# Soundness and Completeness Proofs by Coinductive Methods

## Abstract

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.

### Keywords

Codatatypes Lazy evaluation First-order logic Soundness Completeness Gentian systems Proof assistants Isabelle/HOL## 1 Introduction

Gödel’s completeness theorem [23] is a major result about first-order logic (FOL). It forms the basis of results and techniques in various areas, including mathematical logic, automated deduction, and program verification. It can be stated as follows: If a set of formulas is satisfied by all structures, then it has a proof. The theorem enjoys many accounts in the literature that generalize and simplify the original proof; indeed, a textbook on mathematical logic would be incomplete without a proof of this fundamental theorem.

*Logic for Computer Science*[22], Pfenning, a fellow “syntactician,” notes the following [41]:

In the context of completeness, the “superficially more elegant semantic arguments” are proofs that rely on Hilbert systems. These systems have several axioms but only one or two deduction rules, providing minimal support for presenting the structure of proofs or for modeling proof search. A proof of completeness based on Hilbert systems follows theAll too often, proof-theoretic methods are neglected in favor of shorter, and superficially more elegant semantic arguments. [In contrast, in Gallier’s book] the treatment of the proof theory of the Gentzen system is oriented towards computation with proofs. For example, a pseudo-Pascal version of a complete search procedure for first-order cut-free Gentzen proofs is presented.

*Henkin style*: It employs a heavy bureaucratic apparatus to establish facts about deduction and conservative language extensions, culminating in a highly nonconstructive step: an application of Zorn’s lemma to extend any syntactically consistent set of formulas to a maximally consistent one, from which a model is produced.

In contrast, a proof of completeness based on more elaborate Gentzen or tableau systems follows the *Beth–Hintikka style* [31]. It performs a search that builds either a finite deduction tree yielding a proof (or refutation, depending on the system) or an infinite tree from which a countermodel (or model) can be extracted. Such completeness proofs have an intuitive content that stresses the tension of the argument: The deduction system systematically tries to prove the goal; a failure yields, at the limit, a countermodel.

A tableau is a set of elements, called nodes, partially ordered and classified into levels as explained below. With each node is associated a finite set of formulas. We shall usually identify a given node with its associated set of formulas; this is somewhat imprecise (since in fact the same set of formulas can be associated with different nodes) but will not cause confusion.

Each node belongs to a unique level, which is labeled by some natural number. There is just one node of level 0, called the initial node of the tableau. Each node at level \(n+1\) is a successor of a unique node, which must be of level

n.

In textbooks, at best the trees are defined rigorously (e.g., as prefix-closed sets), but the reasoning is performed informally, disregarding the original definition and relying on the intuitive notion of trees, as Gallier does. One could argue that trees are intuitive and do not need a formal treatment, but the same holds for the syntax of formulas, which is treated very rigorously in most of the textbooks.

The main contribution of this article is a rigorous Beth–Hintikka-style proof of the completeness theorem, based on a Gentzen system. The potentially infinite trees are captured by codatatypes (also called coinductive datatypes or final coalgebras) [29]. Another novel aspect of the proof is its modularity: The core tree construction argument is isolated from the proof system and concrete formula syntax, with the concrete syntactic details concealed behind an abstract *Herbrandness* assumption (Sect. 3). This assumption can be verified in concrete cases (by performing the standard Herbrand model construction) for a wide range of Gentzen and tableau systems for FOL, various flavors of FOL (e.g., with or without predicates, equality, or sorts), and even modal logics with explicit-world Gentzen systems (Sect. 4). This modularization replaces the textbook proofs by analogy. The core of the argument amounts to reasoning about a functional program over lazy data structures.

A second contribution of this article is an application of the same coinductive machinery (infinite trees and streams and corecursive functions between them) to some interesting recent results from the automated deduction literature: the soundness of infinite (including cyclic) proofs for FOL with inductive definitions and related logics, studied by Brotherston et al. [12, 13, 14, 15, 16]. For formalizing these results, we follow rather closely the abstract constructions of Brotherston et al. [16], except that we use coinduction and corecursion to streamline the development. The presentation follows the same path as for completeness: The main result is stated abstractly (Sect. 5) and instantiated by concrete examples (Sect. 6).

The proofs of the abstract results are formalized in Isabelle/HOL (Sect. 7). The definitions of infinite trees and paths rely on a new definitional package for codatatypes [6, 11], which automates the derivation of characteristic theorems from high-level specifications of types and functions. Through Isabelle’s code generator [25], the corecursive construction gives rise to a Haskell program that implements a semidecision procedure for validity instantiable with various proof systems, yielding *verified* sound and complete provers.

The completeness proof has been applied to the formalization of optimized translations between sorted and unsorted FOL [4, 7]. The soundness proofs of these translations rest on the Löwenheim–Skolem theorem, a corollary of a slightly generalized version of the completeness theorem. The previous formal proofs of the completeness theorem, including two in Isabelle, support a more restrictive logic than many-sorted FOL (Sect. 8).

An earlier version of this article was presented at the IJCAR 2014 conference in Vienna, Austria, under a different title [9]. The article considerably extends the conference paper with infinite-proof soundness (Sects. 5, 6) as a second application of coinductive methods. It also provides more details about the Beth–Hintikka-style proof of the completeness theorem (Sects. 3, 4).

*Conventions* Isabelle/HOL [39] is a proof assistant based on classical higher-order logic (HOL) with Hilbert choice, the axiom of infinity, and rank-1 polymorphism. It is the logic of Gordon’s original HOL system [24] and of its many successors. HOL notations are similar to those of functional programming languages, but they also include many traditional symbols and syntaxes from mathematics, notably to denote simply typed sets. We refer to Nipkow and Klein [38, Part 1] for a modern introduction. In this article, the logic is viewed not as a formal system but rather as a framework for expressing mathematics, much like set theory is employed by working mathematicians. In keeping with the standard semantics of HOL, types \(\alpha \) are identified with sets.

## 2 Preliminaries on First-Order Logic

The soundness and completeness results we formalize in this article apply to an array of variations of first-order logic and beyond (modal logics, separation logic, etc.). To give some concrete grounding for the forthcoming abstract development, we recall basic (unsorted) first-order logic and its extension with inductive predicates.

