$\begin{split}\newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{#1[] \vdash #2 \lra #3}} \newcommand{\WEVT}[3]{\mbox{#1[] \vdash #2 \lra}\\ \mbox{ #3}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}$

# Early history of Coq¶

## Historical roots¶

Coq is a proof assistant for higher-order logic, allowing the development of computer programs consistent with their formal specification. It is the result of about ten years [1] of research of the Coq project. We shall briefly survey here three main aspects: the logical language in which we write our axiomatizations and specifications, the proof assistant which allows the development of verified mathematical proofs, and the program extractor which synthesizes computer programs obeying their formal specifications, written as logical assertions in the language.

The logical language used by Coq is a variety of type theory, called the Calculus of Inductive Constructions. Without going back to Leibniz and Boole, we can date the creation of what is now called mathematical logic to the work of Frege and Peano at the turn of the century. The discovery of antinomies in the free use of predicates or comprehension principles prompted Russell to restrict predicate calculus with a stratification of types. This effort culminated with Principia Mathematica, the first systematic attempt at a formal foundation of mathematics. A simplification of this system along the lines of simply typed λ-calculus occurred with Church’s Simple Theory of Types. The λ-calculus notation, originally used for expressing functionality, could also be used as an encoding of natural deduction proofs. This Curry-Howard isomorphism was used by N. de Bruijn in the Automath project, the first full-scale attempt to develop and mechanically verify mathematical proofs. This effort culminated with Jutting’s verification of Landau’s Grundlagen in the 1970’s. Exploiting this Curry-Howard isomorphism, notable achievements in proof theory saw the emergence of two type-theoretic frameworks; the first one, Martin-Löf’s Intuitionistic Theory of Types, attempts a new foundation of mathematics on constructive principles. The second one, Girard’s polymorphic λ-calculus $$F_\omega$$, is a very strong functional system in which we may represent higher-order logic proof structures. Combining both systems in a higher-order extension of the Automath language, T. Coquand presented in 1985 the first version of the Calculus of Constructions, CoC. This strong logical system allowed powerful axiomatizations, but direct inductive definitions were not possible, and inductive notions had to be defined indirectly through functional encodings, which introduced inefficiencies and awkwardness. The formalism was extended in 1989 by T. Coquand and C. Paulin with primitive inductive definitions, leading to the current Calculus of Inductive Constructions. This extended formalism is not rigorously defined here. Rather, numerous concrete examples are discussed. We refer the interested reader to relevant research papers for more information about the formalism, its meta-theoretic properties, and semantics. However, it should not be necessary to understand this theoretical material in order to write specifications. It is possible to understand the Calculus of Inductive Constructions at a higher level, as a mixture of predicate calculus, inductive predicate definitions presented as typed PROLOG, and recursive function definitions close to the language ML.

Automated theorem-proving was pioneered in the 1960’s by Davis and Putnam in propositional calculus. A complete mechanization (in the sense of a semidecision procedure) of classical first-order logic was proposed in 1965 by J.A. Robinson, with a single uniform inference rule called resolution. Resolution relies on solving equations in free algebras (i.e. term structures), using the unification algorithm. Many refinements of resolution were studied in the 1970’s, but few convincing implementations were realized, except of course that PROLOG is in some sense issued from this effort. A less ambitious approach to proof development is computer-aided proof-checking. The most notable proof-checkers developed in the 1970’s were LCF, designed by R. Milner and his colleagues at U. Edinburgh, specialized in proving properties about denotational semantics recursion equations, and the Boyer and Moore theorem-prover, an automation of primitive recursion over inductive data types. While the Boyer-Moore theorem-prover attempted to synthesize proofs by a combination of automated methods, LCF constructed its proofs through the programming of tactics, written in a high-level functional meta-language, ML.

The salient feature which clearly distinguishes our proof assistant from say LCF or Boyer and Moore’s, is its possibility to extract programs from the constructive contents of proofs. This computational interpretation of proof objects, in the tradition of Bishop’s constructive mathematics, is based on a realizability interpretation, in the sense of Kleene, due to C. Paulin. The user must just mark his intention by separating in the logical statements the assertions stating the existence of a computational object from the logical assertions which specify its properties, but which may be considered as just comments in the corresponding program. Given this information, the system automatically extracts a functional term from a consistency proof of its specifications. This functional term may be in turn compiled into an actual computer program. This methodology of extracting programs from proofs is a revolutionary paradigm for software engineering. Program synthesis has long been a theme of research in artificial intelligence, pioneered by R. Waldinger. The Tablog system of Z. Manna and R. Waldinger allows the deductive synthesis of functional programs from proofs in tableau form of their specifications, written in a variety of first-order logic. Development of a systematic programming logic, based on extensions of Martin-Löf’s type theory, was undertaken at Cornell U. by the Nuprl team, headed by R. Constable. The first actual program extractor, PX, was designed and implemented around 1985 by S. Hayashi from Kyoto University. It allows the extraction of a LISP program from a proof in a logical system inspired by the logical formalisms of S. Feferman. Interest in this methodology is growing in the theoretical computer science community. We can foresee the day when actual computer systems used in applications will contain certified modules, automatically generated from a consistency proof of their formal specifications. We are however still far from being able to use this methodology in a smooth interaction with the standard tools from software engineering, i.e. compilers, linkers, run-time systems taking advantage of special hardware, debuggers, and the like. We hope that Coq can be of use to researchers interested in experimenting with this new methodology.

 [1] At the time of writing, i.e. 1995.

## Versions 1 to 5¶

Note

This summary was written in 1995 together with the previous section and formed the initial version of the Credits chapter.

A more comprehensive description of these early versions is available in the following subsections, which come from a document written in September 2015 by Gérard Huet, Thierry Coquand and Christine Paulin.

A first implementation of CoC was started in 1984 by G. Huet and T. Coquand. Its implementation language was CAML, a functional programming language from the ML family designed at INRIA in Rocquencourt. The core of this system was a proof-checker for CoC seen as a typed λ-calculus, called the Constructive Engine. This engine was operated through a high-level notation permitting the declaration of axioms and parameters, the definition of mathematical types and objects, and the explicit construction of proof objects encoded as λ-terms. A section mechanism, designed and implemented by G. Dowek, allowed hierarchical developments of mathematical theories. This high-level language was called the Mathematical Vernacular. Furthermore, an interactive Theorem Prover permitted the incremental construction of proof trees in a top-down manner, subgoaling recursively and backtracking from dead-ends. The theorem prover executed tactics written in CAML, in the LCF fashion. A basic set of tactics was predefined, which the user could extend by his own specific tactics. This system (Version 4.10) was released in 1989. Then, the system was extended to deal with the new calculus with inductive types by C. Paulin, with corresponding new tactics for proofs by induction. A new standard set of tactics was streamlined, and the vernacular extended for tactics execution. A package to compile programs extracted from proofs to actual computer programs in CAML or some other functional language was designed and implemented by B. Werner. A new user-interface, relying on a CAML-X interface by D. de Rauglaudre, was designed and implemented by A. Felty. It allowed operation of the theorem-prover through the manipulation of windows, menus, mouse-sensitive buttons, and other widgets. This system (Version 5.6) was released in 1991.

