Vericert Manual

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Vericert Manual

Table of Contents

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1 Introduction

Vericert translates C code into a hardware description language called Verilog, which can then be synthesised into hardware, to be placed onto a field-programmable gate array (FPGA) or application-specific integrated circuit (ASIC).

images/toolflow

Figure 1.1: Current design of Vericert, where HTL is an intermediate language representing a finite state machine with data-path (FSMD) and Verilog is the target language.

The design shown in Figure Figure 1.1 shows how Vericert leverages an existing verified C compiler called CompCert to perform this translation.

Copyright (C) 2019-2022 Yann Herklotz.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.

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2 Building Vericert

To build Vericert, the provided Makefile can be used. External dependencies are needed to build the project, which can be pulled in automatically with nix using the provided default.nix and shell.nix files.

The project is written in Coq, a theorem prover, which is extracted to OCaml so that it can then be compiled and executed. The dependencies of this project are the following:

These dependencies can be installed manually, or automatically through Nix.

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2.1 Downloading Vericert and CompCert

CompCert is added as a submodule in the ‘lib/CompCert’ directory. It is needed to run the build process below, as it is the one dependency that is not downloaded by nix, and has to be downloaded together with the repository. To clone CompCert together with this project, and check it out at the correct revision, you can run: 

git clone -b v1.2.2 --recursive https://github.com/ymherklotz/vericert

If the repository is already cloned, you can run the following command to make sure that CompCert is also downloaded and the correct branch is checked out: 

git checkout v1.2.2
git submodule update --init

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2.2 Setting up Nix

Nix is a package manager that can create an isolated environment so that the builds are reproducible. Once nix is installed, it can be used in the following way.

To open a shell which includes all the necessary dependencies, one can use: 

nix-shell

which will open a shell that has all the dependencies loaded.

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2.3 Makefile build

If the dependencies were installed manually, or if one is in the ‘nix-shell’, the project can be built by running: 

make -j8

and installed locally, or under the ‘PREFIX’ location using: 

make install

Which will install the binary in ‘./bin/vericert’ by default. However, this can be changed by changing the ‘PREFIX’ environment variable, in which case the binary will be installed in ‘$PREFIX/bin/vericert’.

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2.4 Testing

To test out vericert you can try the following examples which are in the test folder using the following: 

./bin/vericert test/loop.c -o loop.v
./bin/vericert test/conditional.c -o conditional.v
./bin/vericert test/add.c -o add.v

Or by running the test suite using the following command: 

make test

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3 Using Vericert

Vericert can be used to translate a subset of C into Verilog. As a simple example, consider the following C file (main.c): 

void matrix_multiply(int first[2][2], int second[2][2], int multiply[2][2]) {
    int sum = 0;
    for (int c = 0; c < 2; c++) {
        for (int d = 0; d < 2; d++) {
            for (int k = 0; k < 2; k++) {
                sum = sum + first[c][k]*second[k][d];
            }
            multiply[c][d] = sum;
            sum = 0;
        }
    }
}

int main() {
    int f[2][2] = {{1, 2}, {3, 4}};
    int s[2][2] = {{5, 6}, {7, 8}};
    int m[2][2] = {{0, 0}, {0, 0}};

    matrix_multiply(f, s, m);
    return m[1][1];
}

It can be compiled using the following command, assuming that vericert is somewhere on the path. 

vericert main.c -o main.v

The Verilog file contains a top-level test-bench, which can be given to any Verilog simulator to simulate the hardware, which should give the same result as executing the C code. Using Icarus Verilog as an example: 

iverilog -o main_v main.v

When executing, it should therefore print the following: 

$ ./main_v
finished: 50

This gives the same result as executing the C in the following way: 

$ gcc -o main_c main.c
$ ./main_c
$ echo $?
50

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3.1 Man pages

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3.1.1 NAME

vericert - A formally verified high-level synthesis tool.