### 2.1 Classical First-Order Logic

We fix a first-order language: a countably infinite set \({\textsf {var}}\) of variables \(x,\, y,\, z\) and countable sets \({\textsf {fsym}}\) and \({\textsf {psym}}\) of function symbols *f* and predicate symbols *p* together with an assignment \({\textsf {ar}}: {\textsf {fsym}}\mathrel {\uplus } {\textsf {psym}}\rightarrow {\textsf {nat}}\) of numeric arities. Terms \(t \in {\textsf {term}}\) are symbolic expressions built inductively from variables by application of function symbols \(f \in {\textsf {fsym}}\) to tuples of arguments whose lengths respect the arities: \(f (t_1,\ldots ,t_{{\textsf {ar}}\,f})\). Atoms \(a \in {\textsf {atom}}\) are symbolic expressions of the form \(p (t_1,\ldots ,t_{{\textsf {ar}}\,p})\), where \(p \in {\textsf {psym}}\) and \(t_1,\ldots ,t_{{\textsf {ar}}\,p} \in {\textsf {term}}\).

To distinguish between the first-order language of study and the HOL metalanguage, we use the constructor names \({\textsf {Neg}}\), \({\textsf {Conj}}\), and \({\textsf {All}}\) for the former, keeping the traditional symbols \(\lnot \), \(\wedge \), and \(\forall \) for the latter. We often write *a* instead of \({\textsf {Atm}}\;a\), thus pretending that atoms are included in formulas.

*n*-ary \(f \in {\textsf {fsym}}\) and a predicate \(P_{p} : \textit{S}^{n} \rightarrow {\textsf {bool}}\) for each

*n*-ary \(p \in {\textsf {psym}}\). The notions of interpretation of a term

*t*and satisfaction of a formula \(\varphi \) by a structure \(\mathscr {S}\) with respect to a variable valuation \(\xi : {\textsf {var}}\rightarrow \textit{S}\) are defined in the standard way. For terms (by structural recursion):

*x*to

*a*and all other variables

*y*to \(\xi \;y\).

We define the notion of a structure \(\mathscr {S}\) satisfying a formula \(\varphi \), written \(\mathscr {S}\models \varphi \), to mean satisfaction with respect to all valuations: \(\forall \xi .\;\mathscr {S}\models _\xi \varphi \).

A sequent is a pair \(\Gamma \mathrel {\rhd }\Delta \) of finite sets of formulas. Satisfaction is extended to sequents: \(\mathscr {S}\models \Gamma \mathrel {\rhd }\Delta \) iff \((\forall \varphi \mathbin {\in }\Gamma .\;\mathscr {S}\models \varphi ) \Rightarrow (\exists \psi \mathbin {\in }\Delta .\;\mathscr {S}\models \psi )\). We can think of \(\Gamma \mathrel {\rhd }\Delta \) as an implication between the conjunction of the formulas of \(\Gamma \) and the disjunction of the formulas of \(\Delta \). If \(\mathscr {S}\models \Gamma \mathrel {\rhd }\Delta \), we also call \(\mathscr {S}\) a *model* of \(\Gamma \mathrel {\rhd }\Delta \). By contrast, if \(\mathscr {S}\not \models \Gamma \mathrel {\rhd }\Delta \), we call \(\mathscr {S}\) a *countermodel* of \(\Gamma \mathrel {\rhd }\Delta \)

Since here sequents are sets, we do not need the structural rules (weakening, contraction, and exchange).

In a proof, the rules are applied from bottom to top. One chooses a formula from either side of the sequent, the eigenformula, and applies a rule according to the topmost connective or quantifier. For a given choice of eigenformula, at most one rule is applicable. The aim of applying the rules is to prove the sequent by building a finite derivation tree whose branches are closed by an axiom (Ax). Provability of a sequent in this system is denoted by prefixing \(\vdash \) to the sequent (e.g., \(\vdash \Gamma \mathrel {\rhd }\Delta \)).

### 2.2 First-Order Logic with Inductive Predicates

*inductive*. Moreover, we fix, for each \(p \in {\textsf {ipsym}}\), a set \({\textsf {ind}}_p\) of Horn clauses specifying

*p*, so that each element of \({\textsf {ind}}_p\) is of the form \({\textsf {Imp}}\;({\textsf {Conj}}\;\psi _1\;\ldots \;\psi _m\;\varphi _1\;\ldots \;\varphi _n)\;\varphi \), where

\({\textsf {Conj}}\) denotes the conjunction of multiple formulas

^{1};\(\varphi \) is an atom of the form \(p(t_1,\ldots ,t_{{\textsf {ar}}\,p})\);

each \(\varphi _i\) is an atom of the form \(p'(t_1,\ldots ,t_{{\textsf {ar}}\,p'})\) for some \(p' \in {\textsf {ipsym}}\);

each \(\psi _j\) is a formula not containing any \(p' \in {\textsf {ipsym}}\).

A first-order structure \(\mathscr {S}= \bigl (\textit{S},\, (F_{f})_{f \,\in \, {\textsf {fsym}}},\, (P_{p})_{p \,\in \, {\textsf {psym}}}\bigr )\) is said to be *inductive* (with respect to \({\textsf {ipsym}}\) and \(({\textsf {ind}}_p)_{p \in {\textsf {ipsym}}}\)) if the interpretation of each \(p \in {\textsf {ipsym}}\) is indeed inductive, i.e., the family \((P_{p})_{p \,\in \, {\textsf {ipsym}}}\) constitutes the least fixpoint of all the inductive Horn clauses interpreted in \(\mathscr {S}\), meaning that \((P_p)_{p \in {\textsf {ipsym}}}\) is the least family of predicates (with respect to component-wise implication) such that \(\mathscr {S}\models \chi \) for all \(\chi \in \bigcup _{p \in {\textsf {psym}}} {\textsf {ind}}_p\).

### Example 1

- \({\textsf {ind}}_{{\textsf {even}}}\) consists of the following clauses:
\({{\textsf {even}}}(0)\);

\({\textsf {Imp}}\;({{\textsf {even}}}(x))\;({{\textsf {even}}}({{\textsf {Suc}}}({{\textsf {Suc}}}(x))))\);

- \({\textsf {ind}}_{{\textsf {odd}}}\) consists of the following clauses:
\({{\textsf {odd}}}({{\textsf {Suc}}}(0))\);

\({\textsf {Imp}}\;({{\textsf {odd}}}(x))\;({{\textsf {odd}}}({{\textsf {Suc}}}({{\textsf {Suc}}}(x))))\).