Coq was ported to the new implementation Caml-light of X. Leroy and D. Doligez by D. de Rauglaudre (Version 5.7) in 1992. A new version of Coq was then coordinated by C. Murthy, with new tools designed by C. Parent to prove properties of ML programs (this methodology is dual to program extraction) and a new user-interaction loop. This system (Version 5.8) was released in May 1993. A Centaur interface CTCoq was then developed by Y. Bertot from the Croap project from INRIA-Sophia-Antipolis.

In parallel, G. Dowek and H. Herbelin developed a new proof engine, allowing the general manipulation of existential variables consistently with dependent types in an experimental version of Coq (V5.9).

The version V5.10 of Coq is based on a generic system for manipulating terms with binding operators due to Chet Murthy. A new proof engine allows the parallel development of partial proofs for independent subgoals. The structure of these proof trees is a mixed representation of derivation trees for the Calculus of Inductive Constructions with abstract syntax trees for the tactics scripts, allowing the navigation in a proof at various levels of details. The proof engine allows generic environment items managed in an object-oriented way. This new architecture, due to C. Murthy, supports several new facilities which make the system easier to extend and to scale up:

• User-programmable tactics are allowed
• It is possible to separately verify development modules, and to load their compiled images without verifying them again - a quick relocation process allows their fast loading
• A generic parsing scheme allows user-definable notations, with a symmetric table-driven pretty-printer
• Syntactic definitions allow convenient abbreviations
• A limited facility of meta-variables allows the automatic synthesis of certain type expressions, allowing generic notations for e.g. equality, pairing, and existential quantification.

In the Fall of 1994, C. Paulin-Mohring replaced the structure of inductively defined types and families by a new structure, allowing the mutually recursive definitions. P. Manoury implemented a translation of recursive definitions into the primitive recursive style imposed by the internal recursion operators, in the style of the ProPre system. C. Muñoz implemented a decision procedure for intuitionistic propositional logic, based on results of R. Dyckhoff. J.C. Filliâtre implemented a decision procedure for first-order logic without contraction, based on results of J. Ketonen and R. Weyhrauch. Finally C. Murthy implemented a library of inversion tactics, relieving the user from tedious definitions of “inversion predicates”.

Rocquencourt, Feb. 1st 1995
Gérard Huet

### Version 1¶

This software is a prototype type-checker for a higher-order logical formalism known as the Theory of Constructions, presented in his PhD thesis by Thierry Coquand, with influences from Girard's system F and de Bruijn's Automath. The metamathematical analysis of the system is the PhD work of Thierry Coquand. The software is mostly the work of Gérard Huet. Most of the mathematical examples verified with the software are due to Thierry Coquand.

The programming language of the CONSTR software (as it was called at the time) was a version of ML adapted from the Edinburgh LCF system and running on a LISP backend. The main improvements from the original LCF ML were that ML was compiled rather than interpreted (Gérard Huet building on the original translator by Lockwood Morris), and that it was enriched by recursively defined types (work of Guy Cousineau). This ancestor of CAML was used and improved by Larry Paulson for his implementation of Cambridge LCF.

Software developments of this prototype occurred from late 1983 to early 1985.

Version 1.10 was frozen on December 22nd 1984. It is the version used for the examples in Thierry Coquand's thesis, defended on January 31st 1985. There was a unique binding operator, used both for universal quantification (dependent product) at the level of types and functional abstraction (λ) at the level of terms/proofs, in the manner of Automath. Substitution (λ-reduction) was implemented using de Bruijn's indexes.

Version 1.11 was frozen on February 19th, 1985. It is the version used for the examples in the paper: T. Coquand, G. Huet. Constructions: A Higher Order Proof System for Mechanizing Mathematics [CH85].

Christine Paulin joined the team at this point, for her DEA research internship. In her DEA memoir (August 1985) she presents developments for the lambo function – $$\text{lambo}(f)(n)$$ computes the minimal $$m$$ such that $$f(m)$$ is greater than $$n$$, for $$f$$ an increasing integer function, a challenge for constructive mathematics. She also encoded the majority voting algorithm of Boyer and Moore.

### Version 2¶

The formal system, now renamed as the Calculus of Constructions, was presented with a proof of consistency and comparisons with proof systems of Per Martin Löf, Girard, and the Automath family of N. de Bruijn, in the paper: T. Coquand and G. Huet. The Calculus of Constructions [CH86b].

An abstraction of the software design, in the form of an abstract machine for proof checking, and a fuller sequence of mathematical developments was presented in: T. Coquand, G. Huet. Concepts Mathématiques et Informatiques Formalisés dans le Calcul des Constructions [CH86a].

Version 2.8 was frozen on December 16th, 1985, and served for developing the examples in the above papers.

This calculus was then enriched in version 2.9 with a cumulative hierarchy of universes. Universe levels were initially explicit natural numbers. Another improvement was the possibility of automatic synthesis of implicit type arguments, relieving the user of tedious redundant declarations.

Christine Paulin wrote an article Algorithm development in the Calculus of Constructions [Moh86]. Besides lambo and majority, she presents quicksort and a text formatting algorithm.

Version 2.13 of the Calculus of Constructions with universes was frozen on June 25th, 1986.

A synthetic presentation of type theory along constructive lines with ML algorithms was given by Gérard Huet in his May 1986 CMU course notes Formal Structures for Computation and Deduction. Its chapter Induction and Recursion in the Theory of Constructions was presented as an invited paper at the Joint Conference on Theory and Practice of Software Development TAPSOFT’87 at Pise in March 1987, and published as Induction Principles Formalized in the Calculus of Constructions [Hue88].

### Version 3¶

This version saw the beginning of proof automation, with a search algorithm inspired from PROLOG and the applicative logic programming programs of the course notes Formal structures for computation and deduction. The search algorithm was implemented in ML by Thierry Coquand. The proof system could thus be used in two modes: proof verification and proof synthesis, with tactics such as AUTO.

The implementation language was now called CAML, for Categorical Abstract Machine Language. It used as backend the LLM3 virtual machine of Le Lisp by Jérôme Chailloux. The main developers of CAML were Michel Mauny, Ascander Suarez and Pierre Weis.

V3.1 was started in the summer of 1986, V3.2 was frozen at the end of November 1986. V3.4 was developed in the first half of 1987.

