Clausal Analysis of First-order Proof Schemata

TL;DR

This paper explores a novel approach to cut-elimination in proof schemata using lazy instantiation of proof links, advancing the understanding of schematic cut-elimination and its relation to characteristic clause sets.

Contribution

It introduces a lazy instantiation technique for proof links in proof schemata, enabling analysis of cut-elimination's effects on characteristic clause sets.

Findings

Lazy instantiation allows local evaluation of proof links.

Characteristic clause sets are subsumed during reductive cut-elimination.

Results extend to proof schemata, aiding complete cut-elimination methods.

Abstract

Proof schemata are a variant of LK-proofs able to simulate various induction schemes in first-order logic by adding so called proof links to the standard first-order LK-calculus. Proof links allow proofs to reference proofs thus giving proof schemata a recursive structure. Unfortunately, applying reductive cut- elimination is non-trivial in the presence of proof links. Borrowing the concept of lazy instantiation from functional programming, we evaluate proof links locally allowing reductive cut-elimination to proceed past them. Though, this method cannot be used to obtain cut-free proof schemata, we nonetheless obtain important results concerning the schematic CERES method, that is a method of cut-elimination for proof schemata based on resolution. In "Towards a clausal analysis of cut-elimination", it was shown that reductive cut-elimination transforms a given LK-proof in such a way that a subsumption relation holds between the pre- and post-transformation characteristic clause sets, i.e. the clause set representing the cut-structure of an LK-proof. Let CL(A') be the characteristic clause set of a normal form A' of an LK-proof A that is reached by performing reductive cut-elimination on A without atomic cut elimination. Then CL(A') is subsumed by all characteristic clause sets extractable from any application of reductive cut-elimination to A. Such a normal form is referred to as an ACNF top and plays an essential role in methods of cut-elimination by resolution. These results can be extended to proof schemata through our "lazy instantiation" of proof links, and provides an essential step toward a complete cut-elimination method for proof schemata.

Figures

Equations

$$\underset{i=0}{\overset{0}{\bigvee }}P(i)\equiv$$

$$c{l}^{\psi ,\varnothing }(0)\equiv$$

$$c{l}^{\psi ,\varnothing }(n+1)\equiv$$

$$c{l}^{\phi ,\Omega (0)}(0)\equiv$$

$$c{l}^{\phi ,\Omega (n+1)}(n+1)\equiv$$

$$\begin{array}[]{lll}\vdash\alpha\leq\alpha&\vdash\alpha\leq g(\alpha)&0=f(\alpha),0=f(g(\alpha))\vdash\\ 1=f(\alpha),1=f(g(\alpha))\vdash&\beta\leq\alpha,f(\beta)<1\vdash f(\alpha)<0,0=f(\alpha)&f(\alpha)<0\vdash\\ f(g(\alpha))<0\vdash&0\leq\alpha\vdash f(\alpha)<1,f(\alpha)=1\end{array}$$

$${CL}^t(\{A\})=\left\{A⊢,⊢A\right\}$$

$${CL}^t(\mathcal{A}\cup {\mathcal{A}}^{\prime })={CL}^t(\mathcal{A})\times {CL}^t({\mathcal{A}}^{\prime })$$

$$\nu (k)=$$

$$\pi =$$

$$\mathcal{A}=\left\{\Big{\langle}P_{1},t_{P_{1}}^{1},a_{1}^{1}\Big{\rangle},\cdots,\Big{\langle}P_{1},t_{P_{1}}^{\alpha_{1}},a_{1}^{\alpha_{1}}\Big{\rangle},\cdots,\Big{\langle}P_{\beta},t_{P_{\beta}}^{1},a_{\beta}^{1}\Big{\rangle},\cdots,\left\langle P_{\beta},t_{P_{\beta}}^{\alpha_{\beta}},a_{\beta}^{\alpha_{\beta}}\right\rangle\right\}$$

$$\mathcal{A}{\downarrow }_{\alpha }=\bigcup _{⟨P,t_P,a⟩\in \mathcal{A}}⟨P,t_P,a⟩\downarrow \alpha .$$

$$\left\{⟨P,\lambda r.(\alpha +\hat{S}(r)),n+1⟩\right\}{\downarrow }_5=\left\{P(\alpha +0),P(\alpha +s(0)),\cdots ,P(\alpha +s^5(0))\right\}$$

Full text

0.1 Introduction

The schematic CERES (Cut Elimination by RESolution) method of cut-elimination was developed for a primitive recursively defined first-order sequent calculus, the LKS-calculus [13]. Note that LKS-proofs have a free parameter, which when instantiated results in an LK-proof. The method is based on CERES which is an alternative to the reductive cut-elimination method of Gentzen [14]. It relies on the extraction of a characteristic clause set $C$ from a sequent calculus proof $\varphi$, where $C$ represents the global cut structure of $\varphi$. $C$ is always unsatisfiable and thus, refutable using resolution. A refutation $R$ of $C$ can be used to construct a proof $\varphi^{\prime}$ ending with the same sequent as $\varphi$, $\textit{es}(\varphi)$, but containing only atomic cuts. Note that removal of atomic cuts is computationally inexpensive. The formulas of $\textit{es}(\varphi)$ can be proven by proof projections, which are small cut-free proofs algorithmically extracted from $\varphi$. Note that proof projections also contain ancestors of the cut formulas. That is the end sequents of the proof projections are of the form $\textit{es}(\varphi)\cup c$ for some $c\in C$. Attaching the projections to the leaves of $R$ and adding the necessary contractions results in $\varphi^{\prime}$. Benefits of the CERES method are its non-elementary speed-up over the Gentzen method [7] and its “global” approach to the elimination of cuts. A global approach is essential for analysis of proofs with an inductive argument.

The LKS-calculus is based on propositional schemata introduced by Aravantinos et al. [1, 2]. Propositional schemata have been used in the field of inductive theorem proving [3] to define more expressive classes of inductive formula whose satisfiability is decidable. The proofs of the LKS-calculus are used to construct proof schemata which are denoted using a finite ordered list of proof schema components. The components consist of a proof symbol, an LKS-proof and an LKE-proof. The LKE-calculus is essentially the LK-calculus with a rudimentary equational rule for dealing with recursively defined functions and predicates. In addition, LKS-proofs are allowed to “call” other proofs through proof links. To maintain well foundedness, a constraint was added to proof schemata that an LKS-proof can only contain proof links to proof schema components lower in the ordering. Also, no “free” proof links are allowed, i.e. proofs links must point to proofs within its proof schema. It was shown that proof schemata, under these constraints, can be used to express certain inductive arguments [13]. One way to think about proof schemata is as a countable set of LK-proofs. Thus, cut-elimination would be the elimination of cuts from a countable set of proofs simultaneously. One can imagine why such a problem would benefit from a global approach.

Induction has been a stumbling block for classical cut-elimination methods, i.e. reductive cut-elimination à la Gentzen, because passing cuts over the inference rule for induction can be unsound. Note that this is still an issue for the more exotic (non-reductive) methods such as CERES because the cut ancestor relation is not well defined [13]. Though, there have been systems developed to get around the issue, such as the work of Mcdowell and Miller [17], and Brotherston and Simpson [8]. These systems manage to eliminate cuts but without some of the benefits of cut-elimination like the subformula property, i.e. every formula occurring in the derivation is a subformula of some formula in the end sequent. This property admits the construction of Herbrand sequents and other objects which are essential in proof analysis. Proof schemata preserve the ancestor relation by representing an inductive argument through recursion indexed by a free parameter. Though, not completely satisfactory for reductive methods, i.e. pushing cuts through proof links is non-trivial, the preservation of the ancestor relation benefits a more global approach to the problem. The schematic CERES method [13] takes advantage of this property and results in a “proof” with the subformula property, though for a smaller class of inductive arguments than other methods. Problematically, the application of the method to proof schemata is not trivial.