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3.1.2 SYNOPSYS

vericert [ OPTION ]… [ FILE ]…

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3.1.3 DESCRIPTION

  1. HLS Options:
    –no-hls

    Do not use HLS and generate standard flow

    –simulate

    Simulate the result with the Verilog semantics

    –debug-hls

    Add debug logic to the Verilog

    –initialise-stack

    initialise the stack to all 0s

  2. HLS Optimisations:
    -fschedule

    Schedule the resulting hardware [off]

    -fif-conversion

    If-conversion optimisation [off]

  3. General options:
    -stdlib <dir>

    Set the path of the Compcert run-time library

    -v

    Print external commands before invoking them

    -timings

    Show the time spent in various compiler passes

    -version

    Print the version string and exit

    -target <value>

    Generate code for the given target

    -conf <file>

    Read configuration from file

    @<file>

    Read command line options from <file>

  4. Tracing Options:
    -dprepro

    Save C file after preprocessing in <file>.i

    -dparse

    Save C file after parsing and elaboration in <file>.parsed.c

    -dc

    Save generated C in <file>.compcert.c

    -dclight

    Save generated Clight in <file>.light.c

    -dcminor

    Save generated Cminor in <file>.cm

    -drtl

    Save RTL at various optimization points in <file>.rtl.<n>

    -drtlblock

    Save RTLBlock <file>.rtlblock

    -dhtl

    Save HTL before Verilog generation <file>.htl

    -dltl

    Save LTL after register allocation in <file>.ltl

    -dmach

    Save generated Mach code in <file>.mach

    -dasm

    Save generated assembly in <file>.s

    -dall

    Save all generated intermediate files in <file>.<ext>

    -sdump

    Save info for post-linking validation in <file>.json

    -o <file>

    Generate output in <file>

  5. Diagnostic options:
    -Wall

    Enable all warnings

    -W<warning>

    Enable the specific <warning>

    -Wno-<warning>

    Disable the specific <warning>

    -Werror

    Make all warnings into errors

    -Werror=<warning>

    Turn <warning> into an error

    -Wno-error=<warning>

    Turn <warning> into a warning even if -Werror is specified

    -Wfatal-errors

    Turn all errors into fatal errors aborting the compilation

    -fdiagnostics-color

    Turn on colored diagnostics

    -fno-diagnostics-color

    Turn of colored diagnostics

    -fmax-errors=<n>

    Maximum number of errors to report

    -fdiagnostics-show-option

    Print the option name with mappable diagnostics

    -fno-diagnostics-show-option

    Turn of printing of options with mappable diagnostics

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3.1.4 AUTHOR

Written by Yann Herklotz, Michalis Pardalos, James Pollard, Nadesh Ramanathan and John Wickerson.

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4 Unreleased Features

The following are unreleased features in Vericert that are currently being worked on and have not been completely proven correct yet. Currently this includes features such as:

This page gives some preliminary information on how the features are implemented and how the proofs for the features are being done. Once these features are properly implemented, they will be added to the proper documentation.

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4.1 Scheduling

Scheduling is an optimisation which is used to run various instructions in parallel that are independent to each other.

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4.2 Operation Chaining

Operation chaining is an optimisation that can be added on to scheduling and allows for the sequential execution of instructions in a clock cycle, while executing other instructions in parallel in the same clock cycle.

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4.3 If-conversion

If-conversion is an optimisation which can turn code with simple control flow into a single block (called a hyper-block), using predicated instructions.

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4.4 Functions

Functions are currently only inlined in Vericert, however, we are working on a proper interface to integrate function calls into the hardware.

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5 Coq Style Guide

This style guide was taken from Silveroak, it outlines code style for Coq code in this repository. There are certainly other valid strategies and opinions on Coq code style; this is laid out purely in the name of consistency. For a visual example of the style, see the example at the bottom of this file.

