Sequent Calculus, Derivability, Provability. Gödel's Completeness Theorem

Fifth of a series of articles laying down the bases for classical first order model theory. This paper presents multiple themes: first it introduces sequents, rules and sets of rules for a first order language L as L-dependent types. Then defines derivability and provability according to a set of rules, and gives several technical lemmas binding all those concepts. Following that, it introduces a fixed set D of derivation rules, and proceeds to convert them to Mizar functorial cluster registrations to give the user a slick interface to apply them. The remaining goals summon all the definitions and results introduced in this series of articles. First: D is shown to be correct and having the requisites to deliver a sensible definition of Henkin model (see [18]). Second: as a particular application of all the machinery built thus far, the satisfiability and Gödel completeness theorems are shown when restricting to countable languages. The techniques used to attain this are inspired from [18], then heavily modified with the twofold goal of embedding them into the more flexible framework of a variable ruleset here introduced, and of proving completeness of a set of rules more sparing than the one there used; in particular the simpler ruleset allowed to avoid the definition and tractation of free occurence of a literal, a fact which, along with shortening proofs, is remarkable in its own right. A preparatory account of some of the ideas used in the proofs given here can be found in [15].Mathematics Department "G. Castelnuovo", Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011, doi: 10.2478/v10037-011-0025-2.Marco B. Caminati. Definition of first order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms. Formalized Mathematics, 19(3):169-178, 2011, doi: 10.2478/v10037-011-0026-1.Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011, doi: 10.2478/v10037-011-0027-0.Marco B. Caminati. Free interpretation, quotient interpretation and substitution of a letter with a term for first order languages. Formalized Mathematics, 19(3):193-203, 2011, doi: 10.2478/v10037-011-0028-z.M. B. Caminati. Yet another proof of Goedel's completeness theorem for first-order classical logic. Arxiv preprint arXiv:0910.2059, 2009.Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part I. Formalized Mathematics, 2(5):683-687, 1991.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Springer, 1994.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Marta Pruszyńska and Marek Dudzicz. On the isomorphism between finite chains. Formalized Mathematics, 9(2):429-430, 2001.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

Aspects of the constructive omega rule within automated deduction

In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.De begränsningar av formella system som uppdagades av Gödels ofullständighetsteorem år 1931 innebär att Peanoaritmetikens konsistens endast kan bevisas med hjälp av fundamentala principer som inte kan formaliseras inom systemet. Trots att Hilberts finitistiska metoder inte kunde producera ett konsistensbevis, så fortsatte sökandet efter ett bevis med konstruktiva metoder. För att ett bevis skall vara meningsfullt borde principerna som används vara mera pålitliga än de element som betvivlas inom teorin. Avhandlingens titel hänvisar till ett citat av Gentzen då han motiverar behovet av konsistensbevis för första ordningens aritmetik. Gentzen själv producerade fyra konsistensbevis och analyserade hur väl dessa stämde överens med Hilberts program. Gentzen använde konstruktiva metoder i sina bevis, men det debatteras huruvida dessa metoder kan anses vara finitistiska. Det tredje och mest kända beviset presenterar en reduktion av härledningar av kontradiktioner. Med hjälp av transfinit induktion visas att reduktionsprocessen terminerar i en enkel härledning som konstateras vara omöjlig. Därför kan det inte finnas någon härledning av en kontradiktion. Avhandlingen undersöker och jämför Gentzens bevis från olika aspekter. Konsistensen av intuitionistisk Heytingaritmetik bevisas både i sekvenskalkyl och i naturlig deduktion. Det tidigare beviset är i Gentzens anda och innehåller ett snittelimineringsbevis för kalkylen och en syntaktisk studie av den aritmetiska delen av systemet. Det senare beviset påminner om ett normaliseringsbevis och visar terminering med hjälp av en vektortilldelning.Gödelin vuonna 1931 jullkaisemista epätäydellisyyslauseista seurausi rajoituksia formaalisille järjestelmille: Niiden mukaan Peano-aritmetiikan ristiriidattomuus voidaan todistaa ainoastaan periaatteilla, jotka eivät ole formalisoitavissa järjestelmän itsensä sisällä. Vaikka Hilbertin finitistisillä menetelmillä ei siksi pystytty tuottamaan konsistenssitodistusta, todistuksen etsiminen jatkui konstruktiivisillä menetelmillä. Jotta todistus olisi mielekäs, siinä käytettyjen periaatteiden oli oltava luotettavampia kuin teorian itsensä sisältämät periaatteet. Väitöskirjan otsikko viittaa Gentzenin kirjoitukseen, jossa hän perustelee ensimmäisen kertaluvun aritmetiikan konsistenssitodistuksen tarvetta. Gentzen itse laati neljä sellaista konsistenssitodistusta ja analysoi, missä määrin ne olivat yhdenmukaisia Hilbertin ohjelman kanssa. Gentzen käytti konstruktiivisia menetelmiä todistuksissaan ja on paljon väitelty kysymys, voidaanko näitä menetelmiä pitää finitistisinä. Kolmannessa ja tunnetuimassa Gentzenin todistuksessa esitetään ristiriitaisuuksien päättelyn reduktiomenetelmä. Transfiniittistä induktiota käyttämällä osoitetaan, että reduktioprosessi päättyy yksinkertaiseen päättelyyn, jollainen on erikseen todettu mahdottomaksi. Tämän vuoksi ristiriitaa ei voida päätellä. Väitöskirjassa selvitetään ja vertaillaan Gentzenin todistuksia eri näkökulmista. Intuitionistisen Heyting-aritmetiikan ristiriidattomuus osoitetaan sekä sekvenssikalkyylissä että luonnollisessa päättelyssä. Ensimmäinen todistus seuraa Gentzenin henkeä ja siinä sovelletaan ns. leikkaussäänön eliminointitodistusta sekä syntaktista analyysia järjestelmän aritmeettisesta osasta. Jälkimmäinen todistus muistuttaa luonnollisen päättelyn normalisointitodistusta ja näyttää reduktion päättymisen vektorimäärityksen avulla