A method for the algorithmic extraction of the characteristic clause set from proof schemata exists, but like the proof schemata it is primitive recursively defined. As shown in [10] refuting such a clause set is non-trivial even for a simple mathematically meaningful statement. This analysis was performed with the aid of automated theorem provers. Part of the issue is that the refutation must also be recursive. To simplify the construction of a recursive refutation, the substitutions are separated from the resolution rule and are given their own recursive definition. Essentially, we define the structure separately from the term instantiations. Thus, the result of the method is a substitution schema and a resolution refutation schema, both having a free parameter, which when instantiated can be combined to construct an LK-derivation. An LK-proof is not constructed being that schematic proof projections are not extracted. Nonetheless, the resulting LK-derivation contains enough information to allow extraction of a Herbrand system [13], that is a schematic version of a Herbrand sequent. Though, this separation simplifies the definitions and construction it requires a mapping to exists between the two schemata, thus, more complex refutations/term instantiation become difficult to represent using the method. This issue was highlighted in [9]. Thus, finding a canonical structure [20] for resolution refutations of proof schemata would remove the problem of mapping the substitutions to the refutation structure.

It has been shown that characteristic clause sets extracted from LK-proofs are of a specific shape [6, 7, 20], thus resulting in interesting clausal subsumption properties. If we consider an alternative reductive cut-elimination method, which does not eliminate atomic cuts, it is possible to prove the existence of a subsumption relation between the characteristic clause sets extracted at different steps during the application of reductive cut-elimination to an LK-proof [6, 7]. The characteristic clause set of the final proof, all atomic cuts as high as possible in the proof structure, is subsumed by all the extracted clauses sets from previous steps [6, 7]. Essentially, there is a canonical clause set associated with the CERES method [20]. What is most important about this canonical clause set is that there is only one way to refute it. The structure of this refutation is recursive and fits perfectly into the current language for representing schematic resolution refutations [11]. Problematically, we do not know if the subsumption properties hold for schematic characteristic clause sets. In the case of propositional proof schema, a weakening rule was added to the resolution calculus allowing the construction of the so called Top Atomic Cut Normal Form (ACNF${}^{\text{top}}$) [11, 20]. Note that resolution for propositional logic does not require substitutions and thus this provides a decidable and complete method for propositional proof schemata. While it seems intuitive that the result would carry over, the original clausal analysis of first-order sequent calculus proofs used the Gentzen method [14] to construct the subsumption relation between clause sets. As we have already pointed out, reductive cut-elimination and proof links do not get along.

In this paper we develop a proof schema transformation which allows one to “unroll” a proof link without instantiating the free parameter. The transformation preserves syntactic equivalence insomuch as for every instantiation of the resulting proof schema there is an instantiation of the original proof schema which results in an identical LK-proof. This transformation allows us to perform the Gentzen method on proof schemata because every time we get to a proof link we apply the transformation. Though, this does not result in a method which leads to cut-free proofs, it allows us to extend the clausal analysis results of [6, 7] to proof schemata and allows us to use the clause set of the Top Atomic Cut Normal Form (ACNF${}^{\text{top}}$). This implies that we can use the schematic resolution refutation structure discussed in [11] for every schematic characteristic clause set.

This work is part of a research program into computational proof analysis, which, so far, has lead to the analysis of Fürstenberg’s proof concerning the infinitude of primes [4]. The proof includes an inductive argument and as mentioned earlier the standard CERES method cannot be applied. A recursive formalization of the proof was constructed allowing for a informal clausal proof analysis and the discovery of a relationship between Euclid’s proof and Fürstenberg’s proof. Though, the theory of proof schemata is not yet sufficiently developed to handle such a complex proof formally, the results of this paper greatly advances the state of the art in the field. Though, investigations of proof schemata have so far been driven by the above research program, the applications are not limited to this particular type of proof analysis. They also serve as a compact way to store a formal proof whilst allowing analysis and transformations to take place. We plan to investigate alternative uses of the formalism in future work.

The rest of this paper is as follows: In Section 0.2 we discuss the necessary background knowledge needed for the results. In Section 0.3 we introduce the concept of lazy instantiation. In Section 0.4 we use lazy instantiation to perform clause analysis of proof schema and show that every schematic clause set can be subsumed by a sequence of clause sets in top form corresponding to the instantiations of the schematic clause set. In Section 0.5, we conclude the paper and discuss future work and open problems.

0.2 Preliminaries

Due to the maturity of Gentzen’s reductive cut-elimination and the sequent calculus we refrain from giving an introduction to the material. Unconventional uses of the method and/or calculus will be addressed when necessary to understanding. For more details, there exists numerous publications addressing his results ([7, 14, 19] to name a few) as well as the basics of the sequent calculus. One point we would like to address concerning reductive cut-elimination as considered in this paper is that we will use the generalized rewrite system introduced in [6] allowing application of the rewrite rules to any cut in the proof, not just to the upper-most cut. This generalized rewrite rule system $\mathcal{R}$ will be referred to as reductive cut-elimination. When an application of reductive cut-elimination must be made explicit, we write $\psi\rightarrow_{\mathcal{R}}\chi$ where $\rightarrow_{\mathcal{R}}$ is a binary relation on LK-derivations and $\psi$ and $\chi$ are LK-derivations. The meaning of $\psi\rightarrow_{\mathcal{R}}\chi$ is $\psi$ can be transformed into $\chi$ using the rewrite rule system $\mathcal{R}$. By $\rightarrow_{\mathcal{R}}^{*}$, we mean the reflexive and transitive closure of $\rightarrow_{\mathcal{R}}$.

0.2.1 The Atomic Cut Normal Form (top)

(ACNF${}^{(\text{top})}$)

The Atomic Cut Normal Form of an LK-calculus proof was introduced in [5] as the result of the CERES method. A proof transformed to ACNF still has cuts, but only rank reduction has to be applied to the proof to get a truly cut-free proof, i.e. all cuts are atomic. Rank reduction operations are the cheapest reduction rules of $\mathbf{R}$ which may be applied to an LK-proof, thus, a proof in ACNF can be thought of as essentially cut-free. In this work, we consider a special ACNF, the ACNF${}^{\text{top}}$. An ACNF${}^{\text{top}}$ is an ACNF where all cuts are shifted to the top of the proof. An easy way to transform a given proof $\varphi$ into ACNF${}^{\text{top}}$ is to apply generalized reductive cut-elimination without the following transformation rules

[TABLE]

being applied to $\varphi$. A full explanation can be found in [6, 20].

0.2.2 Proof Schemata: Language, Calculus, and Interpretation

To give a formal construction of proof schemata we extend the classical first-order term language. Following the syntax of a to be published paper concerning proof schemata [16], we partition the set of function symbols $\mathbf{P}$ into two categories, uninterpreted function symbols $\mathbf{P_{u}}$ and defined function symbols $\mathbf{P_{d}}$. Together with a countable set of schematic variable symbols these symbols are used to construct the $\iota$ sort of the term language. We will denote defined function symbol using $\hat{\cdot}$ in order to distinguish them from uninterpreted function symbols. Defined function symbols are added to the language to allow the definition of primitive recursive functions within the object language. Analogously, we allow uninterpreted predicate symbols and defined predicate symbols.

In addition to the $\iota$ sort, we add an $\omega$ sort where every term normalizes to numerals (hence, the only uninterpreted function symbols are [math] and $s(\cdot)$). We will denote numerals by lowercase Greek letters, i.e. $\alpha$, $\beta$, $\gamma$, etc. There is also a countable set $\mathcal{N}$ of parameter symbols of type $\omega$. For this work, we will only need a single parameter which in most cases we denote by $n$. We use $k,k^{\prime}$ to represent $\omega$-terms containing the parameter. This parameter symbol, referred to as the free parameter, is used to index LKS-derivations. The set of all $\omega$-terms will be denoted by

$\textstyle\Omega$

$\textstyle\Omega$

$\textstyle\Omega$

$\textstyle\Omega$ . Schematic variables are of type $\omega\to\iota$, i.e. when the schematic variable $x$ is instantiated by a numeral $\alpha$, $x(\alpha)$, it is to be interpreted as a first-order variable of type $\iota$. Hence, schematic variable symbols describe infinite sequences of distinct variables.

