) :
compare a b = CompOpp (compare b a).
Lemma compare_eq_iff (a b : ascii) : compare a b = Eq -> a = b.
Definition ltb (a b : ascii) : bool :=
if compare a b is Lt then true else false.
Definition leb (a b : ascii) : bool :=
if compare a b is Gt then false else true.
Lemma leb_antisym (a b : ascii) :
leb a b = true -> leb b a = true -> a = b.
Lemma leb_total (a b : ascii) : leb a b = true \/ leb b a = true.
Infix "?=" := compare : char_scope.
Infix "<?" := ltb : char_scope.
Infix "<=?" := leb : char_scope.

# Concrete syntax

Ascii characters can be represented in scope char_scope as follows:

- "c" represents itself if c is a character of code < 128,
- """" is an exception: it represents the ascii character 34 (double quote),
- "nnn" represents the ascii character of decimal code nnn.

For instance, both "065" and "A" denote the character `uppercase A', and both "034" and """" denote the character `double quote'.
Notice that the ascii characters of code >= 128 do not denote stand-alone utf8 characters so that only the notation "nnn" is available for them (unless your terminal is able to represent them, which is typically not the case in coqide).
Definition ascii_of_byte (b : byte) : ascii
:= let '(b0, (b1, (b2, (b3, (b4, (b5, (b6, b7))))))) := Byte.to_bits b in
Ascii b0 b1 b2 b3 b4 b5 b6 b7.
Definition byte_of_ascii (a : ascii) : byte
:= let (b0, b1, b2, b3, b4, b5, b6, b7) := a in
Byte.of_bits (b0, (b1, (b2, (b3, (b4, (b5, (b6, b7))))))).
Lemma ascii_of_byte_of_ascii x : ascii_of_byte (byte_of_ascii x) = x.
Lemma byte_of_ascii_of_byte x : byte_of_ascii (ascii_of_byte x) = x.
Lemma ascii_of_byte_via_N x : ascii_of_byte x = ascii_of_N (Byte.to_N x).
Lemma ascii_of_byte_via_nat x : ascii_of_byte x = ascii_of_nat (Byte.to_nat x).
Lemma byte_of_ascii_via_N x : Some (byte_of_ascii x) = Byte.of_N (N_of_ascii x).
Lemma byte_of_ascii_via_nat x : Some (byte_of_ascii x) = Byte.of_nat (nat_of_ascii x).
Module Export AsciiSyntax.
End AsciiSyntax.
Local Open Scope char_scope.
Example Space := " ".
Example DoubleQuote := """".
Example Beep := "007".