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5.1 Code organization

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5.1.2 Import statements

  • ‘Require Import’ statements should all go at the top of the file, followed by file-wide ‘Import’ statements.
    • =Import=s often contain notations or typeclass instances that might override notations or instances from another library, so it’s nice to highlight them separately.
  • One ‘Require Import’ statement per line; it’s easier to scan that way.
  • ‘Require Import’ statements should use "fully-qualified" names (e.g. ‘Require Import Coq.ZArith.ZArith’ instead of ‘Require Import ZArith’).
    • Use the ‘Locate’ command to find the fully-qualified name!
  • ‘Require Import’’s should go in the following order:
    1. Standard library dependencies (start with ‘Coq.’)
    2. External dependencies (anything outside the current project)
    3. Same-project dependencies
  • ‘Require Import’’s with the same root library (the name before the first ‘.’) should be grouped together. Within each root-library group, they should be in alphabetical order (so ‘Coq.Lists.List’ before ‘Coq.ZArith.ZArith’).

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5.1.3 Notations and scopes

  • Any file-wide ‘Local Open Scope’’s should come immediately after the =Import=s (see example).
    • Always use ‘Local Open Scope’; just ‘Open Scope’ will sneakily open the scope for those who import your file.
  • Put notations in their own separate modules or files, so that those who import your file can choose whether or not they want the notations.
    • Conflicting notations can cause a lot of headache, so it comes in very handy to leave this flexibility!

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5.2 Formatting

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5.2.1 Line length

  • Maximum line length 80 characters.
    • Many Coq IDE setups divide the screen in half vertically and use only half to display source code, so more than 80 characters can be genuinely hard to read on a laptop.

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5.2.2 Whitespace and indentation

  • No trailing whitespace.
  • Spaces, not tabs.
  • Files should end with a newline.
    • Many editors do this automatically on save.
  • Colons may be either "English-spaced", with no space before the colon and one space after (‘x: nat’) or "French-spaced", with one space before and after (‘x : nat’).
  • Default indentation is 2 spaces.
    • Keeping this small prevents complex proofs from being indented ridiculously far, and matches IDE defaults.
  • Use 2-space indents if inserting a line break immediately after:
    • ‘Proof.’
    • ‘fun <...> =>’
    • ‘forall <...>,’
    • ‘exists <....>,’
  • The style for indenting arguments in function application depends on where you make a line break. If you make the line break immediately after the function name, use a 2-space indent. However, if you make it after one or more arguments, align the next line with the first argument: 
    (Z.pow
   1 2)
(Z.pow 1 2 3
       4 5 6)
  • ‘Inductive’ cases should not be indented. Example: 
    Inductive Foo : Type :=
| FooA : Foo
| FooB : Foo
.
  • ‘match’ or ‘lazymatch’ cases should line up with the "m" in ‘match’ or "l" in ‘lazymatch’, as in the following examples: 
    match x with
| 3 => true
| _ => false
end.

lazymatch x with
| 3 => idtac
| _ => fail "Not equal to 3:" x
end.

repeat match goal with
       | _ => progress subst
       | _ => reflexivity
       end.

do 2 lazymatch goal with
     | |- context [eq] => idtac
     end.

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5.3 Definitions and Fixpoints

  • It’s okay to leave the return type of ‘Definition’’s and ‘Fixpoint’’s implicit (e.g. Definition x := 5 instead of Definition x : nat := 5) when the type is very simple or obvious (for instance, the definition is in a file which deals exclusively with operations on ‘Z’).

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5.4 Inductives

  • The ‘.’ ending an ‘Inductive’ can be either on the same line as the last case or on its own line immediately below. That is, both of the following are acceptable: 
    Inductive Foo : Type :=
| FooA : Foo
| FooB : Foo
.
Inductive Foo : Type :=
| FooA : Foo
| FooB : Foo.

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5.5 Lemma/Theorem statements

  • Generally, use ‘Theorem’ for the most important, top-level facts you prove and ‘Lemma’ for everything else.
  • Insert a line break after the colon in the lemma statement.
  • Insert a line break after the comma for ‘forall’ or ‘exist’ quantifiers.
  • Implication arrows (‘->’) should share a line with the previous hypothesis, not the following one.
  • There is no need to make a line break after every ‘->’; short preconditions may share a line.