Formula schemata (a generalization of formulas including defined functions and predicates) are defined as usual by induction starting with a countable set of uninterpreted predicate symbols and defined predicate symbols. To deal with defined functions and predicates, we add to our interpretation a set of convergent rewrite rules, $\mathcal{E}$, of the form $\hat{f}(\bar{t})\rightarrow E$, where $\bar{t}$ contains no defined symbols, and either $\hat{f}$ is a function symbol of range $\iota$ and $E$ is a term or $\hat{f}$ is a predicate symbol and E is a formula. We extend the LK-calculus with an inference rule applying this set of rewrite rules as an equational theory. This extended calculus is referred to as LKE.

Definition 1** (LKE).**

Let $\mathcal{E}$ be a given equational theory. LKE is an extension of LK with the $\mathcal{E}$ inference rule

$S(t)$

$\mathcal{E}$

$S(t^{\prime})$

---

where the term $t$ in the sequent $S$ is replaced by a term $t^{\prime}$ for $\mathcal{E}\models t=t^{\prime}$.

Example 1**.**

We can add iterated $\lor$ and $\land$ ( the defined predicates are abbreviated as $\bigvee$ and $\bigwedge$ ) to the term language using the following rewrite rules:

[TABLE]

Formula schemata are used to build schematic sequents $\Delta\vdash\Gamma$, where $\Delta$ and $\Gamma$ are multi-sets of formula schemata. Schematic sequents are to be interpreted in the standard way. Note that the size of $\Delta$ and $\Gamma$ cannot depend on the free parameter $n$.

Schematic proofs are a finite ordered list of proof schema components that can interact with each other. This interaction is defined using so-called proof links, a 0-ary inference rule we add to LKE-calculus: Let $S(k,\bar{x})$ be a sequent where $\bar{x}$ is a vector of schematic variables. By $S(k,\bar{t})$ we denote $S(k,\bar{x})$ where $\bar{x}$ is replaced by $\bar{t}$ respectively, where $\bar{t}$ is a vector of terms of appropriate type. Furthermore, we assume a countably infinite set of proof symbols $\mathcal{B}$ denoted by $\varphi,\psi,\varphi_{i},\psi_{j}$. The expression

$(\varphi,k,\bar{t})$

$S(k,\bar{t})$

is called a proof link with the intended meaning that there is a proof called $\varphi$ with the end-sequent $S(k,\bar{x})$. For k\in\mathchoice{\ooalign{\kern 0.46pt\displaystyle\Omega$\cr\kern 0.33pt$\displaystyle\Omega$\cr\kern 0.2pt$\displaystyle\Omega$\cr$\displaystyle\Omega}}{\ooalign{\kern 0.46pt\textstyle\Omega$\cr\kern 0.33pt$\textstyle\Omega$\cr\kern 0.2pt$\textstyle\Omega$\cr$\textstyle\Omega}}{\ooalign{\kern 0.46pt\scriptstyle\Omega$\cr\kern 0.33pt$\scriptstyle\Omega$\cr\kern 0.2pt$\scriptstyle\Omega$\cr$\scriptstyle\Omega}}{\ooalign{\kern 0.46pt\scriptscriptstyle\Omega$\cr\kern 0.33pt$\scriptscriptstyle\Omega$\cr\kern 0.2pt$\scriptscriptstyle\Omega$\cr$\scriptscriptstyle\Omega}}, let $\mathcal{V}(k)$ be the set of parameters in $k$. We refer to a proof link as an $E$-proof link if $\mathcal{V}(k)\subseteq E$. Note that in this work $E=\left\{n\right\}$ or $E=\emptyset$.

Definition 2** (LKS).**

LKS is an extension of LKE, where proof links may appear at the leaves of a proof.

The class of all LKS-proofs will be denoted by (read little yus). For two LKS-proofs (LKE-proof) $\varphi$ and $\psi$, we say that $\varphi\equiv_{syn}\psi$ if they are syntactically the same. We will denote a subLKS-proof (LKE-proof) $\nu$ of an LKS-proof (LKE-proof) proof $\varphi$ by $\varphi.\nu$. If the subLKS-proof $\varphi.\nu$ is a $\left\{n\right\}$-proof link $(\psi,k,\overline{x})$ then we define $\mathcal{P}(\varphi.\nu)=\psi$ and $\mathcal{I}(\varphi.\nu)=k$. If $\varphi.\nu$ is not a proof link $\mathcal{P}(\varphi.\nu)$ and $\mathcal{I}(\varphi.\nu)$ do nothing. The set of proof links in an LKS-proof $\varphi$ is $\mathit{PL}(\varphi)$. We also define substitution of subproofs, that is let $\varphi,\psi$ be LKS-proof and $\varphi.\nu$ a subLKS-proof such that $\textit{es}(\psi)=\textit{es}(\varphi.\nu)$, then we can construct a new LKS-proof $\varphi\left[\psi\right]_{\nu}$ where $\varphi.\nu$ is replaced by $\psi$. We also allow term substitutions for the parameters of an LKS-proof.

Definition 3**.**

Let \nu\in\mbox{\includegraphics[scale={.045}]{yus}}\makebox[1.42271pt][l]{} be an LKS-proof. A parameter substitution is a term substitution \sigma:\mathcal{N}\rightarrow\mathchoice{\ooalign{\kern 0.46pt\displaystyle\Omega$\cr\kern 0.33pt$\displaystyle\Omega$\cr\kern 0.2pt$\displaystyle\Omega$\cr$\displaystyle\Omega}}{\ooalign{\kern 0.46pt\textstyle\Omega$\cr\kern 0.33pt$\textstyle\Omega$\cr\kern 0.2pt$\textstyle\Omega$\cr$\textstyle\Omega}}{\ooalign{\kern 0.46pt\scriptstyle\Omega$\cr\kern 0.33pt$\scriptstyle\Omega$\cr\kern 0.2pt$\scriptstyle\Omega$\cr$\scriptstyle\Omega}}{\ooalign{\kern 0.46pt\scriptscriptstyle\Omega$\cr\kern 0.33pt$\scriptscriptstyle\Omega$\cr\kern 0.2pt$\scriptscriptstyle\Omega$\cr$\scriptscriptstyle\Omega}}, such that $\nu\sigma$ is a new LKS-proof with the substitutions applied to every $\omega$-term containing a parameter.

Definition 4** (Proof Schema Component).**

Let $\psi\in\mathcal{B}$ and $n\in\mathcal{N}$. A proof schema component $\mathbf{C}$ is a triple $(\psi,\pi,\nu(k))$ where $\pi$ is an LKS-proof only containing $\emptyset$-proof links and $\nu(k)$ is an LKS-proof containing $\left\{n\right\}$-proof links. The end-sequents of the proofs are $S(0,\bar{x})$ and $S(k,\bar{x})$, respectively. Given a proof schema component $\mathbf{C}=(\psi,\pi,\nu(k))$ we define $\mathbf{C}.1=\psi$, $\mathbf{C}.2=\pi$, and $\mathbf{C}.3=\nu(k)$.

If $\nu(k)$ of a proof schema component $(\psi,\pi,\nu(k))$ contains a proof link to $\psi$ it will be referred to as cyclic, otherwise it is acyclic.