Thierry Coquand held a post-doctoral position in Cambridge University in 1986-87, where he developed a variant implementation in SML, with which he wrote some developments on fixpoints in Scott's domains.

### Version 4¶

This version saw the beginning of program extraction from proofs, with two varieties of the type Prop of propositions, indicating constructive intent. The proof extraction algorithms were implemented by Christine Paulin-Mohring.

V4.1 was frozen on July 24th, 1987. It had a first identified library of mathematical developments (directory exemples), with libraries Logic (containing impredicative encodings of intuitionistic logic and algebraic primitives for booleans, natural numbers and list), Peano developing second-order Peano arithmetic, Arith defining addition, multiplication, euclidean division and factorial. Typical developments were the Knaster-Tarski theorem and Newman's lemma from rewriting theory.

V4.2 was a joint development of a team consisting of Thierry Coquand, Gérard Huet and Christine Paulin-Mohring. A file V4.2.log records the log of changes. It was frozen on September 1987 as the last version implemented in CAML 2.3, and V4.3 followed on CAML 2.5, a more stable development system.

V4.3 saw the first top-level of the system. Instead of evaluating explicit quotations, the user could develop his mathematics in a high-level language called the mathematical vernacular (following Automath terminology). The user could develop files in the vernacular notation (with .v extension) which were now separate from the ml sources of the implementation. Gilles Dowek joined the team to develop the vernacular language as his DEA internship research.

A notion of sticky constant was introduced, in order to keep names of lemmas when local hypotheses of proofs were discharged. This gave a notion of global mathematical environment with local sections.

Another significant practical change was that the system, originally developed on the VAX central computer of our lab, was transferred on SUN personal workstations, allowing a level of distributed development. The extraction algorithm was modified, with three annotations Pos, Null and Typ decorating the sorts Prop and Type.

Version 4.3 was frozen at the end of November 1987, and was distributed to an early community of users (among those were Hugo Herbelin and Loic Colson).

V4.4 saw the first version of (encoded) inductive types. Now natural numbers could be defined as:

[source, coq]
Inductive NAT : Prop = O : NAT | Succ : NAT->NAT.


These inductive types were encoded impredicatively in the calculus, using a subsystem rec due to Christine Paulin. V4.4 was frozen on March 6th 1988.

Version 4.5 was the first one to support inductive types and program extraction. Its banner was Calcul des Constructions avec Réalisations et Synthèse. The vernacular language was enriched to accommodate extraction commands.

The verification engine design was presented as: G. Huet. The Constructive Engine. Version 4.5. Invited Conference, 2nd European Symposium on Programming, Nancy, March 88. The final paper, describing the V4.9 implementation, appeared in: A perspective in Theoretical Computer Science, Commemorative Volume in memory of Gift Siromoney, Ed. R. Narasimhan, World Scientific Publishing, 1989.

Version 4.5 was demonstrated in June 1988 at the YoP Institute on Logical Foundations of Functional Programming organized by Gérard Huet at Austin, Texas.

Version 4.6 was started during the summer of 1988. Its main improvement was the complete rehaul of the proof synthesis engine by Thierry Coquand, with a tree structure of goals.

Its source code was communicated to Randy Pollack on September 2nd 1988. It evolved progressively into LEGO, proof system for Luo's formalism of Extended Calculus of Constructions.

The discharge tactic was modified by Gérard Huet to allow for inter-dependencies in discharged lemmas. Christine Paulin improved the inductive definition scheme in order to accommodate predicates of any arity.

Version 4.7 was started on September 6th, 1988.

This version starts exploiting the CAML notion of module in order to improve the modularity of the implementation. Now the term verifier is identified as a proper module Machine, which the structure of its internal data structures being hidden and thus accessible only through the legitimate operations. This machine (the constructive engine) was the trusted core of the implementation. The proof synthesis mechanism was a separate proof term generator. Once a complete proof term was synthesized with the help of tactics, it was entirely re-checked by the engine. Thus there was no need to certify the tactics, and the system took advantage of this fact by having tactics ignore the universe levels, universe consistency check being relegated to the final type-checking pass. This induced a certain puzzlement in early users who saw, after a successful proof search, their QED followed by silence, followed by a failure message due to a universe inconsistency…

The set of examples comprise set theory experiments by Hugo Herbelin, and notably the Schroeder-Bernstein theorem.

Version 4.8, started on October 8th, 1988, saw a major re-implementation of the abstract syntax type constr, separating variables of the formalism and metavariables denoting incomplete terms managed by the search mechanism. A notion of level (with three values TYPE, OBJECT and PROOF) is made explicit and a type judgement clarifies the constructions, whose implementation is now fully explicit. Structural equality is speeded up by using pointer equality, yielding spectacular improvements. Thierry Coquand adapts the proof synthesis to the new representation, and simplifies pattern matching to first-order predicate calculus matching, with important performance gain.

A new representation of the universe hierarchy is then defined by Gérard Huet. Universe levels are now implemented implicitly, through a hidden graph of abstract levels constrained with an order relation. Checking acyclicity of the graph insures well-foundedness of the ordering, and thus consistency. This was documented in a memo Adding Type:Type to the Calculus of Constructions which was never published.

The development version is released as a stable 4.8 at the end of 1988.

Version 4.9 is released on March 1st 1989, with the new "elastic" universe hierarchy.

The spring of 1989 saw the first attempt at documenting the system usage, with a number of papers describing the formalism:

• Metamathematical Investigations of a Calculus of Constructions, by Thierry Coquand [Coq89],
• Inductive definitions in the Calculus of Constructions, by Christine Paulin-Mohrin,
• Extracting Fω's programs from proofs in the Calculus of Constructions, by Christine Paulin-Mohring* [PM89],
• The Constructive Engine, by Gérard Huet [Hue89],

as well as a number of user guides:

• A short user's guide for the Constructions, Version 4.10, by Gérard Huet
• A Vernacular Syllabus, by Gilles Dowek.
• The Tactics Theorem Prover, User's guide, Version 4.10, by Thierry Coquand.

Stable V4.10, released on May 1st, 1989, was then a mature system, distributed with CAML V2.6.

In the mean time, Thierry Coquand and Christine Paulin-Mohring had been investigating how to add native inductive types to the Calculus of Constructions, in the manner of Per Martin-Löf's Intuitionistic Type Theory. The impredicative encoding had already been presented in: F. Pfenning and C. Paulin-Mohring. Inductively defined types in the Calculus of Constructions [PPM89]. An extension of the calculus with primitive inductive types appeared in: T. Coquand and C. Paulin-Mohring. Inductively defined types [CP90].

This led to the Calculus of Inductive Constructions, logical formalism implemented in Versions 5 upward of the system, and documented in: C. Paulin-Mohring. Inductive Definitions in the System Coq - Rules and Properties [PM93b].