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5.6 Proofs and tactics

  • Use the ‘Proof’ command (lined up vertically with ‘Lemma’ or ‘Theorem’ it corresponds to) to open a proof, and indent the first line after it 2 spaces.
  • Very small proofs (where ‘Proof. <tactics> Qed.’ is <= 80 characters) can go all in one line.
  • When ending a proof, align the ending statement (‘Qed’, ‘Admitted’, etc.) with ‘Proof’.
  • Avoid referring to autogenerated names (e.g. ‘H0’, ‘n0’). It’s okay to let Coq generate these names, but you should not explicitly refer to them in your proof. So ‘intros; my_solver’ is fine, but ‘intros; apply H1; my_solver’ is not fine.
    • You can force a non-autogenerated name by either putting the variable before the colon in the lemma statement (‘Lemma foo x : ...’ instead of ‘Lemma foo : forall x, ...’), or by passing arguments to ‘intros’ (e.g. ‘intros ? x’ to name the second argument ‘x’)
  • This way, the proof won’t break when new hypotheses are added or autogenerated variable names change.
  • Use curly braces ‘{}’ for subgoals, instead of bullets.
  • Never write tactics with more than one subgoal focused. This can make the proof very confusing to step through! If you have more than one subgoal, use curly braces.
  • Consider adding a comment after the opening curly brace that explains what case you’re in (see example).
    • This is not necessary for small subgoals but can help show the major lines of reasoning in large proofs.
  • If invoking a tactic that is expected to return multiple subgoals, use ‘[ | ... | ]’ before the ‘.’ to explicitly specify how many subgoals you expect.
    • Examples: ‘split; [ | ].’ ‘induction z; [ | | ].’
    • This helps make code more maintainable, because it fails immediately if your tactic no longer solves as many subgoals as expected (or unexpectedly solves more).
  • If invoking a string of tactics (composed by ‘;’) that will break the goal into multiple subgoals and then solve all but one, still use ‘[ ]’ to enforce that all but one goal is solved.
    • Example: ‘split; try lia; [ ]’.
  • Tactics that consist only of ‘repeat’-ing a procedure (e.g. ‘repeat match’, ‘repeat first’) should factor out a single step of that procedure a separate tactic called ‘<tactic name>_step’, because the single-step version is much easier to debug. For instance: 
    Ltac crush_step :=
  match goal with
  | _ => progress subst
  | _ => reflexivity
  end.
Ltac crush := repeat crush_step.

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5.7 Naming

  • Helper proofs about standard library datatypes should go in a module that is named to match the standard library module (see example).
    • This makes the helper proofs look like standard-library ones, which is helpful for categorizing them if they’re genuinely at the standard-library level of abstraction.
  • Names of modules should start with capital letters.
  • Names of inductives and their constructors should start with capital letters.
  • Names of other definitions/lemmas should be snake case.

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5.8 Example

A small standalone Coq file that exhibits many of the style points. 

(*
 * Vericert: Verified high-level synthesis.
 * Copyright (C) 2021 Name <[email protected]>
 *
 * <License...>
 *)

  Require Import Coq.Lists.List.
  Require Import Coq.micromega.Lia.
  Require Import Coq.ZArith.ZArith.
  Import ListNotations.
  Local Open Scope Z_scope.

  (* Helper proofs about standard library integers (Z) go within [Module Z] so
     that they match standard-library Z lemmas when used. *)
  Module Z.
    Lemma pow_3_r x : x ^ 3 = x * x * x.
    Proof. lia. Qed. (* very short proofs can go all on one line *)

    Lemma pow_4_r x : x ^ 4 = x * x * x * x.
    Proof.
      change 4 with (Z.succ (Z.succ (Z.succ (Z.succ 0)))).
      repeat match goal with
             | _ => rewrite Z.pow_1_r
             | _ => rewrite Z.pow_succ_r by lia
             | |- context [x * (?a * ?b)] =>
               replace (x * (a * b)) with (a * b * x) by lia
             | _ => reflexivity
             end.
    Qed.
  End Z.
  (* Now we can access the lemmas above as Z.pow_3_r and Z.pow_4_r, as if they
     were in the ZArith library! *)

  Definition bar (x y : Z) := x ^ (y + 1).