Definition 5** (Proof Schema [13]).**

Let $\mathbf{C}_{1},\cdots,\mathbf{C}_{m}$ be proof schema components such that $\mathbf{C}_{i}.1$ is distinct for $1\leq i\leq m$ and $n\in\mathcal{N}$. Let the end sequents of $\mathbf{C}_{1}$ be $S(0,\bar{x})$ and $S(k,\bar{x})$. We define $\Psi=\left\langle\mathbf{C}_{1},\cdots,\mathbf{C}_{m}\right\rangle$ as a proof schema if $\mathbf{C}_{i}.3$ only contains $\left\{n\right\}$-proof links to $\mathbf{C}_{i}.1$ or $\mathbf{C}_{j}.1$ for $1\leq i<j\leq m$. The $\left\{n\right\}$-proof links are of the following form:

$(\mathbf{C}_{i}.1,k^{\prime},\bar{a})$

$S^{\prime}(k^{\prime},\bar{a})$

---

$(\mathbf{C}_{j}.1,t,\bar{b})$

$S^{\prime\prime}(t,\bar{b})$

---

where t\in\mathchoice{\ooalign{\kern 0.46pt\displaystyle\Omega$\cr\kern 0.33pt$\displaystyle\Omega$\cr\kern 0.2pt$\displaystyle\Omega$\cr$\displaystyle\Omega}}{\ooalign{\kern 0.46pt\textstyle\Omega$\cr\kern 0.33pt$\textstyle\Omega$\cr\kern 0.2pt$\textstyle\Omega$\cr$\textstyle\Omega}}{\ooalign{\kern 0.46pt\scriptstyle\Omega$\cr\kern 0.33pt$\scriptstyle\Omega$\cr\kern 0.2pt$\scriptstyle\Omega$\cr$\scriptstyle\Omega}}{\ooalign{\kern 0.46pt\scriptscriptstyle\Omega$\cr\kern 0.33pt$\scriptscriptstyle\Omega$\cr\kern 0.2pt$\scriptscriptstyle\Omega$\cr$\scriptscriptstyle\Omega}} such that $\mathcal{V}(t)\in\left\{n\right\}$, $k^{\prime}$ is a sub-term of $k$, and $\bar{a}$ and $\bar{b}$ are vectors of terms from the appropriate sort. $S^{\prime}(k^{\prime},\bar{a})$ and $S^{\prime\prime}(t,\bar{b})$ are the end sequents of components $\mathbf{C}_{i}$ and $\mathbf{C}_{j}$ respectively. We call $S(k,\bar{x})$ the end sequent of $\Psi$ and assume an identification between the formula occurrences in the end sequents of the proof schema components so that we can speak of occurrences in the end sequent of $\Psi$. The class of all proof schemata will be denoted by (read big yus).

For any proof schema \Phi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}, such that $\Phi=\left\langle\mathbf{C}_{1},\cdots,\mathbf{C}_{m}\right\rangle$ we define $|\Phi|=m$ and $\Phi.i=\mathbf{C}_{i}$ for $1\leq i\leq m$. Note that instead of using proof schema pair [13, 16] to define proof schemata we use proof schema components. The only difference is that proof schema components make the name explicit. All results concerning proof schemata built from proof schema pairs carry over for our above definition.

Definition 6**.**

Let \Phi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{} such that $\Phi$ contains $(\chi,\pi,\nu(k))$. Let $\nu.\gamma\in PL(\nu(k))$. We call $\nu.\gamma$ an incoming link if $\mathcal{P}(\nu(k).\gamma)=\chi$, otherwise, we call it an outgoing link.

Example 2**.**

Let us consider the proof schema $\Phi=\left\langle(\varphi,\pi,\nu(k))\right\rangle$. The proof schema uses one defined function symbol $\hat{S}(\cdot)$ which is used to convert terms of the $\omega$ sort to the $\iota$ sort, i.e. $\mathcal{E}=\left\{\hat{S}(k+1)=s(\hat{S}(k))\ ;\ \hat{S}(0)=0\ ;\ k+s(l)=s(k+l)\right\}$. The proofs $\pi$ and $\nu(k)$ are as follows:

$\pi=$**

$P(\alpha+0)\vdash P(\alpha+0)$

$w\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+0)$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+\hat{S}(0))$

---

$\nu(k)=$**

$(\varphi,n,\alpha)$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+\hat{S}(n))$

       $P(s(\alpha+\hat{S}(n)))\vdash P(s(\alpha+\hat{S}(n)))$

$\to\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),P(\alpha+\hat{S}(n))\to P(s(\alpha+\hat{S}(n)))\vdash P(s(\alpha+\hat{S}(n)))$

$\forall\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(s(\alpha+\hat{S}(n)))$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(\alpha+s(\hat{S}(n)))$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(\alpha+\hat{S}(n+1))$

$c\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+\hat{S}(n+1))$

---

Note that $\pi$ contains no proof links, while $\nu(k)$ contains $\{n\}$-proof links.

For our clausal analysis, we need to consider a specific type of proof schema which we will refer to as an accumulating proof schema. For a given proof schema we can easily construct the corresponding accumulating proof schema by adding a new proof component to the beginning of the proof schema. The additional component allows incremental unfolding of the proof schema (see Section 0.3).

Definition 7**.**

Let \Phi=\left\langle\mathbf{C}_{1},\cdots,\mathbf{C}_{m}\right\rangle\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}. We define the accumulating proof schema of $\Phi$, as $\Phi^{a}=\left\langle A,\mathbf{C}_{1},\cdots,\mathbf{C}_{m}\right\rangle$, where $A=\left(\chi,\pi,\nu(k)\right)$, $\chi$ is a distinct proof symbol from those of $\Phi$, $\pi=\mathbf{C}_{1}.2$, and $\nu(k)$ is defined as follows:

\begin{array}[]{c}(\mathbf{C}_{1}.1,k,\bar{t})\end{array}

\begin{array}[]{c}S(k,\bar{t})\end{array}

---

where $S(k,\bar{t})$ is the end sequent of $\mathbf{C}_{1}.3$.

Definition 8** (Evaluation of proof schema [13]).**

We define the rewrite rules for proof links

$(\varphi,0,\bar{t})$

$\Rightarrow\pi$

$S(0,\bar{t})$

---

$(\varphi,k,\bar{t})$

$\Rightarrow\nu(k)$

$S(k,\bar{t})$

---

for all proof schema components $\mathbf{C}=(\varphi,\pi,\nu(k))$. We define $\mathbf{C}\downarrow_{\alpha}$ as a normal form of

$(\varphi,\alpha,\bar{t})$

$S(\alpha,\bar{t})$

---

where $\alpha\in\mathbb{N}$, under the rewrite system just given extended with rewrite rules for defined function and predicate symbols. Further, we define $\Phi\downarrow_{\alpha}=\mathbf{C}_{1}\downarrow_{\alpha}$ for a proof schema $\Phi=\langle\mathbf{C}_{1},\ldots,\mathbf{C}_{m}\rangle$.

Example 3**.**

Let $\Phi$ be the proof schema of example 2 and $\Phi^{a}$ be the corresponding accumulating proof schema. For $1\in\mathbb{N}$ we can write down $\Phi^{a}\downarrow_{1}$:

$P(s(\alpha+S(0)))\vdash P(s(\alpha+S(0)))$       

$P(\alpha+0)\vdash P(\alpha+0)$

$w\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+0)$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+S(0))$

$\to\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),P(\alpha+S(0))\to P(s(\alpha+S(0)))\vdash P(s(\alpha+S(0)))$

$\forall\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(s(\alpha+S(0)))$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(\alpha+s(S(0)))$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x)),\forall x.P(x)\to P(s(x))\vdash P(\alpha+S(0+1))$

$c\colon l$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+S(0+1))$

$\mathcal{E}$

$P(\alpha+0),\forall x.P(x)\to P(s(x))\vdash P(\alpha+S(1))$

---

Proposition 1** (Soundness of proof schemata [13]).**

Let $\Phi=\langle\mathbf{C}_{1},\ldots,\mathbf{C}_{m}\rangle$ be a proof schema, and let $\alpha\in\mathbb{N}$. Then there exists a LK-proof of $\mathbf{C}_{1}\downarrow_{\alpha}$.

If $S(k,\bar{t})$ is the end sequent of $\Phi$ then Proposition 1 essentially states that $\mathbf{C}_{1}\downarrow_{\alpha}$ is an LK-proof of $S(\alpha,\bar{t})\downarrow$ where by $\downarrow$ we refer to normalization of the defined symbols in $S(\alpha,\bar{t})$.