The last version of CONSTR is Version 4.11, which was last distributed in the spring of 1990. It was demonstrated at the first workshop of the European Basic Research Action Logical Frameworks In Sophia Antipolis in May 1990.

### Version 5¶

At the end of 1989, Version 5.1 was started, and renamed as the system Coq for the Calculus of Inductive Constructions. It was then ported to the new stand-alone implementation of ML called Caml-light.

In 1990 many changes occurred. Thierry Coquand left for Chalmers University in Göteborg. Christine Paulin-Mohring took a CNRS researcher position at the LIP laboratory of École Normale Supérieure de Lyon. Project Formel was terminated, and gave rise to two teams: Cristal at INRIA-Roquencourt, that continued developments in functional programming with Caml-light then OCaml, and Coq, continuing the type theory research, with a joint team headed by Gérard Huet at INRIA-Rocquencourt and Christine Paulin-Mohring at the LIP laboratory of CNRS-ENS Lyon.

Chetan Murthy joined the team in 1991 and became the main software architect of Version 5. He completely rehauled the implementation for efficiency. Versions 5.6 and 5.8 were major distributed versions, with complete documentation and a library of users' developments. The use of the RCS revision control system, and systematic ChangeLog files, allow a more precise tracking of the software developments.

September 2015 +
Thierry Coquand, Gérard Huet and Christine Paulin-Mohring.

## Versions 6¶

### Version 6.1¶

The present version 6.1 of Coq is based on the V5.10 architecture. It was ported to the new language Objective Caml by Bruno Barras. The underlying framework has slightly changed and allows more conversions between sorts.

The new version provides powerful tools for easier developments.

Cristina Cornes designed an extension of the Coq syntax to allow definition of terms using a powerful pattern matching analysis in the style of ML programs.

Amokrane Saïbi wrote a mechanism to simulate inheritance between types families extending a proposal by Peter Aczel. He also developed a mechanism to automatically compute which arguments of a constant may be inferred by the system and consequently do not need to be explicitly written.

Yann Coscoy designed a command which explains a proof term using natural language. Pierre Crégut built a new tactic which solves problems in quantifier-free Presburger Arithmetic. Both functionalities have been integrated to the Coq system by Hugo Herbelin.

Samuel Boutin designed a tactic for simplification of commutative rings using a canonical set of rewriting rules and equality modulo associativity and commutativity.

Finally the organisation of the Coq distribution has been supervised by Jean-Christophe Filliâtre with the help of Judicaël Courant and Bruno Barras.

Lyon, Nov. 18th 1996
Christine Paulin

### Version 6.2¶

In version 6.2 of Coq, the parsing is done using camlp4, a preprocessor and pretty-printer for CAML designed by Daniel de Rauglaudre at INRIA. Daniel de Rauglaudre made the first adaptation of Coq for camlp4, this work was continued by Bruno Barras who also changed the structure of Coq abstract syntax trees and the primitives to manipulate them. The result of these changes is a faster parsing procedure with greatly improved syntax-error messages. The user-interface to introduce grammar or pretty-printing rules has also changed.

Eduardo Giménez redesigned the internal tactic libraries, giving uniform names to Caml functions corresponding to Coq tactic names.

Bruno Barras wrote new, more efficient reduction functions.

Hugo Herbelin introduced more uniform notations in the Coq specification language: the definitions by fixpoints and pattern matching have a more readable syntax. Patrick Loiseleur introduced user-friendly notations for arithmetic expressions.

New tactics were introduced: Eduardo Giménez improved the mechanism to introduce macros for tactics, and designed special tactics for (co)inductive definitions; Patrick Loiseleur designed a tactic to simplify polynomial expressions in an arbitrary commutative ring which generalizes the previous tactic implemented by Samuel Boutin. Jean-Christophe Filliâtre introduced a tactic for refining a goal, using a proof term with holes as a proof scheme.

David Delahaye designed the tool to search an object in the library given its type (up to isomorphism).

Henri Laulhère produced the Coq distribution for the Windows environment.

Finally, Hugo Herbelin was the main coordinator of the Coq documentation with principal contributions by Bruno Barras, David Delahaye, Jean-Christophe Filliâtre, Eduardo Giménez, Hugo Herbelin and Patrick Loiseleur.

Orsay, May 4th 1998
Christine Paulin

### Version 6.3¶

The main changes in version V6.3 were the introduction of a few new tactics and the extension of the guard condition for fixpoint definitions.

B. Barras extended the unification algorithm to complete partial terms and fixed various tricky bugs related to universes.

D. Delahaye developed the AutoRewrite tactic. He also designed the new behavior of Intro and provided the tacticals First and Solve.

J.-C. Filliâtre developed the Correctness tactic.

E. Giménez extended the guard condition in fixpoints.

H. Herbelin designed the new syntax for definitions and extended the Induction tactic.

P. Loiseleur developed the Quote tactic and the new design of the Auto tactic, he also introduced the index of errors in the documentation.

C. Paulin wrote the Focus command and introduced the reduction functions in definitions, this last feature was proposed by J.-F. Monin from CNET Lannion.

Orsay, Dec. 1999
Christine Paulin

## Versions 7¶

### Summary of changes¶

The version V7 is a new implementation started in September 1999 by Jean-Christophe Filliâtre. This is a major revision with respect to the internal architecture of the system. The Coq version 7.0 was distributed in March 2001, version 7.1 in September 2001, version 7.2 in January 2002, version 7.3 in May 2002 and version 7.4 in February 2003.

Jean-Christophe Filliâtre designed the architecture of the new system. He introduced a new representation for environments and wrote a new kernel for type checking terms. His approach was to use functional data-structures in order to get more sharing, to prepare the addition of modules and also to get closer to a certified kernel.

Hugo Herbelin introduced a new structure of terms with local definitions. He introduced “qualified” names, wrote a new pattern matching compilation algorithm and designed a more compact algorithm for checking the logical consistency of universes. He contributed to the simplification of Coq internal structures and the optimisation of the system. He added basic tactics for forward reasoning and coercions in patterns.

David Delahaye introduced a new language for tactics. General tactics using pattern matching on goals and context can directly be written from the Coq toplevel. He also provided primitives for the design of user-defined tactics in Caml.

Micaela Mayero contributed the library on real numbers. Olivier Desmettre extended this library with axiomatic trigonometric functions, square, square roots, finite sums, Chasles property and basic plane geometry.

Jean-Christophe Filliâtre and Pierre Letouzey redesigned a new extraction procedure from Coq terms to Caml or Haskell programs. This new extraction procedure, unlike the one implemented in previous version of Coq is able to handle all terms in the Calculus of Inductive Constructions, even involving universes and strong elimination. P. Letouzey adapted user contributions to extract ML programs when it was sensible. Jean-Christophe Filliâtre wrote coqdoc, a documentation tool for Coq libraries usable from version 7.2.