Syntactical consistency proofs for term induction revisited

GIn [6] wird von Gerhard Gentzen die Widerspruchsfreiheit der Peano Arithmetik erster Stufe PA bewiesen. Die Methode geht dabei folgendermaßen vor: Man deniert einen simplen Teil SPA der Peano Arithmetik (SPA enthält im speziellen keine Anwendung des Induktionsschemas) und zeigt zuerst die von SPA. Der Rest des Arguments verläuft indirekt. Man nimmt an, dass PA einen Widerspruch ableitet und zeigt das dessen Deduktion zu einer Deduktion in SPA transformiert werden kann, was der Widerspruchsfreiheit von SPA widerspricht. Diese Transformation verläuft wie folgt: Jeder Deduktion in PA wird eine Ordinalzahl (oder genauer, ein Ordinalzahlterm eines Ordinalzahlnotations Systems) zugeordnet, diese wird der Rang der Deduktion genannt. Dann wird gezeigt, dass es zu jeder Deduktion eines Widerspruches (die nicht in SPA verläuft) eine Deduktion (ebenfalls eines Widerspruches) gibt die einen kleineren Rang hat. Diese Methode benötigt daher die Wohlfundiertheit des verwendeten Ordinalzahlnotations Systems (in diesem Fall bis "0). Bei näherer Betrachtung von Gentzens Methode fällt auf, dass sie lediglich folgende Eigenschaften von PA verwendet: 1. Alle Axiome von PA sind Allsätze oder Instanzen des Induktionsschemas φ(0) ^ ∀ x[φ(x) -> φ(x + 1)] -> ∀ xφ(x): 2. Alle geschlossenen Terme sind beweisbar (in SPA) gleich zu einem Term der lediglich aus 0 und dem Symbol der Nachfolgerfunktion aufgebaut ist. Dies erlaubt eine Verallgemeinerung von Gentzens Methode. In dieser Diplomarbeit werden wir daher Theorien Tf = (Tf )0 [ (Tf )Ind betrachten die folgende Eigenschaften erfüllen: 1. (Tf )0 besteht lediglich aus Allsätzen. 2. (Tf )Ind beinhaltet alle Instanzen des Induktionsschemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): 3. Der simple Teil von Tf beweist für jeden geschlossenen Term t, dass t gleich einem Term _t ist der lediglich aus den symbolen c1; :::; cm und f aufgebaut ist. Die Widerspruchsfreiheit von Tf kann nun, wie in [6] für PA, relativ zu ihrem simplen Teil (wo Induktion wie zuvor bei Gentzen nicht möglich ist) gezeigt werden. Eine Konsequenz dieses Resultates ist das folgende Korollar. Korollar. Sei φ(a) quantorenfrei und Tf widerspruchsfrei. Wenn Tf j= 9xφ(x), dann (Tf )0 j= 9xφ(x). Insbesondere ist Tf _0 1 -konservativ über (Tf )0. Es scheint mir als wäre die Methode, die von Kurt Schütte in seinem Widerspruchsfreiheitsbeweis von PA verwendet wird, eine gänzlich andere. Schütte, Tait und Andere verwenden Kalküle mit unendlichen Deduktionsregeln um, in einem gewissen Sinne, die Beweistheoretische Ordinalzahl einer Theorie zu berechnen. Dies erfolgt über eine Transformation der endlichenDeduktionen der Theorie (in der Logik erster Stufe) in Deduktionen ineinem unendlichen Kalkül, das Schnittelimination erlaubt. Im Gegensatz zu Gentzens Methode hat die von Schütte eine enge Beziehung zu den beweistheoretischenOrdinalzahlen. Auf die Unterschiede der beiden Methoden wird nicht weiter eingegangen werden. Anstatt dieses Vergleiches wird lediglich eine Variante von Taits Methode dazu verwendet die _1 1-Ordinalzahl, wie von Wolfram Pohlers in [∀ ] beschrieben, von Theorien TAf (aufgefasst als Taitkalkühl) zu messen. Es wird angenommen das TAf folgende Eigenschaften erfüllt: 1. TAf enthält für jede primitiv rekursive Funktion die denierenden Formeln als Axiome. 2. Weiters enthält TAf alle Instanzen des Schemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): Hierbei ist f ein m-stelliges Symbol einer gleichstelligen primitiv rekursiven Funktion und c1; :::; cl Individuenkonstanten. 