\(P_{{\textsf {even}}}(F_0)\);

\(\forall n \in S.\;P_{{\textsf {even}}}(n) \implies P_{{\textsf {even}}}(F_{{\textsf {Suc}}}(F_{{\textsf {Suc}}}(n)))\);

\(P_{{\textsf {odd}}}(F_{{\textsf {Suc}}}(F_0))\);

\(\forall n \in S.\;P_{{\textsf {odd}}}(n) \implies P_{{\textsf {odd}}}(F_{{\textsf {Suc}}}(F_{{\textsf {Suc}}}(n)))\).

*t*for the variable

*x*in all formulas of \(\Gamma \):

*direct*rules \(\textsc {Even}_0\) and \(\textsc {Even}_{{\textsf {Suc}}}\) (corresponding to the two cases in the inductive specification of \({{\textsf {even}}}\)) are admissible since all inductive structures satisfy the Horn clauses for \({{\textsf {even}}}\). Moreover, the

*inversion*rule \(\textsc {Even}_{\mathrm{split}}\) is admissible by the least fixpoint assumption—the rule expresses that, if \({{\textsf {even}}}(x)\) is known, then it must have been obtained by an application of one of the inductive clauses for \({{\textsf {even}}}\). Similar rules hold for \({{\textsf {odd}}}\):

\(\chi '\) is a variant of \(\chi \) where all the free variables are fresh for \(\Gamma \,\cup \, \Delta \);

\(p(\overline{t})\) is \({\textsf {concl}}(\chi ')\)—with \(\overline{t}\) thus being a tuple of terms \((t_1,\ldots ,t_{{\textsf {ar}}\;p})\);

\(\overline{x}\) is a tuple of distinct variables \((x_1,\ldots ,x_{{\textsf {ar}}\;p})\).

*p*to particular instances.

The inversion rule \(p_{\mathrm{split}}\) is not as powerful as it could be, due to the restriction that the eigenformula, \(p(\overline{x})\), must be a predicate symbol applied to variables. In a FOL\(_{\textsf {ind}}\) variant with equality, this could be strengthened to speak about arbitrary terms instead of variables [13, Section 4.1].

Beyond the additional admissible rules, a crucial insight of Brotherston [12] is that, for inductive structures, a certain type of proof circularity is permissible. It allows the proof trees to have as leaf nodes not only axioms, but also backward links to other sequents in the tree. This insight is useful for automating induction [12]. Brotherston et al. [16] give an abstract, logic-independent argument for why allowing such circularities is sound. In Sect. 5, we show that this argument can be naturally formalized using a coinductive machinery similar to the one we use for classic completeness.

## 3 Abstract Soundness and Completeness

Before studying applications, we first develop an abstract framework that allows us to state and prove soundness and completeness for an unspecified syntax and class of structures and satisfaction relation. The framework is obtained by distilling the corresponding concrete results for FOL.

The second part maps the escape path to a concrete, proof-system-specific countermodel employing a construction due to Herbrand. At the abstract level, we assume a “Herbrand function” that produces countermodels from escape paths. In Sect. 4, we instantiate this function for the Gentzen system introduced in Sect. 2.1.Either there exists a finite derivation tree or there exists an infinite derivation tree containing a suitable escape path.

### 3.1 Sequents and Structures

We abstract away from the syntax of formulas and sequents and the specific rules of the proof system. We fix countable sets \({\textsf {sequent}}\) and \({\textsf {rule}}\) for sequents and rules. Our abstract sequents represent the formal statements in the logic, which can take either the form of concrete sequents or other forms.

*s*. We write \(\models s\) to indicate that

*s*is satisfied by all models in \({\textsf {structure}}\): \(\forall \textit{S} \mathbin \in {\textsf {structure}}.\;\textit{S} \models s\).

### 3.2 Rule Systems

*s*, applying rule

*r*expands

*s*into the sequents \({ ss }\). We can think of sequents as proof goals, each goal being replaced by zero or more subgoals by applying a rule. The triple \(\mathscr {R}= ({\textsf {sequent}},{\textsf {rule}},{\textsf {eff}})\) forms a

*rule system*.

### Example 2

The Gentzen system from Sect. 2.1 can be presented as a rule system. The set \({\textsf {sequent}}\) is the set of sequents, and \({\textsf {rule}}\) consists of the following: a rule \({\textsc {Ax}}_a\) for each atom *a*; rules \(\textsc {Neg}\textsc {L}_\varphi \) and \(\textsc {Neg}\textsc {R}_\varphi \) for each formula \(\varphi \); rules \({\textsc {Conj}}\textsc {L}_{\varphi ,\psi }\) and \({\textsc {Conj}}\textsc {R}_{\varphi ,\psi }\) for each pair of formulas \(\varphi \) and \(\psi \); a rule \({\textsc {All}}\textsc {L}_{x,\varphi ,t}\) for each variable *x*, formula \(\varphi \), and term *t*; and a rule \({\textsc {All}}\textsc {R}_{x,\varphi }\) for each variable *x* and formula \(\varphi \).

### 3.3 Derivation Trees

*a*is the tree’s

*label*and \({ Ts }\) is the finite set of its (immediate)

*subtrees*. The keyword indicates that, unlike for inductive datatypes, this tree formation rule may be iterated an infinite number of times to create infinitely deep objects.

### Remark 3

*step*combines the current sequent and the rule to be applied. Derivation trees are defined as trees labeled by steps:

*s*,

*r*) as representing the proved goal

*s*and the first (backwards) applied rule

*r*. The well-formed derivation trees are captured by the predicate \({\textsf {wf}}: {\textsf {dtree}}\rightarrow {\textsf {bool}}\) defined by the coinductive ruleThe term \(\textsf {{image}}\;f\;A\) denotes the image of set

*A*through function

*f*, and \(\textsf {{fst}}\) is the left projection operator (i.e., \(\textsf {{fst}}\;(x,y) = x\)). The first assumption requires that the rule

*r*from the root be applied to obtain the subtrees’ labels. The second assumption requires that wellformedness holds for the immediate subtrees. The coinductive interpretation of the definition ensures that the iteration of this rule can cover infinite trees; this would not be the case with an inductive interpretation.

*coinduction on the definition of*\({\textsf {wf}}\) proceeds by simply showing that

*P*is also a solution of the same implication:

### 3.4 Proofs

Indeed, the inductive interpretation is the right one for defining \({\textsf {finite}}\), since we want to enforce the well-founded iteration of the rule. (By contrast, a coinductive interpretation would classify all trees as “finite,” which is certainly not desirable.)