Bruno Barras improved the efficiency of the reduction algorithm and the confidence level in the correctness of Coq critical type checking algorithm.

Yves Bertot designed the SearchPattern and SearchRewrite tools and the support for the pcoq interface (http://www-sop.inria.fr/lemme/pcoq/).

Micaela Mayero and David Delahaye introduced Field, a decision tactic for commutative fields.

Christine Paulin changed the elimination rules for empty and singleton propositional inductive types.

Loïc Pottier developed Fourier, a tactic solving linear inequalities on real numbers.

Pierre Crégut developed a new, reflection-based version of the Omega decision procedure.

Claudio Sacerdoti Coen designed an XML output for the Coq modules to be used in the Hypertextual Electronic Library of Mathematics (HELM cf http://www.cs.unibo.it/helm).

A library for efficient representation of finite maps using binary trees contributed by Jean Goubault was integrated in the basic theories.

Pierre Courtieu developed a command and a tactic to reason on the inductive structure of recursively defined functions.

Jacek Chrząszcz designed and implemented the module system of Coq whose foundations are in Judicaël Courant’s PhD thesis.

The development was coordinated by C. Paulin.

Many discussions within the Démons team and the LogiCal project influenced significantly the design of Coq especially with J. Courant, J. Duprat, J. Goubault, A. Miquel, C. Marché, B. Monate and B. Werner.

Intensive users suggested improvements of the system : Y. Bertot, L. Pottier, L. Théry, P. Zimmerman from INRIA, C. Alvarado, P. Crégut, J.-F. Monin from France Telecom R & D.

Orsay, May. 2002
Hugo Herbelin & Christine Paulin

### Details of changes in 7.0 and 7.1¶

Notes:

• items followed by (**) are important sources of incompatibilities
• items followed by (*) may exceptionally be sources of incompatibilities
• items followed by (+) have been introduced in version 7.0

#### Main novelties¶

References are to Coq 7.1 reference manual

• New primitive let-in construct (see sections 1.2.8 and )
• Long names (see sections 2.6 and 2.7)
• New high-level tactic language (see chapter 10)
• Improved search facilities (see section 5.2)
• New extraction algorithm managing the Type level (see chapter 17)
• New rewriting tactic for arbitrary equalities (see chapter 19)
• New tactic Field to decide equalities on commutative fields (see 7.11)
• New tactic Fourier to solve linear inequalities on reals numbers (see 7.11)
• New tactics for induction/case analysis in "natural" style (see 7.7)
• Deep restructuration of the code (safer, simpler and more efficient)
• Export of theories to XML for publishing and rendering purposes (see http://www.cs.unibo.it/helm)

#### Details of changes¶

##### Language: new "let-in" construction¶
• New construction for local definitions (let-in) with syntax [x:=u]t (*)(+)
• Local definitions allowed in Record (a.k.a. record à la Randy Pollack)
##### Language: long names¶
• Each construction has a unique absolute names built from a base name, the name of the module in which they are defined (Top if in coqtop), and possibly an arbitrary long sequence of directory (e.g. "Coq.Lists.PolyList.flat_map" where "Coq" means that "flat_map" is part of Coq standard library, "Lists" means it is defined in the Lists library and "PolyList" means it is in the file Polylist) (+)
• Constructions can be referred by their base name, or, in case of conflict, by a "qualified" name, where the base name is prefixed by the module name (and possibly by a directory name, and so on). A fully qualified name is an absolute name which always refer to the construction it denotes (to preserve the visibility of all constructions, no conflict is allowed for an absolute name) (+)
• Long names are available for modules with the possibility of using the directory name as a component of the module full name (with option -R to coqtop and coqc, or command Add LoadPath) (+)
• Improved conflict resolution strategy (the Unix PATH model), allowing more constructions to be referred just by their base name
##### Language: miscellaneous¶
• The names of variables for Record projections _and_ for induction principles (e.g. sum_ind) is now based on the first letter of their type (main source of incompatibility) (**)(+)
• Most typing errors have now a precise location in the source (+)
• Slightly different mechanism to solve "?" (*)(+)
• More arguments may be considered implicit at section closing (*)(+)
• Bug with identifiers ended by a number greater than 2^30 fixed (+)
• New visibility discipline for Remark, Fact and Local: Remark's and Fact's now survive at the end of section, but are only accessible using a qualified names as soon as their strength expires; Local's disappear and are moved into local definitions for each construction persistent at section closing
##### Language: Cases¶
• Cases no longer considers aliases inferable from dependencies in types (*)(+)
• A redundant clause in Cases is now an error (*)
##### Reduction¶
• New reduction flags "Zeta" and "Evar" in Eval Compute, for inlining of local definitions and instantiation of existential variables
• Delta reduction flag does not perform Zeta and Evar reduction any more (*)
• Constants declared as opaque (using Qed) can no longer become transparent (a constant intended to be alternatively opaque and transparent must be declared as transparent (using Defined)); a risk exists (until next Coq version) that Simpl and Hnf reduces opaque constants (*)
##### New tactics¶
• New set of tactics to deal with types equipped with specific equalities (a.k.a. Setoids, e.g. nat equipped with eq_nat) [by C. Renard]
• New tactic Assert, similar to Cut but expected to be more user-friendly
• New tactic NewDestruct and NewInduction intended to replace Elim and Induction, Case and Destruct in a more user-friendly way (see restrictions in the reference manual)
• New tactic ROmega: an experimental alternative (based on reflexion) to Omega [by P. Crégut]
• New tactic language Ltac (see reference manual) (+)
• New versions of Tauto and Intuition, fully rewritten in the new Ltac language; they run faster and produce more compact proofs; Tauto is fully compatible but, in exchange of a better uniformity, Intuition is slightly weaker (then use Tauto instead) (**)(+)
• New tactic Field to decide equalities on commutative fields (as a special case, it works on real numbers) (+)
• New tactic Fourier to solve linear inequalities on reals numbers [by L. Pottier] (+)
• New tactics dedicated to real numbers: DiscrR, SplitRmult, SplitAbsolu (+)
##### Changes in existing tactics¶
• Reduction tactics in local definitions apply only to the body
• New syntax of the form "Compute in Type of H." to require a reduction on the types of local definitions
• Inversion, Injection, Discriminate, ... apply also on the quantified premises of a goal (using the "Intros until" syntax)
• Decompose has been fixed but hypotheses may get different names (*)(+)
• Tauto now manages uniformly hypotheses and conclusions of the form t=t which all are considered equivalent to True. Especially, Tauto now solves goals of the form H : ~ t = t |- A.
• The "Let" tactic has been renamed "LetTac" and is now based on the primitive "let-in" (+)
• Elim can no longer be used with an elimination schema different from the one defined at definition time of the inductive type. To overload an elimination schema, use "Elim <hyp> using <name of the new schema>" (*)(+)
• Simpl no longer unfolds the recursive calls of a mutually defined fixpoint (*)(+)
• Intro now fails if the hypothesis name already exists (*)(+)
• "Require Prolog" is no longer needed (i.e. it is available by default) (*)(+)
• Unfold now fails on a non unfoldable identifier (*)(+)
• Unfold also applies on definitions of the local context
• AutoRewrite now deals only with the main goal and it is the purpose of Hint Rewrite to deal with generated subgoals (+)
• Redundant or incompatible instantiations in Apply ... with ... are now correctly managed (+)
##### Efficiency¶
• Excessive memory uses specific to V7.0 fixed
• Sizes of .vo files vary a lot compared to V6.3 (from -30% to +300% depending on the developments)
• An improved reduction strategy for lazy evaluation
• A more economical mechanism to ensure logical consistency at the Type level; warning: this is experimental and may produce "universes" anomalies (please report)
• Only identifiers starting with "_" or a letter, and followed by letters, digits, "_" or "'" are allowed (e.g. "$" and "@" are no longer allowed) (*) • A multiple binder like (a:A)(a,b:(P a))(Q a) is no longer parsed as (a:A)(a0:(P a))(b:(P a))(Q a0) but as (a:A)(a0:(P a))(b:(P a0))(Q a0) (*)(+) • A dedicated syntax has been introduced for Reals (e.g 3+1/x) (+) • Pretty-printing of Infix notations fixed. (+) ##### Parsing and grammar extension¶ • More constraints when writing ast • "{...}" and the macros$LIST, $VAR, etc. now expect a metavariable (an identifier starting with$) (*)
• identifiers should starts with a letter or "_" and be followed
by letters, digits, "_" or "'" (other characters are still supported but it is not advised to use them) (*)(+)
• Entry "command" in "Grammar" and quotations (<<...>> stuff) is renamed "constr" as in "Syntax" (+)
• New syntax "[" sentence_1 ... sentence_n"]." to group sentences (useful for Time and to write grammar rules abbreviating several commands) (+)
• The default parser for actions in the grammar rules (and for patterns in the pretty-printing rules) is now the one associated to the grammar (i.e. vernac, tactic or constr); no need then for quotations as in <:vernac:<...>>; to return an "ast", the grammar must be explicitly typed with tag ": ast" or ": ast list", or if a syntax rule, by using <<...>> in the patterns (expression inside these angle brackets are parsed as "ast"); for grammars other than vernac, tactic or constr, you may explicitly type the action with tags ": constr", ": tactic", or ":vernac" (**)(+)
• Interpretation of names in Grammar rule is now based on long names, which allows to avoid problems (or sometimes tricks;) related to overloaded names (+)
##### New commands¶
• New commands "Print XML All", "Show XML Proof", ... to show or export theories to XML to be used with Helm's publishing and rendering tools (see http://www.cs.unibo.it/helm) (by Claudio Sacerdoti Coen) (+)
• New commands to manually set implicit arguments (+)
• "Implicits ident." to activate the implicit arguments mode just for ident
• "Implicits ident [num1 num2 ...]." to explicitly give which
arguments have to be considered as implicit
• New SearchPattern/SearchRewrite (by Yves Bertot) (+)
• New commands "Debug on"/"Debug off" to activate/deactivate the tactic language debugger (+)
• New commands to map physical paths to logical paths (+) - Add LoadPath physical_dir as logical_dir - Add Rec LoadPath physical_dir as logical_dir
##### Changes in existing commands¶
• Generalization of the usage of qualified identifiers in tactics and commands about globals, e.g. Decompose, Eval Delta; Hints Unfold, Transparent, Require