0.2.3 The Characteristic Clause Set

The characteristic clause set of an LK-proof $\varphi$ is extracted by inductively following the formula occurrences of cut ancestors up the proof tree to the leaves. The cut ancestors are sub-formulas of any cut in the given proof. However, in the case of proof schemata, the concept of ancestors and formula occurrence is more complex. A formula occurrence might be an ancestor of a cut formula in one recursive call and in another it might not. Additional machinery is necessary to extract the characteristic clause set from proof schemata. A set $\Omega$ of formula occurrences from the end-sequent of an LKS-proof $\psi$ is called a configuration for $\psi$. A configuration $\Omega$ for $\psi$ is called relevant w.r.t. a proof schema $\Psi$ if $\psi$ is a proof in $\Psi$ and there is a $\gamma\in\mathbb{N}$ such that $\psi$ induces a subproof $\pi\downarrow_{\gamma}$ of $\Psi\downarrow_{\gamma}$ such that the occurrences in $\Omega$ correspond to cut-ancestors below $\pi\downarrow_{\gamma}$ [12]. By $\pi\downarrow_{\gamma}$, we mean substitute the free parameter of $\pi$ with $\gamma\in\mathbb{N}$ and unroll the proof schema to an LKE-proof. We note that the set of relevant cut-configurations can be computed given a proof schema $\Psi$. To represent a proof symbol $\varphi$ and configuration $\Omega$ pairing in a clause set we assign a clause set symbol $cl^{\varphi,\Omega}(k,\bar{x})$ to them, where k\in\mathchoice{\ooalign{\kern 0.46pt\displaystyle\Omega$\cr\kern 0.33pt$\displaystyle\Omega$\cr\kern 0.2pt$\displaystyle\Omega$\cr$\displaystyle\Omega}}{\ooalign{\kern 0.46pt\textstyle\Omega$\cr\kern 0.33pt$\textstyle\Omega$\cr\kern 0.2pt$\textstyle\Omega$\cr$\textstyle\Omega}}{\ooalign{\kern 0.46pt\scriptstyle\Omega$\cr\kern 0.33pt$\scriptstyle\Omega$\cr\kern 0.2pt$\scriptstyle\Omega$\cr$\scriptstyle\Omega}}{\ooalign{\kern 0.46pt\scriptscriptstyle\Omega$\cr\kern 0.33pt$\scriptscriptstyle\Omega$\cr\kern 0.2pt$\scriptscriptstyle\Omega$\cr$\scriptscriptstyle\Omega}}.

Definition 9** (Characteristic clause term [13]).**

Let $\pi$ be an LKS-proof and $\Omega$ a configuration. In the following, by $\Gamma_{\Omega},\Delta_{\Omega}$ and $\Gamma_{C},\Delta_{C}$ we will denote multisets of formulas of $\Omega$- and cut-ancestors respectively. Let $r$ be an inference in $\pi$. We define the clause-set term $\Theta_{r}^{\pi,\Omega}$ inductively:

• •

if $r$ is an axiom of the form $\Gamma_{\Omega},\Gamma_{C},\Gamma\vdash\Delta_{\Omega},\Delta_{C},\Delta$, then $\Theta_{r}^{\pi,\Omega}=\{\Gamma_{\Omega},\Gamma_{C}\vdash\Delta_{\Omega},\Delta_{C}\}$

• •

if $r$ is a proof of the form

$\psi(k,\bar{a})$

$\Gamma_{\Omega},\Gamma_{C},\Gamma\vdash\Delta_{\Omega},\Delta_{C},\Delta$

---

then define $\Omega^{\prime}$ as the set of formula occurrences from $\Gamma_{\Omega},\Gamma_{C}\vdash\Delta_{\Omega},\Delta_{C}$ and $\Theta_{r}^{\pi,\Omega}=\textit{cl}^{\psi,\Omega^{\prime}}(k,\bar{a})$

• •

if $r$ is a unary rule with immediate predecessor $r^{\prime}$, then $\Theta_{r}^{\pi,\Omega}=\Theta_{r^{\prime}}^{\pi,\Omega}$

• •

if $r$ is a binary rule with immediate predecessor $r_{1},r_{2}$, then

• –

if the auxiliary formulas of $r$ are $\Omega$- or cut-ancestors, then $\Theta_{r}^{\pi,\Omega}=\Theta_{r_{1}}^{\pi,\Omega}\oplus\Theta_{r_{2}}^{\pi,\Omega}$

• –

otherwise $\Theta_{r}^{\pi,\Omega}=\Theta_{r_{1}}^{\pi,\Omega}\otimes\Theta_{r_{2}}^{\pi,\Omega}$

Finally, define $\Theta^{\pi,\Omega}=\Theta_{r_{0}}^{\pi,\Omega}$, where $r_{0}$ is the last inference of $\pi$, and $\Theta^{\pi}=\Theta^{\pi,\emptyset}$. $\Theta^{\pi}$ is called the characteristic term of $\pi$.

Note that considering an empty configuration and ignoring the rule for proof links results in the definition of clause set term for LK-proofs [5]. Clause terms can be evaluate to sets of clauses using $|\Theta^{\pi}|$, that is $|\Theta_{1}\oplus\Theta_{2}|=|\Theta_{1}|\cup|\Theta_{2}|$, $|\Theta_{1}\otimes\Theta_{2}|=\{C\circ D\mid C\in|\Theta_{1}|,D\in|\Theta_{2}|\}$, where $C\circ D$ is the sequent whose antecedent is the union of the antecedents of $C$ and $D$ and analogously whose succedent is the union of the succedents of $C$ and $D$.

The characteristic clause term is extracted for each proof symbol in a given proof schema $\Psi$, and together they make the characteristic clause set schema for $\Psi$, $CL(\Psi)$.

Definition 10** (Characteristic Term Schema[13]).**

Let $\Psi=\left\langle\mathbf{C}_{1},\cdots,\mathbf{C}_{m}\right\rangle$ be a proof schema. We define the rewrite rules for clause-set symbols for all proof schema components $\mathbf{C}_{i}$ and configurations $\Omega$ as $cl^{\mathbf{C}_{i},\Omega}(0,\overline{u})\rightarrow\Theta^{\pi_{i},\Omega}$ and $cl^{\mathbf{C}_{i},\Omega}(k,\overline{u})\rightarrow\Theta^{\nu_{i},\Omega}$ where $1\leq i\leq m$. Next, let $\gamma\in\mathbb{N}$ and $cl^{\mathbf{C}_{i},\Omega}\downarrow_{\gamma}$ be the normal form of $cl^{\mathbf{C}_{i},\Omega}(\gamma,\overline{u})$ under the rewrite system just given extended by rewrite rules for defined function and predicate symbols. Then define $\Theta^{\mathbf{C}_{i},\Omega}=cl^{\mathbf{C}_{i},\Omega}$ and $\Theta^{\Psi,\Omega}=cl^{\mathbf{C}_{1},\Omega}$ and finally the characteristic term schema $\Theta^{\Psi}=\Theta^{\Psi,\emptyset}$.

Note that, concerning the characteristic clause schema of a proof schema $\Phi$ the following essential proposition holds.

Proposition 2** ([13, 16]).**

Let $\gamma\in\mathbb{N}$. Then $\Theta^{\Phi}\downarrow_{\gamma}\equiv_{syn}\Theta(\Phi\downarrow_{\gamma})$

Example 4**.**

The following characteristic term schema is taken from the proof analysis of [10] where $\Theta^{\Phi}=cl^{\psi,\emptyset}(n+1)$ and $\Omega(n)\equiv\exists x\forall y\left(\left(\left(x\leq y\right)\rightarrow n+1=f(y)\right)\vee f(y)<n+1\right):$

[TABLE]

The following clause set is $\Theta^{\Phi}\downarrow_{1}$. Equivalence can be checked by referring to the formal proof of [10].

[TABLE]

0.2.4 Top Clause Set

Let us consider the characteristic clause set of an ACNF${}^{\text{top}}$ $\varphi$, that is $\mathit{CL}(\varphi)$. The smallest unsatisfiable subset of $\mathit{CL}(\varphi)$ has a special structure pointed out be S. Wolfsteiner in [20] (Definition 6.1.6) which he called a $\mathbf{TANCF}$. Such a clause set term is generated from the product of atomic clause sets $\left\{A\vdash,\vdash A\right\}$ where $A$ is an atom. Later this structure was used for constructing clause sets from a set of propositional symbols by A. Condoluci [11], the so called top clause set.

