diff options
Diffstat (limited to 'theories/Strings')
-rw-r--r-- | theories/Strings/Ascii.v | 34 | ||||
-rw-r--r-- | theories/Strings/BinaryString.v | 147 | ||||
-rw-r--r-- | theories/Strings/HexString.v | 229 | ||||
-rw-r--r-- | theories/Strings/OctalString.v | 179 | ||||
-rw-r--r-- | theories/Strings/String.v | 34 |
5 files changed, 623 insertions, 0 deletions
diff --git a/theories/Strings/Ascii.v b/theories/Strings/Ascii.v index 5154b75b..31a7fb8a 100644 --- a/theories/Strings/Ascii.v +++ b/theories/Strings/Ascii.v @@ -40,6 +40,40 @@ Proof. decide equality; apply bool_dec. Defined. +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. +Proof. + destruct a, b; simpl. + do 8 (case Bool.eqb_spec; [ intros -> | constructor; now intros [= ] ]). + now constructor. +Qed. + +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. Proof. t_eqb. Qed. +Lemma eqb_sym x y : (x =? y)%char = (y =? x)%char. Proof. t_eqb. Qed. +Lemma eqb_eq n m : (n =? m)%char = true <-> n = m. Proof. t_eqb. Qed. +Lemma eqb_neq x y : (x =? y)%char = false <-> x <> y. Proof. t_eqb. Qed. +Lemma eqb_compat: Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) eqb. +Proof. t_eqb. Qed. + (** * Conversion between natural numbers modulo 256 and ascii characters *) (** Auxiliary function that turns a positive into an ascii by diff --git a/theories/Strings/BinaryString.v b/theories/Strings/BinaryString.v new file mode 100644 index 00000000..6df0a917 --- /dev/null +++ b/theories/Strings/BinaryString.v @@ -0,0 +1,147 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import Ascii String. +Require Import BinNums. +Import BinNatDef. +Import BinIntDef. +Import BinPosDef. + +Local Open Scope positive_scope. +Local Open Scope string_scope. + +Definition ascii_to_digit (ch : ascii) : option N + := (if ascii_dec ch "0" then Some 0 + else if ascii_dec ch "1" then Some 1 + else None)%N. + +Fixpoint pos_bin_app (p q:positive) : positive := + match q with + | q~0 => (pos_bin_app p q)~0 + | q~1 => (pos_bin_app p q)~1 + | 1 => p~1 + end. + +Module Raw. + Fixpoint of_pos (p : positive) (rest : string) : string + := match p with + | 1 => String "1" rest + | p'~0 => of_pos p' (String "0" rest) + | p'~1 => of_pos p' (String "1" rest) + end. + + Fixpoint to_N (s : string) (rest : N) + : N + := match s with + | "" => rest + | String ch s' + => to_N + s' + match ascii_to_digit ch with + | Some v => N.add v (N.double rest) + | None => N0 + end + end. + + Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N) + : to_N (of_pos p rest) base + = to_N rest match base with + | N0 => N.pos p + | Npos v => Npos (pos_bin_app v p) + end. + Proof. + destruct p as [p|p|]; destruct base; try reflexivity; + cbn; rewrite to_N_of_pos; reflexivity. + Qed. +End Raw. + +Definition of_pos (p : positive) : string + := String "0" (String "b" (Raw.of_pos p "")). +Definition of_N (n : N) : string + := match n with + | N0 => "0b0" + | Npos p => of_pos p + end. +Definition of_Z (z : Z) : string + := match z with + | Zneg p => String "-" (of_pos p) + | Z0 => "0b0" + | Zpos p => of_pos p + end. +Definition of_nat (n : nat) : string + := of_N (N.of_nat n). + +Definition to_N (s : string) : N + := match s with + | String s0 (String sb s) + => if ascii_dec s0 "0" + then if ascii_dec sb "b" + then Raw.to_N s N0 + else N0 + else N0 + | _ => N0 + end. +Definition to_pos (s : string) : positive + := match to_N s with + | N0 => 1 + | Npos p => p + end. +Definition to_Z (s : string) : Z + := let '(is_neg, n) := match s with + | String s0 s' + => if ascii_dec s0 "-" + then (true, to_N s') + else (false, to_N s) + | EmptyString => (false, to_N s) + end in + match n with + | N0 => Z0 + | Npos p => if is_neg then Zneg p else Zpos p + end. +Definition to_nat (s : string) : nat + := N.to_nat (to_N s). + +Lemma to_N_of_N (n : N) + : to_N (of_N n) + = n. +Proof. + destruct n; [ reflexivity | apply Raw.to_N_of_pos ]. +Qed. + +Lemma Z_of_of_Z (z : Z) + : to_Z (of_Z z) + = z. +Proof. + cbv [of_Z to_Z]; destruct z as [|z|z]; cbn; + try reflexivity; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Lemma to_nat_of_nat (n : nat) + : to_nat (of_nat n) + = n. +Proof. + cbv [to_nat of_nat]; + rewrite to_N_of_N, Nnat.Nat2N.id; reflexivity. +Qed. + +Lemma to_pos_of_pos (p : positive) + : to_pos (of_pos p) + = p. +Proof. + cbv [of_pos to_pos to_N]; cbn; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Example of_pos_1 : of_pos 1 = "0b1" := eq_refl. +Example of_pos_2 : of_pos 2 = "0b10" := eq_refl. +Example of_pos_3 : of_pos 3 = "0b11" := eq_refl. +Example of_N_0 : of_N 0 = "0b0" := eq_refl. +Example of_Z_0 : of_Z 0 = "0b0" := eq_refl. +Example of_Z_m1 : of_Z (-1) = "-0b1" := eq_refl. +Example of_nat_0 : of_nat 0 = "0b0" := eq_refl. diff --git a/theories/Strings/HexString.v b/theories/Strings/HexString.v new file mode 100644 index 00000000..9ea93c90 --- /dev/null +++ b/theories/Strings/HexString.v @@ -0,0 +1,229 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import Ascii String. +Require Import BinNums. +Import BinNatDef. +Import BinIntDef. +Import BinPosDef. + +Local Open Scope positive_scope. +Local Open Scope string_scope. + +Local Notation "a || b" + := (if a then true else if b then true else false). +Definition ascii_to_digit (ch : ascii) : option N + := (if ascii_dec ch "0" then Some 0 + else if ascii_dec ch "1" then Some 1 + else if ascii_dec ch "2" then Some 2 + else if ascii_dec ch "3" then Some 3 + else if ascii_dec ch "4" then Some 4 + else if ascii_dec ch "5" then Some 5 + else if ascii_dec ch "6" then Some 6 + else if ascii_dec ch "7" then Some 7 + else if ascii_dec ch "8" then Some 8 + else if ascii_dec ch "9" then Some 9 + else if ascii_dec ch "a" || ascii_dec ch "A" then Some 10 + else if ascii_dec ch "b" || ascii_dec ch "B" then Some 11 + else if ascii_dec ch "c" || ascii_dec ch "C" then Some 12 + else if ascii_dec ch "d" || ascii_dec ch "D" then Some 13 + else if ascii_dec ch "e" || ascii_dec ch "E" then Some 14 + else if ascii_dec ch "f" || ascii_dec ch "F" then Some 15 + else None)%N. + +Fixpoint pos_hex_app (p q:positive) : positive := + match q with + | 1 => p~0~0~0~1 + | 2 => p~0~0~1~0 + | 3 => p~0~0~1~1 + | 4 => p~0~1~0~0 + | 5 => p~0~1~0~1 + | 6 => p~0~1~1~0 + | 7 => p~0~1~1~1 + | 8 => p~1~0~0~0 + | 9 => p~1~0~0~1 + | 10 => p~1~0~1~0 + | 11 => p~1~0~1~1 + | 12 => p~1~1~0~0 + | 13 => p~1~1~0~1 + | 14 => p~1~1~1~0 + | 15 => p~1~1~1~1 + | q~0~0~0~0 => (pos_hex_app p q)~0~0~0~0 + | q~0~0~0~1 => (pos_hex_app p q)~0~0~0~1 + | q~0~0~1~0 => (pos_hex_app p q)~0~0~1~0 + | q~0~0~1~1 => (pos_hex_app p q)~0~0~1~1 + | q~0~1~0~0 => (pos_hex_app p q)~0~1~0~0 + | q~0~1~0~1 => (pos_hex_app p q)~0~1~0~1 + | q~0~1~1~0 => (pos_hex_app p q)~0~1~1~0 + | q~0~1~1~1 => (pos_hex_app p q)~0~1~1~1 + | q~1~0~0~0 => (pos_hex_app p q)~1~0~0~0 + | q~1~0~0~1 => (pos_hex_app p q)~1~0~0~1 + | q~1~0~1~0 => (pos_hex_app p q)~1~0~1~0 + | q~1~0~1~1 => (pos_hex_app p q)~1~0~1~1 + | q~1~1~0~0 => (pos_hex_app p q)~1~1~0~0 + | q~1~1~0~1 => (pos_hex_app p q)~1~1~0~1 + | q~1~1~1~0 => (pos_hex_app p q)~1~1~1~0 + | q~1~1~1~1 => (pos_hex_app p q)~1~1~1~1 + end. + +Module Raw. + Fixpoint of_pos (p : positive) (rest : string) : string + := match p with + | 1 => String "1" rest + | 2 => String "2" rest + | 3 => String "3" rest + | 4 => String "4" rest + | 5 => String "5" rest + | 6 => String "6" rest + | 7 => String "7" rest + | 8 => String "8" rest + | 9 => String "9" rest + | 10 => String "a" rest + | 11 => String "b" rest + | 12 => String "c" rest + | 13 => String "d" rest + | 14 => String "e" rest + | 15 => String "f" rest + | p'~0~0~0~0 => of_pos p' (String "0" rest) + | p'~0~0~0~1 => of_pos p' (String "1" rest) + | p'~0~0~1~0 => of_pos p' (String "2" rest) + | p'~0~0~1~1 => of_pos p' (String "3" rest) + | p'~0~1~0~0 => of_pos p' (String "4" rest) + | p'~0~1~0~1 => of_pos p' (String "5" rest) + | p'~0~1~1~0 => of_pos p' (String "6" rest) + | p'~0~1~1~1 => of_pos p' (String "7" rest) + | p'~1~0~0~0 => of_pos p' (String "8" rest) + | p'~1~0~0~1 => of_pos p' (String "9" rest) + | p'~1~0~1~0 => of_pos p' (String "a" rest) + | p'~1~0~1~1 => of_pos p' (String "b" rest) + | p'~1~1~0~0 => of_pos p' (String "c" rest) + | p'~1~1~0~1 => of_pos p' (String "d" rest) + | p'~1~1~1~0 => of_pos p' (String "e" rest) + | p'~1~1~1~1 => of_pos p' (String "f" rest) + end. + + Fixpoint to_N (s : string) (rest : N) + : N + := match s with + | "" => rest + | String ch s' + => to_N + s' + match ascii_to_digit ch with + | Some v => N.add v (N.mul 16 rest) + | None => N0 + end + end. + + Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N) + : to_N (of_pos p rest) base + = to_N rest match base with + | N0 => N.pos p + | Npos v => Npos (pos_hex_app v p) + end. + Proof. + do 4 try destruct p as [p|p|]; destruct base; try reflexivity; + cbn; rewrite to_N_of_pos; reflexivity. + Qed. +End Raw. + +Definition of_pos (p : positive) : string + := String "0" (String "x" (Raw.of_pos p "")). +Definition of_N (n : N) : string + := match n with + | N0 => "0x0" + | Npos p => of_pos p + end. +Definition of_Z (z : Z) : string + := match z with + | Zneg p => String "-" (of_pos p) + | Z0 => "0x0" + | Zpos p => of_pos p + end. +Definition of_nat (n : nat) : string + := of_N (N.of_nat n). + +Definition to_N (s : string) : N + := match s with + | String s0 (String so s) + => if ascii_dec s0 "0" + then if ascii_dec so "x" + then Raw.to_N s N0 + else N0 + else N0 + | _ => N0 + end. +Definition to_pos (s : string) : positive + := match to_N s with + | N0 => 1 + | Npos p => p + end. +Definition to_Z (s : string) : Z + := let '(is_neg, n) := match s with + | String s0 s' + => if ascii_dec s0 "-" + then (true, to_N s') + else (false, to_N s) + | EmptyString => (false, to_N s) + end in + match n with + | N0 => Z0 + | Npos p => if is_neg then Zneg p else Zpos p + end. +Definition to_nat (s : string) : nat + := N.to_nat (to_N s). + +Lemma to_N_of_N (n : N) + : to_N (of_N n) + = n. +Proof. + destruct n; [ reflexivity | apply Raw.to_N_of_pos ]. +Qed. + +Lemma to_Z_of_Z (z : Z) + : to_Z (of_Z z) + = z. +Proof. + cbv [of_Z to_Z]; destruct z as [|z|z]; cbn; + try reflexivity; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Lemma to_nat_of_nat (n : nat) + : to_nat (of_nat n) + = n. +Proof. + cbv [to_nat of_nat]; + rewrite to_N_of_N, Nnat.Nat2N.id; reflexivity. +Qed. + +Lemma to_pos_of_pos (p : positive) + : to_pos (of_pos p) + = p. +Proof. + cbv [of_pos to_pos to_N]; cbn; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Example of_pos_1 : of_pos 1 = "0x1" := eq_refl. +Example of_pos_2 : of_pos 2 = "0x2" := eq_refl. +Example of_pos_3 : of_pos 3 = "0x3" := eq_refl. +Example of_pos_7 : of_pos 7 = "0x7" := eq_refl. +Example of_pos_8 : of_pos 8 = "0x8" := eq_refl. +Example of_pos_9 : of_pos 9 = "0x9" := eq_refl. +Example of_pos_10 : of_pos 10 = "0xa" := eq_refl. +Example of_pos_11 : of_pos 11 = "0xb" := eq_refl. +Example of_pos_12 : of_pos 12 = "0xc" := eq_refl. +Example of_pos_13 : of_pos 13 = "0xd" := eq_refl. +Example of_pos_14 : of_pos 14 = "0xe" := eq_refl. +Example of_pos_15 : of_pos 15 = "0xf" := eq_refl. +Example of_pos_16 : of_pos 16 = "0x10" := eq_refl. +Example of_N_0 : of_N 0 = "0x0" := eq_refl. +Example of_Z_0 : of_Z 0 = "0x0" := eq_refl. +Example of_Z_m1 : of_Z (-1) = "-0x1" := eq_refl. +Example of_nat_0 : of_nat 0 = "0x0" := eq_refl. diff --git a/theories/Strings/OctalString.v b/theories/Strings/OctalString.v new file mode 100644 index 00000000..fe8cc9aa --- /dev/null +++ b/theories/Strings/OctalString.v @@ -0,0 +1,179 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import Ascii String. +Require Import BinNums. +Import BinNatDef. +Import BinIntDef. +Import BinPosDef. + +Local Open Scope positive_scope. +Local Open Scope string_scope. + +Definition ascii_to_digit (ch : ascii) : option N + := (if ascii_dec ch "0" then Some 0 + else if ascii_dec ch "1" then Some 1 + else if ascii_dec ch "2" then Some 2 + else if ascii_dec ch "3" then Some 3 + else if ascii_dec ch "4" then Some 4 + else if ascii_dec ch "5" then Some 5 + else if ascii_dec ch "6" then Some 6 + else if ascii_dec ch "7" then Some 7 + else None)%N. + +Fixpoint pos_oct_app (p q:positive) : positive := + match q with + | 1 => p~0~0~1 + | 2 => p~0~1~0 + | 3 => p~0~1~1 + | 4 => p~1~0~0 + | 5 => p~1~0~1 + | 6 => p~1~1~0 + | 7 => p~1~1~1 + | q~0~0~0 => (pos_oct_app p q)~0~0~0 + | q~0~0~1 => (pos_oct_app p q)~0~0~1 + | q~0~1~0 => (pos_oct_app p q)~0~1~0 + | q~0~1~1 => (pos_oct_app p q)~0~1~1 + | q~1~0~0 => (pos_oct_app p q)~1~0~0 + | q~1~0~1 => (pos_oct_app p q)~1~0~1 + | q~1~1~0 => (pos_oct_app p q)~1~1~0 + | q~1~1~1 => (pos_oct_app p q)~1~1~1 + end. + +Module Raw. + Fixpoint of_pos (p : positive) (rest : string) : string + := match p with + | 1 => String "1" rest + | 2 => String "2" rest + | 3 => String "3" rest + | 4 => String "4" rest + | 5 => String "5" rest + | 6 => String "6" rest + | 7 => String "7" rest + | p'~0~0~0 => of_pos p' (String "0" rest) + | p'~0~0~1 => of_pos p' (String "1" rest) + | p'~0~1~0 => of_pos p' (String "2" rest) + | p'~0~1~1 => of_pos p' (String "3" rest) + | p'~1~0~0 => of_pos p' (String "4" rest) + | p'~1~0~1 => of_pos p' (String "5" rest) + | p'~1~1~0 => of_pos p' (String "6" rest) + | p'~1~1~1 => of_pos p' (String "7" rest) + end. + + Fixpoint to_N (s : string) (rest : N) + : N + := match s with + | "" => rest + | String ch s' + => to_N + s' + match ascii_to_digit ch with + | Some v => N.add v (N.mul 8 rest) + | None => N0 + end + end. + + Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N) + : to_N (of_pos p rest) base + = to_N rest match base with + | N0 => N.pos p + | Npos v => Npos (pos_oct_app v p) + end. + Proof. + do 3 try destruct p as [p|p|]; destruct base; try reflexivity; + cbn; rewrite to_N_of_pos; reflexivity. + Qed. +End Raw. + +Definition of_pos (p : positive) : string + := String "0" (String "o" (Raw.of_pos