*proof*of a sequent

*s*is a finite well-formed derivation tree with

*s*at its root:

### Example 4

Given the instantiation of Example 2, Fig. 1 shows a finite derivation tree for the sequent \({\textsf {All}}\;x\;(p(x)) \mathrel {{\mathrel {\rhd }}}{\textsf {Conj}}\;(p(y))\;(p(z))\) written using the familiar syntax for logical symbols. Figure 2 shows an infinite tree for the same sequent.

### 3.5 Soundness

The soundness theorem follows by induction on the finiteness of trees representing proofs.

### Theorem 5

### 3.6 Infinite Paths and König’s Lemma

### Lemma 6

If the tree *T* is infinite (i.e., non-finite), there exists an infinite path \({\sigma }\) in *T*.

### Proof

*T*. The desired infinite path \({\textsf {p}} : \{T \mathbin \in {\textsf {dtree}}.\;\lnot \;{\textsf {finite}}\;T\} \rightarrow {\textsf {step}}\;{\textsf {stream}}\) can be defined by primitive corecursion over the codatatype of streams: \({\textsf {p}}\;T = {{\textsf {SCons}}}\;({{\textsf {lab}}}\;T)\;({\textsf {p}}\;(f\;T))\). Equivalently, in terms of the selectors:

### Remark 7

Essentially the same proof would hold if we allowed our trees to be infinitely branching, by replacing finite sets with countable sets in the definition of the \({\textsf {tree}}\) codatatype. This may seem counterintuitive if we think of the standard formulation of König’s lemma, but \({\textsf {finite}}\) would no longer mean having a finite number of nodes—it would simply mean “well founded.”

*P*holds for all suffixes of \({ xs }\):

### Lemma 8

*T*is well formed, there exists a stream \(\rho \in ({\textsf {step}}\times \alpha )\;{\textsf {stream}}\) such that its first projection is an infinite path in

*T*(formally, \({\textsf {ipath}}\;T\,(\textsf {{smap}}\;\textsf {{fst}}\;\rho )\)) and \(\rho \) satisfies the predicate

In the above lemma, the assumption is that, for any sequent *s*, rule *r*, and element \(a \in \alpha \) such that the predicate *I* holds for (*s*, *a*), there exists a premise sequent \(s'\) along the backward application of *r* and an element \(a'\) such that *I* again holds for \((s',a')\) and the predicate *P* holds for (*s*, *a*), *r*, and \((s',a')\). The conclusion is that there exists an infinite path in the tree along which *I* and *P* always hold.

The proof is similar to that of Lemma 6, but it requires a slightly more complex function *f*, namely \(f : B \rightarrow B\), where \(B = \{(T,a) \in {\textsf {dtree}}\times \alpha \mid I\;(\textsf {{fst}}\;({{\textsf {lab}}}\;T), a)\}\), such that \( P\;(\textsf {{fst}}\;({{\textsf {lab}}}\;T),\, a)\;r\;(\textsf {{fst}}\;({{\textsf {lab}}}\;T'),\, a')\) holds whenever \((T,a) \in B\) and \((T',a') = f\,(T,a)\). The lemma’s assumption ensures such a choice of *f* is possible.

### 3.7 Escape Paths

*escape path*is a stream of steps that can form an infinite path in a derivation tree. It is defined coinductively as the predicate \({\textsf {epath}}: {\textsf {step}}\;{\textsf {stream}}\rightarrow {\textsf {bool}}\), which requires that every element in the given stream be obtained by applying an existing rule and choosing one of the resulting sequents:The following lemma is easy to prove by coinduction on the definition of \({\textsf {epath}}\).

### Lemma 9

For any stream \({\sigma }\) and tree *T*, if \({\textsf {wf}}\,\,T\) and \({\textsf {ipath}}\;{\sigma }\;T\), then \({\textsf {epath}}\;{\sigma }\).

### Example 10

### 3.8 Countermodel Paths

A countermodel path is a sequence of steps that witnesses the unprovability of a sequent *s*. Any escape path starting at *s* is a candidate for a countermodel path, given that it indicates a way to apply the proof rules without reaching any result. For it to be a genuine countermodel path, all possible proofs must have been attempted. More specifically, whenever a rule becomes enabled along the escape path, it is eventually applied later in the sequence. For FOL with its standard sequents, such paths can be used to produce actual countermodels by interpreting all statements along the path on the left of the sequents as true, and all statements on the right as false.

*r*is

*enabled*in a sequent

*s*if it has an effect (i.e., \({\textsf {eff}}\;r\;s\;{ ss }\) for some \({ ss }\)). This is written \({\textsf {enabled}}\;r\;s\). For any rule

*r*and stream \({\sigma }\):

\({\textsf {taken}}_r\;{\sigma }\) states that

*r*is taken at the start of the stream (i.e., \({\textsf {shead}}\;{\sigma }= (s,r)\) for some*s*);\({\textsf {enabledAt}}_r\;{\sigma }\) states that

*r*is enabled at the beginning of the stream (i.e., if \({\textsf {shead}}\;{\sigma }= (s, r')\), then \({\textsf {enabled}}\;r\;s\));

*P*”) states that

*P*is true for all suffixes of \({ xs }\). Dually, we take \({\textsf {ev}}\;P\;{\sigma }\) (“eventually

*P*”) to mean that

*P*is true for some suffix of \({ xs }\).

*saturated*if, at each point, any enabled rule is taken at a later point:

*countermodel path*for a sequent

*s*is a saturated escape path \({\sigma }\) starting at

*s*:

### Example 11

The escape path presented in Example 10 is not saturated, since the rule \({\textsc {All}}\textsc {L}_{x,p(x),z}\) is enabled starting from the first position but never taken.

### Example 12

The escape path in the tree of Fig. 3 is a countermodel path for \(\forall x.\,p(x) \mathrel {{\mathrel {\rhd }}}q(y)\), assuming that each possible term occurs infinitely often in the sequence \(t_1, t_2, \ldots \). The enabled rules along the escape path are all of the form \({\textsc {All}}\textsc {L}_{x,p(x),\_}\), and they are all always eventually taken.

### 3.9 Completeness

*Availability*:For each sequent, at least one rule is enabled (i.e., \(\forall s.\;\exists r.\;{\textsf {enabled}}\;r\;s\)).*Persistence*:For each sequent, if a rule is enabled but not taken, it remains enabled (i.e., \(\forall s\, r\,r'\, s'\, { ss }.\;{\textsf {enabled}}\;r'\,s \mathrel \wedge r' \not = r \mathrel \wedge {\textsf {eff}}\;r\;s\;{ ss }\mathrel \wedge s'\in \textsf {{set}}\;{ ss }\,\Rightarrow \, {\textsf {enabled}}\;r'\,s'\)).