Definition 11** ([11]).**

Let $A$ be an atom whose predicate symbol is $P\in\mathbf{P_{u}}$, and $\mathcal{A},\mathcal{A}^{\prime}$ be multisets of atoms. We define the operator $\mathit{CL}^{t}(\cdot)$ which maps a set of atoms to a clause set:

[TABLE]

[TABLE]

We refer to $\mathit{CL}^{t}(\mathcal{A})$ as the top clause set with respect to the set of atoms $\mathcal{A}$.

As was shown in both [11, 20], such a clause set is always unsatisfiable, even in the first-order case. We use this concept later to further extend the results presented in Section 0.4.

0.2.5 Clausal Subsumption: Theory and Results

Let $A(\bar{x})$ be a formula with the free variables $\bar{x}=(x_{1},\ldots,x_{n})$ for $n\in\mathbb{N}$. A substitution $\theta=[\bar{x}\backslash\bar{t}]$ is an instantiation of the free variables $x_{1},\ldots,x_{n}$ within a formula $A$ with the terms $\bar{t}=(t_{1},\ldots,t_{n})$ such that $t_{i}$ do not contain $x_{i}$ for $1\leq i\leq n$ as a subterm, i.e. $A(\bar{x})\theta=A(\bar{t})\theta$.

We call a sequent containing only atomic formulas a clause. Moreover, we define the subset relation $\subseteq$ on clauses, i.e. a clause $C$ is a subset of a clause $D$ ($C\subseteq D$) if the atoms in the succedent of $C$ are a subset of the atoms in the succedent of $D$ and analogously the atoms in the antecedent of $C$ are a subset of the atoms in the antecedent of $D$. Now we are able to define the subsumption relation of clauses (see [7]).

Definition 12** (Subsumption).**

Let $C$ and $D$ be clauses. Then $C$ subsumes $D$ ($C\leq_{ss}D$) if there exists a substitution $\theta$ s.t. $C\theta\subseteq D$. Since we define $\subseteq$ with sets we do not have to consider permutation. We say that a set of clauses $\mathcal{C}$ subsumes a set of clauses $\mathcal{D}$ ($\mathcal{C}\leq_{ss}\mathcal{D}$) if for all $D\in\mathcal{D}$ there exists a $C\in\mathcal{C}$ s.t. $C\leq_{ss}D$.

The concept of clausal subsumption was used in [6] derive the theorems that the following two corollaries are based on.

Corollary 1** (Direct corollary of Theorem 6.1 [6]).**

*Let $\varphi$ be an LK-proof and $\psi$ be the **ACNF${}^{\text{top}}$*of $\varphi$ as described in Section 0.2.1. Then $\Theta(\varphi)\leq_{ss}\Theta(\psi)$.

Corollary 2** (Direct corollary of Theorem 6.2 [6]).**

*Let $\varphi$ be an LK-proof and $\psi$ be an **ACNF${}^{\text{top}}$*of $\varphi$ as described in Section 0.2.1. Then there exists a resolution refutation $\gamma$ of $CL(\varphi)$ s.t. $\gamma\leq_{ss}\textit{RES}(\psi)$ where $\textit{RES}(\psi)$ is the resolution proof corresponding to $\psi$.

0.3 Lazy Instantiation of Proof Schema

In this section we define a transformation which allows for a kind of lazy evaluation [15] of proof schemata, that is lazy instantiation. A similar concept was used by C. Sprenger and M. Dam [18] to transform global induction into local induction for a variant of the $\mu$-calculus. Lazy instantiation allows us to apply reductive cut-elimination to proof schemata by moving the cyclic structure (proof links) whenever it impedes the cut-elimination procedure, i.e. auxiliary sequents introduced by proof links do not have a reductive cut-elimination rule. Though one can never eliminate the cuts through lazy instantiation we can get a better understanding of the relationship between proof schemata in there instantiated and uninstantiated forms. Essentially, we extend the clausal analysis of [6] to proof schemata. The lazy instantiation relation is defined as \rightarrow_{L}\,:\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}\rightarrow\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}. We first provide an example transformation before providing a formal definition of the transformation.

Example 5**.**

Consider the following proof schema $\Phi=\left\langle\mathbf{C}_{1}\right\rangle$ where $\mathbf{C}_{1}=(\psi,\pi,\nu(k))$ (note that by $\_$ we represent the empty tuple of terms),

[TABLE]

\begin{array}[]{c}\varphi_{(n+1)}\end{array}       

\begin{array}[]{c}(\psi,n,\_)\end{array}

\begin{array}[]{c}\bigwedge_{i=0}^{n}P(i)\rightarrow Q(i)\vdash\bigwedge_{i=0}^{n}\neg P(i)\vee Q(i)\end{array}

$\wedge:r$

\begin{array}[]{c}\bigwedge_{i=0}^{n+1}P(i)\rightarrow Q(i),P(n+1)\rightarrow Q(n+1)\vdash\left(\bigwedge_{i=0}^{n}\neg P(i)\vee Q(i)\right)\wedge\left(\neg P(n+1)\vee Q(n+1)\right)\end{array}

$\mathcal{E}$

\begin{array}[]{c}\bigwedge_{i=0}^{n+1}P(i)\rightarrow Q(i),P(n+1)\rightarrow Q(n+1)\vdash\bigwedge_{i=0}^{n+1}\neg P(i)\vee Q(i)\end{array}

$\land\colon l$

\begin{array}[]{c}\left(\bigwedge_{i=0}^{n}P(i)\rightarrow Q(i)\right)\wedge P(n+1)\rightarrow Q(n+1)\vdash\bigwedge_{i=0}^{n+1}\neg P(i)\vee Q(i)\end{array}

$\mathcal{E}$

\begin{array}[]{c}\bigwedge_{i=0}^{n+1}P(i)\rightarrow Q(i)\vdash\bigwedge_{i=0}^{n+1}\neg P(i)\vee Q(i)\end{array}

---

where $\varphi_{(n+1)}$ is an LK-proof of $P(n+1)\rightarrow Q(n+1)\vdash\neg P(n+1)\vee Q(n+1).$

[TABLE]

\begin{array}[]{c}P(0)\vdash P(0)\end{array}

$\neg:r$

\begin{array}[]{c}\vdash\neg P(0),P(0)\end{array}

       \begin{array}[]{c}Q(0)\vdash Q(0)\end{array}

$\rightarrow:l$

\begin{array}[]{c}P(0)\rightarrow Q(0)\vdash\neg P(0),Q(0)\end{array}

$\vee:r$

\begin{array}[]{c}P(0)\rightarrow Q(0)\vdash\neg P(0)\vee Q(0)\end{array}

---

We can lazily evaluate the accumulating proof schema $\Phi^{a}$ to $\Psi$, $\Phi^{a}\rightarrow_{L}\Psi$, where $\Psi=\left\langle\mathbf{C}_{1}^{\prime},\mathbf{C}_{2}\right\rangle$, $\mathbf{C}_{1}^{\prime}=(\psi,\pi,\nu(k)^{\prime})$, and $\mathbf{C}_{2}=(\chi,\pi,\nu(k)^{\prime\prime})$. Let $\nu(k).\mu=(\psi,n,\_)$. We define $\nu(k)^{\prime}=\nu(k)\sigma\left[\nu(k)^{\prime\prime}\right]_{\mu}$, where $\sigma=\left\{n\leftarrow n+1\right\}$. The LKS-proof $\nu(k)^{\prime\prime}$ is the same as $\nu(k)$ except $\nu(k)^{\prime\prime}.\mu=(\chi,n,\_)$. The important parts of $\nu(k)^{\prime}$ can be sketched as follows:

\begin{array}[]{c}\varphi_{(n+2)}\end{array}       

\begin{array}[]{c}\varphi_{(n+1)}\end{array}       

\begin{array}[]{c}(\chi,n,\_)\end{array}

\begin{array}[]{c}\bigwedge_{i=0}^{n}P(i)\rightarrow Q(i)\vdash\bigwedge_{i=0}^{n}\neg P(i)\vee Q(i)\end{array}

$\wedge:r$

\begin{array}[]{c}\vdots\end{array}

$c\colon l$

\begin{array}[]{c}\bigwedge_{i=0}^{n+1}P(i)\rightarrow Q(i)\vdash\bigwedge_{i=0}^{n+1}\neg P(i)\vee Q(i)\end{array}

$\land\colon r$

\begin{array}[]{c}\bigwedge_{i=0}^{n+1}P(i)\rightarrow Q(i),P(n+2)\rightarrow Q(n+2)\vdash\left(\bigwedge_{i=0}^{n+1}\neg P(i)\vee Q(i)\right)\wedge\left(\neg P(n+2)\vee Q(n+2)\right)\end{array}

$\mathcal{E}$

\begin{array}[]{c}\vdots\end{array}

$c\colon l$

\begin{array}[]{c}\bigwedge_{i=0}^{n+2}P(i)\rightarrow Q(i)\vdash\bigwedge_{i=0}^{n+2}\neg P(i)\vee Q(i)\end{array}

---

Notice $\Phi^{a}\downarrow_{(s(s(\alpha)))}\equiv\Psi\downarrow_{s(\alpha)}$.