• Require synchronous with Reset; Require's scope stops at Section ending (*)

• For a module indirectly loaded by a "Require" but not exported, the command "Import module" turns the constructions defined in the module accessible by their short name, and activates the Grammar, Syntax, Hint, ... declared in the module (+)

• The scope of the "Search" command can be restricted to some modules (+)

• Final dot in command (full stop/period) must be followed by a blank (newline, tabulation or whitespace) (+)

• Slight restriction of the syntax for Cbv Delta: if present, option [-myconst] must immediately follow the Delta keyword (*)(+)

• SearchIsos currently not supported

• New names for the following commands (+)

Implicit Arguments On -> Set Implicit Arguments Implicit Arguments Off -> Unset Implicit Arguments

Begin Silent -> Set Silent End Silent -> Unset Silent.

##### Tools¶
• coqtop (+)
• Two executables: coqtop.byte and coqtop.opt (if supported by the platform)
• coqtop is a link to the more efficient executable (coqtop.opt if present)
• option -full is obsolete (+)
• do_Makefile renamed into coq_makefile (+)
• New option -R to coqtop and coqc to map a physical directory to a logical one (+)
• coqc no longer needs to create a temporary file
• No more warning if no initialization file .coqrc exists
##### Extraction¶
• New algorithm for extraction able to deal with "Type" (+) (by J.-C. Filliâtre and P. Letouzey)
##### Standard library¶
• New library on maps on integers (IntMap, contributed by Jean Goubault)
• New lemmas about integer numbers [ZArith]
• New lemmas and a "natural" syntax for reals [Reals] (+)
• Exc/Error/Value renamed into Option/Some/None (*)
##### New user contributions¶
• Constructive complex analysis and the Fundamental Theorem of Algebra [FTA] (Herman Geuvers, Freek Wiedijk, Jan Zwanenburg, Randy Pollack, Henk Barendregt, Nijmegen)
• A new axiomatization of ZFC set theory [Functions_in_ZFC] (C. Simpson, Sophia-Antipolis)
• Basic notions of graph theory [GRAPHS-BASICS] (Jean Duprat, Lyon)
• A library for floating-point numbers [Float] (Laurent Théry, Sylvie Boldo, Sophia-Antipolis)
• Formalisation of CTL and TCTL temporal logic [CtlTctl] (Carlos Daniel Luna,Montevideo)
• Specification and verification of the Railroad Crossing Problem in CTL and TCTL [RailroadCrossing] (Carlos Daniel Luna,Montevideo)
• P-automaton and the ABR algorithm [PAutomata] (Christine Paulin, Emmanuel Freund, Orsay)
• Semantics of a subset of the C language [MiniC] (Eduardo Giménez, Emmanuel Ledinot, Suresnes)
• Correctness proofs of the following imperative algorithms: Bresenham line drawing algorithm [Bresenham], Marché's minimal edition distance algorithm [Diff] (Jean-Christophe Filliâtre, Orsay)
• Correctness proofs of Buchberger's algorithm [Buchberger] and RSA cryptographic algorithm [Rsa] (Laurent Théry, Sophia-Antipolis)
• Correctness proof of Stalmarck tautology checker algorithm [Stalmarck] (Laurent Théry, Pierre Letouzey, Sophia-Antipolis)