3. Es wird außerdem angenommen das jedes n 2 N gleich einer Komposition aus fN und den natürlichen Zahlen cN1;:::; cN l ist.Gerhard Gentzen proves the consistency of first-order Peano arithmetic \textbf{PA}. His method works as follows: Define a simple part \textbf{SPA} of peano arithmetic (\textbf{SPA} does in particular not contain induction) and first show the consistency of \textbf{SPA}. Now assume towards a contradiction that \textbf{PA} deducts an contradiction. Show that this deduction can be transformed into a deduction in \textbf{SPA}, this contradicts the consistency of \textbf{SPA}. How to get a deduction in \textbf{SPA}: We assign an ordinal (more exact an ordinal term of an ordinal notation system) to each deduction in \textbf{PA}, called the rank of the deduction. Next show that for each deduction which deducts a contradiction (and is not in \textbf{SPA}) there is a deduction (also deducting and contradiction) with smaller rank. This method requires that the ordinal notation system (which goes up to $\varepsilon_0$) is well-founded. It turns out that Gentzen's method requires only to the following properties of \textbf{PA}: \begin{enumerate} \item All axioms of \textbf{PA} are universal sentences or instances of the induction schema $$\varphi(0) \wedge \forall x [\varphi(x) \rightarrow \varphi(x+1)] \rightarrow \forall x \varphi(x).$$ \item All closed terms are provable equal to a term build up just from $0$ and the symbol of the successor function. \end{enumerate} This allows a slight generalisation of Gentzen's method. In this Diploma Thesis we consider theories $T_f = (T_f)_0 \cup (T_f)_{\text{Ind}}$ with the following properties: \begin{enumerate} \item $(T_f)_0$ contains only universal sentences. \item $(T_f)_{\text{Ind}}$ contains all instances of the general induction schema $$\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x).$$ \item The simple part of $T_f$ proves for every closed term $t$ the equality of $t$ to a term $\bar{t}$ build up just from $c_1,...,c_m$ and $f$. \end{enumerate} As Gentzen did for \textbf{PA}, the consistency of $T_f$ can be shown with respect to their simple part which corresponds to the simple part of Gentzen (also without induction). As a consequence, one gets the following result for all such theories. \begin{cor2} Assume $\varphi(a)$ is quantifier free and $T_f$ consistent.\\ If $T_f \models \exists x \varphi(x)$, then $(T_f)_0 \models \exists x \varphi(x)$.\\ I.e., $T_f$ is $\Sigma_1^0$-conservative over $(T_f)_0$. \end{cor2} It seems that this method is different in an essential way to the method Kurt Sch\"{u}tte uses in his consistency proof of \textbf{PA}. Sch\"{u}tte, Tait and others uses calculi with infinite deduction rules. These methods compute, in some sense, the proof theoretical ordinal of the considered theory by embedding the deductions of the theory (in ordinary first-order logic) in an infinite system which allows cut-elimination. In contrast to Gentzen's method Sch\"{u}tte's and Tait's methods are closely related to the proof theoretical ordinals.\\ We do not provide an analysis of the disparities of both methods. Instead we present the point of view Wolfram Pohlers propose, to measure the $\Pi_1^1$-ordinal of theories $\mathsf{TA}_f$ (presented as a Tait-calculus) satisfying the following conditions: \begin{enumerate} \item $\mathsf{TA}_f$ includes all defining axioms for primitive recursive functions. \item All instances of the schema $$\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x)$$ are included. Here $f$ is an $m$-array primitive recursive function constant and $c_1,...,c_l$ are individual constants. \item Every $n \in \mathbb{N}$ is equal to a composition of $f^\mathbb{N}$ and the elements $c_1^\mathbb{N},...,c_l^\mathbb{N}$. \end{enumerate