### Remark 13

The saturation condition on streams of steps from Sect. 3.8 is stronger than the standard properties of fairness and justice [21]. Fairness would require the rules to be continuously enabled to guarantee that they are eventually taken. Justice is stronger in that it would require the rules to be enabled infinitely often, but not necessarily continuously. Saturation goes further: If a rule is ever enabled, it will certainly be chosen at a later point. Saturation may seem too strong for the task at hand. However, in the presence of Persistence, a rule enabled at some point and never taken will be continuously (in particular, infinitely often) enabled; hence the notions of fairness, justice, and saturation all coincide.

A countermodel path provides an argument against provability. That this argument fully complements provability is the essence of the completeness theorem in its abstract form:

### Lemma 14

### Proof

\({\textsf {stake}}\): \(\alpha \;{\textsf {stream}}\rightarrow {\textsf {nat}}\rightarrow \alpha \;{\textsf {list}}\) maps \(\rho \) and

*n*to the list of the first*n*elements of \(\rho \);\(\textsf {{smap}}\): \((\alpha \rightarrow \beta ) \rightarrow \alpha \;{\textsf {stream}}\rightarrow \beta \;{\textsf {stream}}\) maps a function to every element of the stream;

\({\textsf {nats}}\): \({\textsf {nat}}\;{\textsf {stream}}\) denotes the stream of natural numbers: \(0 \cdot 1 \cdot 2 \cdot 3 \cdot \ldots \,\);

\({\textsf {flat}}\): \((\alpha \;{\textsf {list}})\;{\textsf {stream}}\rightarrow \alpha \;{\textsf {stream}}\) maps a stream of finite nonempty lists to the stream obtained by concatenating those lists;

\({\textsf {sdropWhile}}\): \((\alpha \rightarrow {\textsf {bool}}) \rightarrow \alpha \;{\textsf {stream}}\rightarrow \alpha \;{\textsf {stream}}\) removes the maximal prefix of elements that fulfill a given predicate from a given stream (or returns an irrelevant default value if the predicate holds for all elements).

*s*be a sequent. Using \({\textsf {fenum}}\), we build a derivation tree \(T_0\) labeled with

*s*such that all its infinite paths are saturated. Let \({\textsf {fair}}\) be the subset of \({\textsf {rule}}\;{\textsf {stream}}\) consisting of the fair streams. Clearly, any suffix of an element in \({\textsf {fair}}\) also belongs to \({\textsf {fair}}\). In particular, \({\textsf {fenum}}\) and all its suffixes belong to \({\textsf {fair}}\). Given \(\rho \in {\textsf {fair}}\) and \(s\in {\textsf {sequent}}\), \({\textsf {sdropWhile}}\;(\lambda r.\;\lnot \;{\textsf {enabled}}\;r\;s)\;\rho \) has the form \({{\textsf {SCons}}}\;r\;\rho '\), making

*r*the first enabled rule in \(\rho \). Such a rule exists because, by Availability, at least one rule is enabled at

*s*and, by fairness, all the rules occur in \(\rho \). Since \({\textsf {enabled}}\;r\;s\), we can pick a set of sequents \({ ss }\) such that \({\textsf {eff}}\;r\;s\;{ ss }\). We define \({\textsf {mkTree}}: {\textsf {fair}}\rightarrow {\textsf {sequent}}\rightarrow {\textsf {dtree}}\) corecursively as

*s*, the derivation tree \({\textsf {mkTree}}\;\rho \;s\) is well formed and all its infinite paths are saturated. Wellformedness is obvious because at each point the continuation is built starting with the effect of a rule. For saturation, we show that if rule

*r*is enabled at sequent

*s*and \({\textsf {ipath}}\;({\textsf {mkTree}}\;\rho \;s)\;{\sigma }\), then

*r*appears along \({\sigma }\) (i.e., there exists a sequent \(s'\) such that \((s',r)\) is in \({\sigma }\)). This follows by induction on the position of

*r*in \(\rho \), \({\textsf {pos}}\;r\;\rho \)—formally, the length of the shortest list \(\rho _0\) such that \(\rho = \rho _0 \mathbin {@} {{\textsf {SCons}}}\;r\;\_\), where @ denotes concatenation.

Let \(r'\) be the first rule from \(\rho \) enabled at sequent *s*. If \(r = r'\), then \({\textsf {mkTree}}\;\rho \;s\) has label (*s*, *r*) already. Otherwise, \(\rho \) has the form \(\rho _1 \mathbin {@} [r'] \mathbin {@} \rho '\), with *r* not in \(\rho _1\), hence \({\textsf {pos}}\;r\;\rho ' < {\textsf {pos}}\;r\;\rho \). From the definitions of \({\textsf {ipath}}\) and \({\textsf {mkTree}}\), it follows that \({\textsf {ipath}}\;({\textsf {mkTree}}\;\rho '\;s')\;({\textsf {stail}}\;{\sigma })\) holds for some \(s' \in { ss }\) such that \({\textsf {eff}}\;r\;s'\;{ ss }\). By the induction hypothesis, *r* appears along \({\textsf {stail}}\;{\sigma }\), hence along \({\sigma }\) as desired. In particular, \(T_0 = {\textsf {mkTree}}\;{\textsf {fenum}}\;s\) is well formed and all its infinite paths are saturated.

Finally, if \(T_0\) is finite, it is the desired proof of *s*. Otherwise, by Lemma 6 (König) it has an infinite path. This path is necessarily saturated; by Lemma 9, it is the desired countermodel path. \(\square \)

Lemma 14 captures the abstract essence of arguments from the literature, although this is sometimes hard to grasp under the thick forest of syntactic details and concrete strategies for fair enumeration: A fair tree is constructed, which attempts a proof; in case of failure, the tree exhibits a saturated escape path. By Herbrandness, we immediately obtain completeness:

### Theorem 15

### Proof

Given *s* such that \(\models s\), assume by absurdity that there exists no *T* such that \({\textsf {proof}}\;T\;s\). By Lemma 14, we obtain \({\sigma }\) such that \({\textsf {countermodelPath}}\;s\;{\sigma }\). Then, by Herbrandness, we have \({\textsf {herbrand}}\;s\;{\sigma }\not \models s\), which contradicts \(\models s\). \(\square \)