### Details of changes in 7.2¶

Language

• Automatic insertion of patterns for local definitions in the type of the constructors of an inductive types (for compatibility with V6.3 let-in style)
• Coercions allowed in Cases patterns
• New declaration "Canonical Structure id = t : I" to help resolution of equations of the form (proj ?)=a; if proj(e)=a then a is canonically equipped with the remaining fields in e, i.e. ? is instantiated by e

Tactics

• New tactic "ClearBody H" to clear the body of definitions in local context
• New tactic "Assert H := c" for forward reasoning
• Slight improvement in naming strategy for NewInduction/NewDestruct
• Intuition/Tauto do not perform useless unfolding and work up to conversion

Extraction (details in plugins/extraction/CHANGES or documentation)

• Syntax changes: there are no more options inside the extraction commands. New commands for customization and options have been introduced instead.
• More optimizations on extracted code.
• Extraction tests are now embedded in 14 user contributions.

Standard library

• In [Relations], Rstar.v and Newman.v now axiom-free.
• In [Sets], Integers.v now based on nat
• In [Arith], more lemmas in Min.v, new file Max.v, tail-recursive plus and mult added to Plus.v and Mult.v respectively
• New directory [Sorting] with a proof of heapsort (dragged from 6.3.1 lib)
• In [Reals], more lemmas in Rbase.v, new lemmas on square, square root and trigonometric functions (R_sqr.v - Rtrigo.v); a complementary approach and new theorems about continuity and derivability in Ranalysis.v; some properties in plane geometry such as translation, rotation or similarity in Rgeom.v; finite sums and Chasles property in Rsigma.v

Bugs

• Confusion between implicit args of locals and globals of same base name fixed
• Various incompatibilities wrt inference of "?" in V6.3.1 fixed
• Implicits in infix section variables bug fixed
• Known coercions bugs fixed
• Apply "universe anomaly" bug fixed
• NatRing now working
• "Discriminate 1", "Injection 1", "Simplify_eq 1" now working
• NewInduction bugs with let-in and recursively dependent hypotheses fixed
• Syntax [x:=t:T]u now allowed as mentioned in documentation
• Bug with recursive inductive types involving let-in fixed
• Known pattern-matching bugs fixed
• Known Cases elimination predicate bugs fixed
• Improved errors messages for pattern-matching and projections
• Better error messages for ill-typed Cases expressions

Incompatibilities

• New naming strategy for NewInduction/NewDestruct may affect 7.1 compatibility
• Extra parentheses may exceptionally be needed in tactic definitions.
• Coq extensions written in Ocaml need to be updated (see dev/changements.txt for a description of the main changes in the interface files of V7.2)
• New behaviour of Intuition/Tauto may exceptionally lead to incompatibilities

### Details of changes in 7.3¶

Language

• Slightly improved compilation of pattern-matching (slight source of incompatibilities)
• Record's now accept anonymous fields "_" which does not build projections
• Changes in the allowed elimination sorts for certain class of inductive definitions : an inductive definition without constructors of Sort Prop can be eliminated on sorts Set and Type A "singleton" inductive definition (one constructor with arguments in the sort Prop like conjunction of two propositions or equality) can be eliminated directly on sort Type (In V7.2, only the sorts Prop and Set were allowed)

Tactics

• New tactic "Rename x into y" for renaming hypotheses
• New tactics "Pose x:=u" and "Pose u" to add definitions to local context
• Pattern now working on partially applied subterms
• Ring no longer applies irreversible congruence laws of mult but better applies congruence laws of plus (slight source of incompatibilities).
• Field now accepts terms to be simplified as arguments (as for Ring). This extension has been also implemented using the toplevel tactic language.
• Intuition does no longer unfold constants except "<->" and "~". It can be parameterized by a tactic. It also can introduce dependent product if needed (source of incompatibilities)
• "Match Context" now matching more recent hypotheses first and failing only on user errors and Fail tactic (possible source of incompatibilities)
• Tactic Definition's without arguments now allowed in Coq states
• Better simplification and discrimination made by Inversion (source of incompatibilities)

Bugs

• "Intros H" now working like "Intro H" trying first to reduce if not a product
• Forward dependencies in Cases now taken into account
• Known bugs related to Inversion and let-in's fixed
• Bug unexpected Delta with let-in now fixed

Extraction (details in plugins/extraction/CHANGES or documentation)

• Signatures of extracted terms are now mostly expunged from dummy arguments.
• Haskell extraction is now operational (tested & debugged).

Standard library

• Some additions in [ZArith]: three files (Zcomplements.v, Zpower.v and Zlogarithms.v) moved from plugins/omega in order to be more visible, one Zsgn function, more induction principles (Wf_Z.v and tail of Zcomplements.v), one more general Euclid theorem
• Peano_dec.v and Compare_dec.v now part of Arith.v

Tools

User Contributions

• CongruenceClosure (congruence closure decision procedure) [Pierre Corbineau, ENS Cachan]
• MapleMode (an interface to embed Maple simplification procedures over rational fractions in Coq) [David Delahaye, Micaela Mayero, Chalmers University]
• Presburger: A formalization of Presburger's algorithm [Laurent Thery, INRIA Sophia Antipolis]
• Chinese has been rewritten using Z from ZArith as datatype ZChinese is the new version, Chinese the obsolete one [Pierre Letouzey, LRI Orsay]

Incompatibilities

• Ring: exceptional incompatibilities (1 above 650 in submitted user contribs, leading to a simplification)
• Intuition: does not unfold any definition except "<->" and "~"
• Cases: removal of some extra Cases in configurations of the form "Cases ... of C _ => ... | _ D => ..." (effects on 2 definitions of submitted user contributions necessitating the removal of now superfluous proof steps in 3 different proofs)
• Match Context, in case of incompatibilities because of a now non trapped error (e.g. Not_found or Failure), use instead tactic Fail to force Match Context trying the next clause
• Inversion: better simplification and discrimination may occasionally lead to less subgoals and/or hypotheses and different naming of hypotheses
• Unification done by Apply/Elim has been changed and may exceptionally lead to incompatible instantiations
• Peano_dec.v and Compare_dec.v parts of Arith.v make Auto more powerful if these files were not already required (1 occurrence of this in submitted user contribs)

#### Changes in 7.3.1¶

Bug fixes

• Corrupted Field tactic and Match Context tactic construction fixed
• Invalid argument bug in Exact tactic solved (#1387)
• Colliding bound names bug fixed (#1412)
• Wrong non-recursivity test for Record fixed (#1394)
• Out of memory/seg fault bug related to parametric inductive fixed (#1404)
• Setoid_replace/Setoid_rewrite bug wrt "==" fixed