Hilbert's Metamathematical Problems and Their Solutions

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved

Introduction to Mathematical Logic, Edition 2021

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem

A Survey of Probabilistic Reasoning in Justification Logic

Σε αυτήν τη διπλωματική εργασία μελετούμε την έννοια της επιχειρηματολογίας (justification), αναπαριστάμενη σε ένα λογικό φορμαλισμό. Μελετούμε την επιστημική / δοξαστική αναπαράσταση της justification logic, μίας επέκτασης της κλασικής λογικής (classical logic) με φόρμουλες της μορφής t:F, που μεταφράζονται ως "Το t είναι επιχείρημα που υποδεικνύει την αλήθεια της θέσης (ή την πίστη στη θέση) F.". Παρουσιάζουμε τις βασικές σημασιολογίες της justification logic, συνοδευόμενες από τα αντίστοιχα θεωρήματα ορθότητας και πληρότητας και αναλύουμε πώς εκλαμβάνει η κάθε μία την έννοια της επιχειρηματολογίας. Επίσης, αναλύουμε την έννοια τις επιχειρηματολογίας συνυφασμένη με την έννοια της αβεβαιότητας, παρουσιάζοντας τις θεμελιώδεις probabilistic justification logics. Διατυπώνουμε τις αντίστοιχες σημασιολογίες, μαζί με τα αντίστοιχα θεωρήματα ορθότητας και πληρότητας και εξετάζουμε πώς η κάθε μία λογική αντιλαμβάνεται την αβεβαιότητα στο πλαίσιο της επιχειρηματολογίας. Τέλος, μελετούμε μία νέα σημασιολογία που προτάθηκε και μελετήθηκε εκ των E. Lehmann και T. Studer τα τελευταία τρία χρόνια, ονόματι subset models. Ελέγχουμε πώς τα subset models θα μπορούσαν να συνδυαστούν με τη θεωρία πιθανοτήτων, στην προσπάθεια κατασκευής μίας πιθανοτικής λογικής που διαχωρίζει μεταξύ της αβεβαιότητας υπό το πρίσμα της πειστικότητας του επιχειρήματος, της αβεβαιότητας υπό το πρίσμα της αποδεικτικότητας της θέσης εκ του επιχειρήματος και τις αβεβαιότητας ισχύς της θέσης.In this thesis, we study the notion of justification, interpreted in a logical formalism. Specifically, we study the epistemic/doxastic interpretation of justification logic; i.e., an expansion of classical logic with formulae of the form t:F, which translate as "t is an evidence of the truth of F.". We present the basic semantics for justification logic, along with the corresponding theorems of soundness and completeness, and analyze how each one of them perceives the notion of justification. Moreover, we examine the notion of justification in relation to the notion of uncertainty, by presenting the fundamental probabilistic justification logics. We present the corresponding semantics, accompanied with the corresponding soundness and (sort of) completeness and we investigate how each one of these perceives the uncertainty in the context of justification. Last but not least, we define the subset models, a recent semantics for justification logic proposed and studied by E. Lehmann and T. Studer. We analyze the ontology of justification, as it is expressed in this framework, and we examine how subset models could probably combine with the notion of uncertainty, in a way that distinguishes between the suasiveness of the evidence t, the conclusiveness of evidence t over assertion F, and the certainty of F

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined. The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible. Along with the trivial symbol substitution generally defined in [11], the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in [13] (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]). This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule in [14]. A third result supplied is that if two given languages coincide on the letters of a given FinSequence, their evaluation of it will also coincide. This too will be instrumental in [14] for proving correctness of another rule. Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.Mathematics Department "G. Castelnuovo", Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011, doi: 10.2478/v10037-011-0025-2.Marco B. Caminati. Definition of first order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms. Formalized Mathematics, 19(3):169-178, 2011, doi: 10.2478/v10037-011-0026-1.Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011, doi: 10.2478/v10037-011-0027-0.Marco B. Caminati. Sequent calculus, derivability, provability. Gödel's completeness theorem. Formalized Mathematics, 19(3):205-222, 2011, doi: 10.2478/v10037-011-0029-y.H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Springer, 1994.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Krzysztof Retel. Properties of first and second order cutting of binary relations. Formalized Mathematics, 13(3):361-365, 2005.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Many-argument relations. Formalized Mathematics, 1(4):733-737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990