### Remark 16

*s*has no proof and \({\textsf {eff}}\;r\;s\;{ ss }\), there must exist a sequent \(s'\in { ss }\) that has no proof. (Otherwise, we could compose the proofs of all \(s'\) into a proof of

*s*by applying rule

*r*.) Let \({\textsf {pick}}\;r\;s\;{ ss }\) denote such an \(s'\). We proceed directly to the construction of a saturated escape path as a corecursive predicate \({\textsf {mkPath}}: {\textsf {fair}}\rightarrow \{s \in {\textsf {sequent}}.\;s \text{ has } \text{ no } \text{ proof }\} \rightarrow {\textsf {step}}\;{\textsf {stream}}\) following the same idea as for the previous tree construction (function \({\textsf {mkTree}}\)):

### 3.10 Omitting the Availability Assumption

The abstract completeness result (Lemma 14) assumes Availability and Persistence. Among these assumptions, Persistence is essential: It ensures that the constructed fair path is saturated, meaning that every rule available at any point is eventually applied. Availability can be added later to the system without affecting its behavior by introducing a special “idle” rule.

### Lemma 17

A rule system \(\mathscr {R}= ({\textsf {sequent}},{\textsf {rule}},{\textsf {eff}})\) that fulfills Persistence can be transformed into the rule system \(\mathscr {R}_\mathrm {idle}= ({\textsf {sequent}},{\textsf {rule}}_\mathrm {idle},{\textsf {eff}}_\mathrm {idle})\) that fulfills both Persistence and Availability, with \({\textsf {rule}}_\mathrm {idle}= {\textsf {rule}}\;\cup \;\{{\textsc {Idle}}\}\) and \({\textsf {eff}}_\mathrm {idle}\) behaving like \({\textsf {eff}}\) on \({\textsf {rule}}\) and \({\textsf {eff}}_\mathrm {idle}\;{\textsc {Idle}}\;s\;{ ss }\,\iff \, { ss }= \{s\}\).

### Proof

Availability for the modified system follows from the continuous enabledness of \({\textsc {Idle}}\). Persistence follows from the Persistence of the original system together with the property that \({\textsc {Idle}}\) is continuously enabled and does not alter the sequent. The modified system is equivalent to the original one because \({\textsc {Idle}}\) does not alter the sequent. \(\square \)

### Theorem 18

Let \(\mathscr {R}\) be a rule system that fulfills Persistence and Herbrandness\(_\mathrm {idle}\). Then every sequent that is satisfied by all structures admits a proof in \(\mathscr {R}\).

### Proof

We apply Lemma 14 to the system \(\mathscr {R}_\mathrm {idle}\) to obtain that every sequent admits either a proof or a countermodel path, both in this system. Since \(\mathscr {R}_\mathrm {idle}\) is an extension of \(\mathscr {R}\) with a single rule, any proof in \(\mathscr {R}\) corresponds to a proof in \(\mathscr {R}_\mathrm {idle}\). Conversely, any proof in \(\mathscr {R}_\mathrm {idle}\) can be transformed into a proof in \(\mathscr {R}\) by omitting all applications of \({\textsc {Idle}}\). We thus proved that every sequent admits a proof in \(\mathscr {R}\) or a countermodel path over \(\mathscr {R}_\mathrm {idle}\). We can then apply Herbrandness\(_\mathrm {idle}\), just as we did with Herbrandness for Theorem 18. \(\square \)

### Remark 19

The addition of \({\textsc {Idle}}\) is inspired by, and similarly motivated as, that of idle transitions to Kripke structures in the context of temporal logic, where it is technically convenient to consider only infinite paths.

## 4 Concrete Instances of Soundness and Completeness

The abstract soundness is based on the Local Soundness assumption, which is easy to verify for all the considered instances. Therefore, below we focus on completeness.

### 4.1 Classical First-Order Logic

The abstract completeness proof is parameterized by a rule system. This section concretizes the result for the Gentzen system from Sect. 2.1 to derive the standard completeness theorem. Example 2 recasts it as a rule system; we must verify that it fulfills the Persistence and Herbrandness conditions.

The Gentzen rules are syntax-directed in that they operate on formulas having specific connectives or quantifiers at the top. This is what guarantees Persistence. For example, an application of \({\textsc {Ax}}_{a}\) (which affects only the atom *a*) leaves any potential enabledness of \({\textsc {All}}\textsc {L}_{x,\varphi ,t}\) (which affects only formulas with \({\textsf {All}}\) at the top) unchanged, and vice versa; moreover, \({\textsc {Ax}}_{a}\) does not overlap with \({\textsc {Ax}}_{b}\) for \(a \not = b\). A minor subtlety concerns \({\textsc {All}}\textsc {R}_{x,\varphi }\), which requires the existence of a fresh *y* in order to be enabled. Persistence holds because the sequents are finite, so we can always find a fresh variable in the countably infinite set \({\textsf {var}}\). On the other hand, Availability does not hold; for example, the sequent \(p(x) \mathrel {{\mathrel {\rhd }}}q(x)\) has no enabled rule. Hence, we need Theorem 18 and its Idle rule.

*S*is the set of terms, \({\textsf {term}}\);for each

*n*-ary*f*and*p*and each \(t_1,\ldots ,t_n \in \textit{S}\):

\(F_{f}\,(t_1,\ldots ,t_n) \,=\, f(t_1,\ldots ,t_n)\);

### Lemma 20

\(\mathscr {S}\models _\xi \varphi [t / x]\) iff Open image in new window.

### Lemma 21

The structure \({\textsf {herbrand}}\;(\Gamma \mathrel {{\mathrel {\rhd }}}\Delta )\;{\sigma }\) is a countermodel for \(\Gamma \mathrel {{\mathrel {\rhd }}}\Delta \), meaning that there exists a valuation \(\xi : {\textsf {var}}\rightarrow S\) such that \(\mathscr {S}\not \models _\xi \Gamma \mathrel {{\mathrel {\rhd }}}\Delta \).

### Proof

- 1.
For all atoms

*a*, Open image in new window. - 2.
- 3.
- 4.
- 5.
- 6.
If Open image in new window, then Open image in new window for all

*t*. - 7.
If Open image in new window, then there exists a variable

*y*such that Open image in new window.

- 1.
If Open image in new window, the rule \({\textsc {Ax}}_a\) is enabled in \({\sigma }\) and hence, by saturation, it is eventually taken. This is impossible since this is a rule without premises, whose application would prevent the continuation of the infinite sequence \({\sigma }\).