p "")). +Definition of_N (n : N) : string + := match n with + | N0 => "0o0" + | Npos p => of_pos p + end. +Definition of_Z (z : Z) : string + := match z with + | Zneg p => String "-" (of_pos p) + | Z0 => "0o0" + | Zpos p => of_pos p + end. +Definition of_nat (n : nat) : string + := of_N (N.of_nat n). + +Definition to_N (s : string) : N + := match s with + | String s0 (String so s) + => if ascii_dec s0 "0" + then if ascii_dec so "o" + then Raw.to_N s N0 + else N0 + else N0 + | _ => N0 + end. +Definition to_pos (s : string) : positive + := match to_N s with + | N0 => 1 + | Npos p => p + end. +Definition to_Z (s : string) : Z + := let '(is_neg, n) := match s with + | String s0 s' + => if ascii_dec s0 "-" + then (true, to_N s') + else (false, to_N s) + | EmptyString => (false, to_N s) + end in + match n with + | N0 => Z0 + | Npos p => if is_neg then Zneg p else Zpos p + end. +Definition to_nat (s : string) : nat + := N.to_nat (to_N s). + +Lemma to_N_of_N (n : N) + : to_N (of_N n) + = n. +Proof. + destruct n; [ reflexivity | apply Raw.to_N_of_pos ]. +Qed. + +Lemma to_Z_of_Z (z : Z) + : to_Z (of_Z z) + = z. +Proof. + cbv [of_Z to_Z]; destruct z as [|z|z]; cbn; + try reflexivity; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Lemma to_nat_of_nat (n : nat) + : to_nat (of_nat n) + = n. +Proof. + cbv [to_nat of_nat]; + rewrite to_N_of_N, Nnat.Nat2N.id; reflexivity. +Qed. + +Lemma to_pos_of_pos (p : positive) + : to_pos (of_pos p) + = p. +Proof. + cbv [of_pos to_pos to_N]; cbn; + rewrite Raw.to_N_of_pos; cbn; reflexivity. +Qed. + +Example of_pos_1 : of_pos 1 = "0o1" := eq_refl. +Example of_pos_2 : of_pos 2 = "0o2" := eq_refl. +Example of_pos_3 : of_pos 3 = "0o3" := eq_refl. +Example of_pos_7 : of_pos 7 = "0o7" := eq_refl. +Example of_pos_8 : of_pos 8 = "0o10" := eq_refl. +Example of_N_0 : of_N 0 = "0o0" := eq_refl. +Example of_Z_0 : of_Z 0 = "0o0" := eq_refl. +Example of_Z_m1 : of_Z (-1) = "-0o1" := eq_refl. +Example of_nat_0 : of_nat 0 = "0o0" := eq_refl. diff --git a/theories/Strings/String.v b/theories/Strings/String.v index 2be6618a..be9a10c6 100644 --- a/theories/Strings/String.v +++ b/theories/Strings/String.v @@ -14,6 +14,7 @@ Require Import Arith. Require Import Ascii. +Require Import Bool. Declare ML Module "string_syntax_plugin". (** *** Definition of strings *) @@ -35,6 +36,39 @@ Proof. decide equality; apply ascii_dec. Defined. +Local Open Scope lazy_bool_scope. + +Fixpoint eqb s1 s2 : bool := + match s1, s2 with + | EmptyString, EmptyString => true + | String c1 s1', String c2 s2' => Ascii.eqb c1 c2 &&& eqb s1' s2' + | _,_ => false + end. + +Infix "=?" := eqb : string_scope. + +Lemma eqb_spec s1 s2 : Bool.reflect (s1 = s2) (s1 =? s2)%string. +Proof. + revert s2. induction s1; destruct s2; try (constructor; easy); simpl. + case Ascii.eqb_spec; simpl; [intros -> | constructor; now intros [= ]]. + case IHs1; [intros ->; now constructor | constructor; now intros [= ]]. +Qed. + +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)%string = true. Proof. t_eqb. Qed. +Lemma eqb_sym x y : (x =? y)%string = (y =? x)%string. Proof. t_eqb. Qed. +Lemma eqb_eq n m : (n =? m)%string = true <-> n = m. Proof. t_eqb. Qed. +Lemma eqb_neq x y : (x =? y)%string = false <-> x <> y. Proof. t_eqb. Qed. +Lemma eqb_compat: Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) eqb. +Proof. t_eqb. Qed. + (** *** Concatenation of strings *) Reserved Notation "x ++ y" (right associativity, at level 60). |