  (* example with a painfully manual proof to show case formatting *)
  Lemma bar_upper_bound :
    forall x y a,
      0 <= x <= a -> 0 <= y ->
      0 <= bar x y <= a ^ (y + 1).
  Proof.
    (* avoid referencing autogenerated names by explicitly naming variables *)
    intros x y a Hx Hy. revert y Hy x a Hx.
    (* explicitly indicate # subgoals with [ | ... | ] if > 1 *)
    cbv [bar]; refine (natlike_ind _ _ _); [ | ].
    { (* y = 0 *)
      intros; lia. }
    { (* y = Z.succ _ *)
      intros.
      rewrite Z.add_succ_l, Z.pow_succ_r by lia.
      split.
      { (* 0 <= bar x y *)
        apply Z.mul_nonneg_nonneg; [ lia | ].
        apply Z.pow_nonneg; lia. }
      { (* bar x y < a ^ y *)
        rewrite Z.pow_succ_r by lia.
        apply Z.mul_le_mono_nonneg; try lia;
          [ apply Z.pow_nonneg; lia | ].
        (* For more flexible proofs, use match statements to find hypotheses
           rather than referring to them by autogenerated names like H0. In this
           case, we'll take any hypothesis that applies to and then solves the
           goal. *)
        match goal with H : _ |- _ => apply H; solve [auto] end. } }
  Qed.

  (* Put notations in a separate module or file so that importers can
     decide whether or not to use them. *)
  Module BarNotations.
    Infix "#" := bar (at level 40) : Z_scope.
    Notation "x '##'" := (bar x x) (at level 40) : Z_scope.
  End BarNotations.

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Appendix A Index - Features

Jump to:   S  
Index Entry  Section
S
scheduling: Scheduling
Jump to:   S  

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Appendix B GNU Free Documentation License

Version 1.3, 3 November 2008 
Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
https://fsf.org/

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
  1. PREAMBLE

    The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

    This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

    We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

  2. APPLICABILITY AND DEFINITIONS

    This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

    A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

    A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

    The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.

    The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

    A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not "Transparent" is called "Opaque".

    Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

    The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work’s title, preceding the beginning of the body of the text.

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    A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition.

    The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

  3. VERBATIM COPYING

    You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

    You may also lend copies, under the same conditions stated above, and you may publicly display copies.

  4. COPYING IN QUANTITY

    If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document’s license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

    If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

    If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

    It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

  5. MODIFICATIONS

    You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

    1. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
    2. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
    3. State on the Title page the name of the publisher of the Modified Version, as the publisher.
    4. Preserve all the copyright notices of the Document.
    5. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
    6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
    7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document’s license notice.
    8. Include an unaltered copy of this License.
    9. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
    10. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
    11. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
    12. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
    13. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version.
    14. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section.
    15. Preserve any Warranty Disclaimers.

    If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles.

    You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

    You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

    The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

  6. COMBINING DOCUMENTS

    You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

    The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

    In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements."

  7. COLLECTIONS OF DOCUMENTS

    You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

    You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

  8. AGGREGATION WITH INDEPENDENT WORKS

    A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

    If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

  9. TRANSLATION

    Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

    If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

  10. TERMINATION

    You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.

    However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

    Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

    Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.

  11. FUTURE REVISIONS OF THIS LICENSE

    The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See https://www.gnu.org/copyleft/.

    Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

  12. RELICENSING

    "Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site.

    "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.

    "Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document.

    An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.

    The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

B.1 ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: 

Copyright (C)  YEAR  YOUR NAME.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
Texts.  A copy of the license is included in the section entitled ``GNU
Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with…Texts." line with this: 

with the Invariant Sections being LIST THEIR TITLES, with
the Front-Cover Texts being LIST, and with the Back-Cover Texts
being LIST.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.