Misc

• Ocaml version >= 3.06 is needed to compile Coq from sources
• Simplification of fresh names creation strategy for Assert, Pose and LetTac (#1402)

### Details of changes in 7.4¶

Symbolic notations

• Introduction of a notion of scope gathering notations in a consistent set; a notation sets has been developed for nat, Z and R (undocumented)
• New command "Notation" for declaring notations simultaneously for parsing and printing (see chap 10 of the reference manual)
• Declarations with only implicit arguments now handled (e.g. the argument of nil can be set implicit; use !nil to refer to nil without arguments)
• "Print Scope sc" and "Locate ntn" allows to know to what expression a notation is bound
• New defensive strategy for printing or not implicit arguments to ensure re-type-checkability of the printed term
• In Grammar command, the only predefined non-terminal entries are ident, global, constr and pattern (e.g. nvar, numarg disappears); the only allowed grammar types are constr and pattern; ast and ast list are no longer supported; some incompatibilities in Grammar: when a syntax is a initial segment of an other one, Grammar does not work, use Notation

Library

• Lemmas in Set from Compare_dec.v (le_lt_dec, ...) and Wf_nat.v (lt_wf_rec, ...) are now transparent. This may be source of incompatibilities.
• Syntactic Definitions Fst, Snd, Ex, All, Ex2, AllT, ExT, ExT2, ProjS1, ProjS2, Error, Value and Except are turned to notations. They now must be applied (incompatibilities only in unrealistic cases).
• More efficient versions of Zmult and times (30% faster)
• Reals: the library is now divided in 6 parts (Rbase, Rfunctions, SeqSeries, Rtrigo, Ranalysis, Integration). New tactics: Sup and RCompute. See Reals.v for details.

Modules

• Beta version, see doc chap 2.5 for commands and chap 5 for theory

Language

• Inductive definitions now accept ">" in constructor types to declare the corresponding constructor as a coercion.
• Idem for assumptions declarations and constants when the type is mentioned.
• The "Coercion" and "Canonical Structure" keywords now accept the same syntax as "Definition", i.e. "hyps :=c (:t)?" or "hyps :t".
• Theorem-like declaration now accepts the syntax "Theorem thm [x:t;...] : u".
• Remark's and Fact's now definitively behave as Theorem and Lemma: when sections are closed, the full name of a Remark or a Fact has no longer a section part (source of incompatibilities)
• Opaque Local's (i.e. built by tactics and ended by Qed), do not survive section closing any longer; as a side-effect, Opaque Local's now appear in the local context of proofs; their body is hidden though (source of incompatibilities); use one of Remark/Fact/Lemma/Theorem instead to simulate the old behaviour of Local (the section part of the name is not kept though)

ML tactic and vernacular commands

• "Grammar tactic" and "Grammar vernac" of type "ast" are no longer supported (only "Grammar tactic simple_tactic" of type "tactic" remains available).
• Concrete syntax for ML written vernacular commands and tactics is now declared at ML level using camlp4 macros TACTIC EXTEND et VERNAC COMMAND EXTEND.
• "Check n c" now "n:Check c", "Eval n ..." now "n:Eval ..."
• Proof with T (no documentation)
• SearchAbout id - prints all theorems which contain id in their type

Tactic definitions

• Static globalisation of identifiers and global references (source of incompatibilities, especially, Recursive keyword is required for mutually recursive definitions).
• New evaluation semantics: no more partial evaluation at definition time; evaluation of all Tactic/Meta Definition, even producing terms, expect a proof context to be evaluated (especially "()" is no longer needed).
• Debugger now shows the nesting level and the reasons of failure

Tactics

• Equality tactics (Rewrite, Reflexivity, Symmetry, Transitivity) now understand JM equality
• Simpl and Change now apply to subterms also
• "Simpl f" reduces subterms whose head constant is f
• Double Induction now referring to hypotheses like "Intros until"
• "Inversion" now applies also on quantified hypotheses (naming as for Intros until)
• NewDestruct now accepts terms with missing hypotheses
• NewDestruct and NewInduction now accept user-provided elimination scheme
• NewDestruct and NewInduction now accept user-provided introduction names
• Omega could solve goals such as ~x<y |- x>=y but failed when the hypothesis was unfolded to x < y -> False. This is fixed. In addition, it can also recognize 'False' in the hypothesis and use it to solve the goal.
• Coercions now handled in "with" bindings
• "Subst x" replaces all occurrences of x by t in the goal and hypotheses when an hypothesis x=t or x:=t or t=x exists
• Fresh names for Assert and Pose now based on collision-avoiding Intro naming strategy (exceptional source of incompatibilities)
• LinearIntuition (no documentation)
• Unfold expects a correct evaluable argument
• Clear expects existing hypotheses

Extraction (See details in plugins/extraction/CHANGES and README):

• An experimental Scheme extraction is provided.
• Concerning Ocaml, extracted code is now ensured to always type-check, thanks to automatic inserting of Obj.magic.
• Experimental extraction of Coq new modules to Ocaml modules.

Proof rendering in natural language

Miscellaneous

• Printing Coercion now used through the standard keywords Set/Add, Test, Print
• "Print Term id" is an alias for "Print id"
• New switch "Unset/Set Printing Symbols" to control printing of symbolic notations
• Two new variants of implicit arguments are available
• Unset/Set Contextual Implicits tells to consider implicit also the arguments inferable from the context (e.g. for nil or refl_eq)
• Unset/Set Strict Implicits tells to consider implicit only the arguments that are inferable in any case (i.e. arguments that occurs as argument of rigid constants in the type of the remaining arguments; e.g. the witness of an existential is not strict since it can vanish when applied to a predicate which does not use its argument)

Incompatibilities

• "Grammar tactic ... : ast" and "Grammar vernac ... : ast" are no longer supported, use TACTIC EXTEND and VERNAC COMMAND EXTEND on the ML-side instead
• Transparency of le_lt_dec and co (leads to some simplification in proofs; in some cases, incompatibilites is solved by declaring locally opaque the relevant constant)
• Opaque Local do not now survive section closing (rename them into Remark/Lemma/... to get them still surviving the sections; this renaming allows also to solve incompatibilites related to now forbidden calls to the tactic Clear)
• Remark and Fact have no longer (very) long names (use Local instead in case of name conflict)

Bugs

• Improved localisation of errors in Syntactic Definitions
• Induction principle creation failure in presence of let-in fixed (#1459)
• Inversion bugs fixed (#1427 and #1437)
• Omega bug related to Set fixed (#1384)
• Type-checking inefficiency of nested destructuring let-in fixed (#1435)
• Improved handling of let-in during holes resolution phase (#1460)

Efficiency

• Implementation of a memory sharing strategy reducing memory requirements by an average ratio of 3.