- 6.
If Open image in new window and

*t*is a term, \({\textsc {All}}\textsc {L}_{x,\varphi ,t}\) is enabled in \({\sigma }\) and hence eventually taken, ensuring that Open image in new window.

*t*. Hence, by the induction hypothesis, \(\mathscr {S}\models _{\xi } \varphi [t / x]\). By Lemma 20, \(\mathscr {S}\models _{\xi [x \leftarrow t]} \varphi \) for all

*t*; that is, \(\mathscr {S}\models _\xi {\textsf {All}}\;x\;\varphi \). The second fact, concerning Open image in new window, follows similarly from property 7. \(\square \)

We have thus obtained:

### Corollary 22

A sequent is provable by a (finite) proof in the Gentzen system of classical FOL iff it is satisfied by all FOL structures.

### Remark 23

The rule \({\textsc {All}}\textsc {L}\) stores, in the left context, a copy of the universal formula \({\textsf {All}}\;x\;\varphi \) when applied backwards. This is crucial for concrete completeness since a fair enumeration should try all the *t* instances of the universally quantified variable *x*, which requires Availability of \({\textsf {All}}\;x\;\varphi \) even after its use. If we labeled \({\textsc {All}}\textsc {L}\) as \({\textsc {All}}\textsc {L}_{x,\varphi }\) instead of \({\textsc {All}}\textsc {L}_{x,\varphi ,t}\), thereby delegating the choice of *t* to the nondeterminism of \({\textsf {eff}}\), the system would still be persistent as required by the abstract completeness proof, but Lemma 21 (and hence concrete completeness) would not hold—property 6 from the lemma’s proof would fail.

### 4.2 Further Instances

Theorem 18 is applicable to classical FOL Gentzen systems from the literature, in several variants: with sequent components represented as lists, multisets, or sets, one-sided or two-sided, and so on. This includes the systems G\('\), GCNF\('\), G, and G\(_{\text {=}}\) from Gallier [22] and G1, G2, G3, GS1, GS2, and GS3 from Troelstra and Schwichtenberg [51]. Persistence is easy to check. The syntax-independent part of the argument is provided by Theorem 18, while an ad hoc step analogous to Lemma 21 is required to build a concrete countermodel from a countermodel path to complete the proof.

Several FOL refutation systems based on tableaux or resolution are instances of the abstract theorem, providing that we read the abstract notion of “proof” as “refutation” and “countermodel” as “model.” Nondestructive tableaux [26]—including those presented in Bell and Machover [1] and in Fitting [20]—are usually persistent when regarded as derivation systems. After an application of Theorem 18, the ad hoc argument for interpreting the abstract model is similar to that for Gentzen systems (Lemma 21).

The use of labels for worlds (\(w{,}\,w'\)) and the bookkeeping of the accessibility relation *R* makes it possible to recast the rule so that either no facts (as above) or only resilient facts are ever assumed about the surrounding context. The resulting proof system fulfills Persistence, enabling Theorem 18. The Kripke countermodel construction is roughly as for classical FOL Gentzen systems.

## 5 Abstract Infinite-Proof Soundness

In the previous sections, completeness is established by analyzing the interplay between the existence of *finite* proof trees (representing valid proofs) and certain *infinite* proof trees (used to produce countermodels). In this section, we look into the question: When can *infinite* proof trees be accepted as valid proofs?

We give a coinductive account of the abstract development of Brotherston et al. [16], in a slightly more general form since we work with arbitrary infinite proofs, which may be acyclic. We start by recalling some motivation and intuition about cyclic proofs, in the context of the FOL\(_{\textsf {ind}}\) logic from Sect. 2.2. It is clear that the rules of the FOL\(_{\textsf {ind}}\) Gentzen system are (locally) sound for inductive structures. Hence, any finite proof tree built using these rules represents a valid proof, in that its root sequent is known to be satisfied by all inductive structures. But other proofs are permissible too.

*L*“back” to the root of the tree (decorated with

*L*). A cyclic proof indicates that, when

*L*is reached, the (backward) proof continues with the sequent that

*L*points to. Thus, the intended proof corresponding to the cyclic tree from Fig. 4 is the infinite tree from Fig. 5.

### 5.1 Soundness of Infinite Proof Trees

a set \({\textsf {marker}}\) of items called

*markers*;a

*marking function*\({\textsf {mark}}: \{(s,r,s') \in {\textsf {sequent}}\times {\textsf {rule}}\times {\textsf {sequent}}\mid \exists { ss }.\;{\textsf {eff}}\;s\;r\;{ ss }\mathrel \wedge s' \in { ss }\} \rightarrow ({\textsf {marker}}\times {\textsf {bool}}\times {\textsf {marker}})\;\textsf {{set}}\);an ordinal \({{\textsf {ord}}}\), with < and \(\le \) denoting its strict and nonstrict order relations;

a

*height*function \({\textsf {height}}: {\textsf {marker}}\times {\textsf {structure}}\rightarrow {{\textsf {ord}}}\).

*r*is thought of as being applied to

*s*and then one of the resulting sequents, \(s'\), being chosen. In \((M,b,M')\), the Boolean value

*b*being \(\textsf {{False}}\) means “stay” and

*b*being \(\textsf {{True}}\) means “decrease.” The “stay” and “decrease” directives refer to how the height function \({\textsf {height}}\) should evolve along this rule application: Given a sequent

*s*, a countermodel

*S*for it and a rule

*r*that yields a set of sequents \({ ss }\) when applied to

*s*, there should exist a sequent \(s' \in { ss }\) and a countermodel \(S'\) of \(s'\) such that, for all \((M,b,M') \in {\textsf {mark}}\;(s,r,s')\), when moving from (

*M*,

*S*) to \((M',S')\):

the height should at least stay the same (or decrease) when

*b*says “stay”;the height should decrease when

*b*says “decrease.”

Descent:For allSandssuch that \(S \not \models s\) and allr, \({ ss }\) such that \({\textsf {eff}}\;s\;r\;{ ss }\), there exist \(S'\) and \(s' \in { ss }\) such that \(S' \not \models s'\) and \({{\textsf {descent}}}\;(s,S)\;r\;(s',S')\).

### Remark 24

The Descent condition is a strengthening of the Local Soundness condition from Sect. 3.5: Removing the “and \({{\textsf {descent}}}\;(s,S)\;r\;(s',S')\)” part yields the contrapositive of Local Soundness. Essentially, Descent is a form of Local Soundness with the additional requirement that the marking function’s directives must be respected.