For the formal definition we use $\nu(k)^{\prime}$ and $\nu(k)^{\prime\prime}$ as in example 5 even though $\nu(k)$ is not directly discussed.

Definition 13**.**

Let \Phi,\Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}, such that $\mathit{PL}(\Phi.1)$ does not have incoming links. We say that $\Phi$ lazily instantiates to $\Psi$, $\Phi\rightarrow_{L}\Psi$, if

• •

$\Psi.i=\Phi.i$* for $1<i\leq|\Phi|$,*

• •

$\Psi.1=(\Phi.1.1,\Phi.1.2,\nu(k)^{\prime})$**

where $\nu(k)^{\prime}$ is constructed from $\Phi.1.3$ as follows: First construct $\nu(k)^{\prime\prime}=\Phi.1.3\sigma$ where $\sigma=\{n\leftarrow n+1\}$ is a parameter substitution. Then we construct $\nu(k)^{\prime}$ from $\nu(k)^{\prime\prime}$ by replacing every proof link $(\chi,k^{\prime},\bar{t})\in\textit{PL}(\nu(k)^{\prime\prime})$ at position $\gamma$ with the appropriate LKS-proof, that is if $\Phi.i=(\chi,\pi_{i},\nu_{i}(k))$ then we substitute $\nu(k)^{\prime\prime}\left[\nu_{i}(k)(k^{\prime})\right]_{\gamma}$ and $\nu(k)^{\prime\prime}\left[\pi_{i}\right]_{\gamma}$ if $k^{\prime}=0$.

Remark 1**.**

Note that if $\Phi$ is accumulating then $\mathit{PL}(\Phi.1)$ does not have incoming link.

In general it will be necessary to lazily instantiate a proof schema several times. In fact, we will consider all proof schemata that are producible by lazy instantiations.

Definition 14**.**

Let \Phi,\Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}. We define the set of lazily reachable proof schemata from a proof schema $\Phi$, $\mathit{lr}(\Phi)$ by the following inductive construction:

• •

$\Phi\in\mathit{lr}(\Phi)$**

• •

if $\Psi\in\mathit{lr}(\Phi)$ and $\Psi\rightarrow_{L}\Gamma$, then $\Gamma\in\mathit{lr}(\Phi)$

Definition 15**.**

Let \Phi,\Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}. We define the set of reductively reachable proof schemata from a proof schemata $\Phi$, $\mathit{rr}(\Phi)$ by the following inductive construction ($\rightarrow_{R}$ is define at the beginning of Section 0.2) :

• •

$\Phi\in\mathit{rr}(\Phi)$**

• •

if $\Psi\in\mathit{rr}(\Phi)$ and $\Psi\rightarrow_{\mathcal{R}}\Gamma$, then $\Gamma\in\mathit{rr}(\Phi)$

Definition 16**.**

Let \Phi,\Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}. We define the lr-chain length $|\Psi|^{\Phi}_{lr}$(respectively the rr-chain length $|\Psi|^{\Phi}_{rr}$) inductively as follows:

• •

if $\Psi=\Phi$, then $|\Psi|^{\Phi}_{rr}=|\Psi|^{\Phi}_{lr}=0$,

• •

if $\Gamma\rightarrow_{L}\Psi$ or $\Gamma\rightarrow_{R}\Psi$ , then $|\Psi|^{\Phi}_{lr}=|\Gamma|^{\Phi}_{lr}+1$ $(|\Psi|^{\Phi}_{rr}=|\Gamma|^{\Phi}_{rr}+1)$

• •

$|\Psi|^{\Phi}_{lr}=|\Psi|^{\Phi}_{rr}=\infty$* otherwise.*

An important property of this transformation is that, when applied to accumulating proof schemata $\Phi$, for every proof schema in $\Psi\in\mathit{lr}(\Phi)$, there are only finitely many evaluations of $\Phi$ that are not equivalent to an evaluation of $\Psi$. In Example 5 there is no evaluation of $\Psi$ equivalent to $\Phi\downarrow_{1}$.

Lemma 1**.**

Let \Phi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{} be accumulating and $\Psi\in\mathit{lr}(\Phi)$. Then for all $\alpha\in\mathbb{N}$ there exists $\beta\in\mathbb{N}$ such that $\Psi\downarrow_{\alpha}\equiv_{syn}\Phi\downarrow_{\beta}$.

Proof.

By induction on $|\Psi|^{\Phi}_{lr}$. ∎

Lemma 1 tells us that we can use lazy instantiation to change the structure of a given proof schema $\Phi$ into a proof schema $\Psi$ without completely losing the logical meaning, that is equivalence is only lost for finitely many LK-proofs. More specifically, every evaluation of $\Psi$ is equivalent, syntactically, to an evaluation of $\Phi$.

Definition 17**.**

Let \Phi,\Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{} such that $\Psi\in\mathit{lr}(\Phi)$ and $\Phi$ is accumulating. We define a numeral pair $(\alpha,\beta)$ as a pair of natural numbers such that $\Psi\downarrow_{\alpha}\equiv_{syn}\Phi\downarrow_{\beta}$. The set of all numeral pairs for $\Psi$ and $\Phi$ is $\mathit{np}(\Psi,\Phi)$.

The next lemma is a direct result of Lemma 1.

Lemma 2**.**

Let \Phi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{} be an accumulating proof schema and \Psi,\Psi^{\prime}\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{} be proof schemata. Furthermore, let $\Psi\in\mathit{lr}(\Phi)$ be lazily reachable from $\Phi$, $\Psi^{\prime}\in\mathit{rr}(\Psi)$ be reductively reachable from $\Psi$, and $(\alpha,\beta)\in\mathit{np}(\Psi,\Phi)$ be a numeral pair of $\Psi$ and $\Phi$. Then there exists an LK-proof $\chi$ such that $\chi\in\mathit{rr}(\Phi\downarrow_{\beta})$ and $\Psi^{\prime}\downarrow_{\alpha}\equiv_{syn}\chi$.

Proof.

Consider application of reductive cut-elimination rules which only reduce the cuts of $\Psi.1.3$. This means if a rule would require passing a cut through a proof link of $\Psi.1.3$ it cannot be applied. Being that $\Psi$ and $\Phi$ are equivalent when instantiated by $(\hat{n},\hat{m})$ and by definition $\Psi.1$ is acyclic, application of rules transforming $\Psi.1.3$ can be repeated in $\Phi$ without influencing unwanted parts of the proof. The rest of the proof is an induction on $|\Psi^{\prime}|^{\Psi}_{rr}$. ∎

0.4 Clausal Analysis of Proof Schemata

Before we discuss the relationship between the various concepts and results, as shown in Figure 1, we introduce a concept closely related to the ACNF${}^{\text{top}}$, that is the atom set schema and its clause set [11]. Intuitively, an atom set schema of a clause set is just a list of the atoms found in a clause set schema. Though, not exactly the same as the clause set extracted from an ACNF${}^{\text{top}}$, the clause set of an atom set schema, when instantiated, is subsumed by the clause set of the corresponding ACNF${}^{\text{top}}$.

0.4.1 Atom Set Schema

This concept was introduced by A. Condoluci in his master thesis concerning schematic propositional logic [11]. We extend his definition to first-order logic by considering the predicate symbols modulo tuples of terms rather than just an arithmetic term. Note that, these terms form a schema which is easily extractable from a proof schema or characteristic clause set schema. We refer to this schema as a predicate term schema. In some cases the predicate term schema may be describable by a case distinction, i.e. the terms do not contain the free parameter.

Definition 18** (Predicate Term Schema ).**

Let $P\in\mathbf{P_{u}}$ be an $\alpha$-ary predicate symbol. Then the predicate term schema of $P$ is a mapping $t_{P}:\mathbb{N}\rightarrow\iota^{\alpha}$ from numerals to $\alpha$-tuples of terms such that for $\beta\in\mathbb{N}$, $P(t_{P}(\beta))$ is a well-formed atom. we assume that, if the $\alpha$-tuple contains a defined function symbol it will be evaluated and normalized.

Definition 19** (Extension of Definition 78 [11]).**

An atom set schema $\mathcal{A}$ is a finite set of 3-tuples

[TABLE]

where $P_{i}\in\mathbf{P_{u}}$ for $1\leq i\leq\beta$, $t_{P_{i}}^{j}$ is a predicate term schema of $P_{i}$ for $1\leq j\leq\alpha_{i}$, and $a_{i}^{j}$ is an arithmetic term.

Example 6**.**

The atom set schema $\left\{\left\langle P,\lambda r.(\alpha+\hat{S}(r)),n+1\right\rangle\right\}$ was extracted from Example 2.

Definition 20** (Extension of Definition 79 [11]).**

For every $\alpha\in\mathbb{N}$, we define the evaluation of an atom set schema, where $\left\langle P,t_{P},a\right\rangle\downarrow{\alpha}=\left\{P(t_{P}(0)),P(t_{P}(1)),\cdots,P(t_{P}(a\left[n\setminus\alpha\right]))\right\}$, as:

[TABLE]

Example 7**.**

Using the atom set schema of Example 6, we can derive the following evaluation:

[TABLE]

An atom set schema can always be extracted from a clause set term schema.

Lemma 3** (Completeness for atom set schemata [11]).**

For every clause set term schema T, there exists an atom set schema for T.

Though, this lemma was proven for schematic propositional logic, it can easily be extended by constructing the corresponding predicate term schema for each use of a given predicate symbol.

We introduced the concept of lazy instantiation to deal with precisely the following problem faced during application of reductive cut elimination to proof schemata. Note that this may result in additional predicate and term pairings which do not exists in the proof schema, but more importantly, the extraction method presented in [11] guarantees that no pairings, of which are presented in the given proof schema, are missing. This property allows us to further extend our subsumption results, as can be seen in Figure 1 point (x).

Definition 21** (Top clause set schemata [11]).**

Let $\mathcal{A}$ be an atom set schema. We denote the corresponding top clause set schema by $\mathit{CL}^{t}(\mathcal{A})$.

Definition 22** (Semantics of top clause set schemata [11]).**

For $\alpha\in\mathbb{N}$, $\mathit{CL}^{t}(\mathcal{A})\downarrow_{\alpha}=\mathit{CL}^{t}(\mathcal{A}\downarrow_{\alpha})$.

In [11], it was shown that a propositional top clause set schemata can be refutated schematically (Lemma 10). Problematically, in the first-order case, a substitution schema must also be defined [13]. If a substitution schema exists for a given first-order top clause set schema a schematic refutation can be defined. Though, the specification of such a substitution schema remains an open problem. In Figure 1, construction of the substitution schema and construction of the refutation (see [11]) are point (xii).

0.4.2 Clausal Analysis

The main result of this paper is as follows:

Theorem 1**.**

Let \Psi\in\mbox{\includegraphics[scale={.045}]{byus}}\makebox[1.42271pt][l]{}{} be an accumulating proof schema, $\Phi\in lr(\Psi)$ be lazily reachable from $\Psi$, and $\Xi\in rr(\Phi)$ reductively reachable from $\Phi$. Then for all $\alpha\in\mathbb{N}$ there exists a numeral $\beta\in\mathbb{N}$ and an ACNF{}^{\text{top}}$$\chi, such that $\Theta(\psi)\leq_{ss}\Theta(\chi)$ for all $\psi\in rr(\Psi\downarrow_{\beta})$, and $\Theta^{\Xi}\downarrow_{\alpha}\leq_{ss}\Theta(\chi)\leq_{ss}\mathit{CL}^{t}(\mathcal{A}^{\Theta^{\Xi}})\downarrow_{\alpha}$, where $\mathcal{A}^{\Theta^{\Xi}}$ is an atom set schema.

Proof.

The statement $\Theta^{\Xi}\downarrow_{\alpha}\leq_{ss}\Theta(\chi)$ is a consequence of Lemma 2 and Corollary 1. The statement $\Theta(\chi)\leq_{ss}\mathit{CL}^{t}(\mathcal{A}^{\Theta^{\Xi}})\downarrow_{\alpha}$ is a result of Lemma 3 and the extraction method of [11]. ∎

In Figure 1, we present the entire process concerning how one goes from a proof schema $\Phi$ to a refutation. Note, as we mention before, the existence of a refutation for a first-order top clause set schema is dependent on the existence of a substitution schema [13], of which the construction is still an open problem, that is whether a substitution schema exists for every first-order top clause set. We leave this question to future work. The first step in Figure 1, (i), constructs an accumulating proof schema $\Phi^{A}$ from $\Phi$. This adds an additional component which “accumulates” the lazily instantiated proof links. After the construction of an accumulating proof schema $\Phi^{A}$, we can either instantiate it (iii) or apply lazy instantiation (ii) and construct a proof schema $\Phi^{\prime}$. The point of lazy instantiation is to get around situation like the following:

\begin{array}[]{c}(\varphi_{l},t,\bar{x})\end{array}

\begin{array}[]{c}C,\Delta\vdash\Gamma\end{array}

     

\begin{array}[]{c}(\varphi_{j},t^{\prime},\bar{x})\end{array}

\begin{array}[]{c}\Delta^{\prime}\vdash\Gamma^{\prime},C\end{array}

cut

$\Delta,\Delta^{\prime}\vdash\Gamma,\Gamma^{\prime}$

in which Gentzen rules cannot be applied unless we evaluate the proof links. Note that by Lemma 1, we can instantiate $\Phi^{A}$ and $\Phi^{\prime}$ such that they are equivalent. The heart of our result is that lazy instantiation allows us to apply reductive cut elimination to an arbitrarily large but finite portion of $\Phi^{A}$. This results in a proof schema $\Psi$. In turn, we can apply the same reductive cut elimination steps $\Phi^{A}\downarrow_{\alpha}$ resulting in an LK-proof $\psi$ whose clause set subsumes the evaluation of the clause set schema of $\Psi$ at $\beta$. Furthermore, by extending the results of [6], we are able to show subsumption properties between the clause set schema of $\Psi$ and the clause set of $\psi$, as well as between both clause sets and the clause set of the ACNF${}^{\text{top}}$ of $\psi$.

0.5 Conclusion

In this work we extend results concerning the clausal analysis of LK-proofs to first-order proof schemata [6]. The results of [6] are based on clause set subsumption properties between an LK-proof $\varphi$ and an LK-proof $\psi$ which is the result of applying reductive cut-elimination to $\varphi$, without atomic cut elimination. These results are not easily extendable to proof schemata because cuts cannot be easily passed through proofs links. We define lazy instantiation allowing proof links to unfold and allowing reductive cut-elimination to continue. Thus, we were able to derive clause set subsumption results for arbitrary applications of reductive cut-elimination to proof schemata. Furthermore, using the structural results of [11, 20], we were able to strengthen these results and reduce the problem of constructing an ACNF schema (see [13]) to find a substitution schema [13]. As discussed in [9, 10], defining the structure of the resolution refutation is the hardest part of the proof analysis using schematic CERES [13]. For future work, we plan to investigate how the introduced concepts can be applied to other issues concerning proof schemata. Also, we plan to investigate ways one can simplify the construction of substitution schemata, possibly through the use of automated theorem proving methods [10].

Acknowledgment

We would like to thank Alexander Leitsch for reviewing and providing suggestions.