*b*in \({ bs }\), have the property that \((M,b,M') \in {\textsf {mark}}\;(s,r,s')\); in other words, the streams \({ bs }\) and \({ Ms }\) represent choices of the sets of markers for \({\sigma }\).

*iproof*, is then defined as a good, well-formed tree having that sequent at the root:

### Theorem 25

### Proof

*s*and

*T*such that \({\textsf {iproof}}\;T\;s\) and assume by absurdity that \(\not \models s\), meaning there exists

*S*such that \(S \not \models s\). To obtain a contradiction, we proceed as follows:

- 1.Applying Lemma 8 (Sect. 3.6) for \(\alpha = {\textsf {structure}}\), \(I = \lambda (s,\textit{S}).\;\textit{S} \not \models s\) and \(P = {{\textsf {descent}}}\) to the Descent assumption, we obtain a stream \(\delta \) of step–structure pairs (i.e., of sequent–rule–structure triples) \(((s,r),\textit{S})\), such that, at each point in the stream, the structure \(\textit{S}\) is a countermodel for the sequent
*s*and \({{\textsf {descent}}}\) holds for any two consecutive elements. Formally, we have \({\textsf {alwCmodDesc}}\;\delta \) (“always countermodel and descending”), where \({\textsf {alwCmodDesc}}\) is defined as and the step components of \(\delta \) form an infinite path in*T*: \({\textsf {ipath}}\;T\;(\textsf {{smap}}\;\textsf {{fst}}\;\delta )\). - 2.Using the goodness of
*T*, we obtain the streams \({ bs }\) and \({ Ms }\) that follow a suffix \({\sigma }'\) of \(\textsf {{smap}}\;\textsf {{fst}}\;\delta \):where \({{@}}\) denotes concatenation of a list and a stream.$$\begin{aligned} \exists {\sigma }'\;{\sigma }''\;{ bs }\;{ Ms }.\;\,\textsf {{smap}}\;\textsf {{fst}}\;\delta = {\sigma }'' {{@}}\, {\sigma }' \mathrel \wedge {\textsf {follow}}\;{ bs }\;{ Ms }\;{\sigma }' \mathrel \wedge {\textsf {infDecr}}\;{ bs }\end{aligned}$$ - 3.Taking the suffix \(\delta '\) of \(\delta \) that corresponds to the suffix \({\sigma }'\) of its step component \(\textsf {{smap}}\;\textsf {{fst}}\;\delta \) (formally, defining \(\delta '\) to be \({\textsf {stake}}\,({{\textsf {length}}}\;{\sigma }')\,\delta \)), we end up with the streams \(\delta '\), \({ bs }\) and \({ Ms }\) such that
\({\textsf {follow}}\;{ bs }\;{ Ms }\;(\textsf {{smap}}\;\textsf {{fst}}\;\delta ')\);

\({\textsf {infDecr}}\;{ bs }\);

\({\textsf {alwCmodDesc}}\;\delta '\) (because \({\textsf {alw}}\) is invariant under suffix).

- 4.Let \({\textsf {zip}}\) denote the pair-zipping of two streams, defined by primitive corecursion as \({\textsf {zip}}\;({{\textsf {SCons}}}\;x\;{ xs })\;({{\textsf {SCons}}}\;y\)\(\;{ ys }) = {{\textsf {SCons}}}\;(x,y)\;({\textsf {zip}}\;{ xs }\;{ ys })\). Taking \({ ks }= \textsf {{smap}}\;{\textsf {height}}\;({\textsf {zip}}\;(\textsf {{smap}}\;\textsf {{fst}}\;\delta ')\;{ Ms })\) (a stream of \({{\textsf {ord}}}\) elements), we have thatThis follows easily by coinduction using the gathered properties for \(\delta '\), \({ bs }\), and \({ Ms }\).
\({ ks }\) is always nonstrictly decreasing: \({\textsf {alw}}\;(\lambda { ks }'.\;{\textsf {shead}}\;{ ks }' \ge {\textsf {shead}}\;({\textsf {stail}}\;{ ks }'))\;{ ks }\);

\({ ks }\) is infinitely often strictly decreasing: \({\textsf {alw}}\;({\textsf {ev}}\;(\lambda { ks }'.\;{\textsf {shead}}\;({\textsf {stail}}\;{ ks }') > {\textsf {shead}}\;{ ks }'))\;{ ks }\).

- 5.From \({ ks }\), we obtain the substream \({ ks }' \) that is always strictly decreasing:which is in contradiction with the wellfoundedness of the order < on the ordinal \({{\textsf {ord}}}\). \(\square \)$$\begin{aligned} {\textsf {alw}}\;(\lambda { ks }'.\;{\textsf {shead}}\;{ ks }' > {\textsf {shead}}\;({\textsf {stail}}\;{ ks }'))\;{ ks }' \end{aligned}$$

*x*in \({ xs }\), there exists an element

*y*occurring later in \({ xs }\) such that \(P\;x\;y\):

*P*and forms a new stream with them. Its definition is neither purely recursive nor purely corecursive. Whereas the call on the ‘then’ branch is corecursive (showing how to produce the first element in the result stream, \({\textsf {shead}}\;{ xs }\)), the call on the ‘else’ branch does not exhibit any explicit productivity. However, the overall function is well-defined (and indeed productive) because, thanks to the choice of the domain, at each point there will be at most a finite number of consecutive calls on the ‘else’ branch, until an element satisfying \(P\;({\textsf {shead}}\;{ xs })\) is met. In summary, the

*recursive*call on the ‘else’ branch concurs with the

*corecursive*call on the ‘then’ branch for the welldefinedness of this function.

*y*such that \(P\;z\;y\). The use of such sophisticated corecursive machinery for this example may seem excessive. But note that \({\textsf {bfilter}}\) is a general-purpose combinator, defined once and usable for arbitrary predicates

*P*.

### 5.2 From Cyclic Trees to Infinite Trees

*T*yields a derivation tree \({\textsf {treeOf}}\;T\) as follows:

*T*is traversed as if recursively producing \({\textsf {treeOf}}\;T\) by the application of the same rules, and when reaching a \({\textsf {Link}}\;L\) one moves to the pointed cyclic tree \({\textsf {pointsTo}}\;L\). Because \({\textsf {pointsTo}}\;L\) can be arbitrarily large (in particular, larger than

*T*), the definition of \({\textsf {treeOf}}: {\textsf {cdtree}}\rightarrow {\textsf {dtree}}\) cannot proceed recursively on the domain (\({\textsf {cdtree}}\)). However, it can naturally proceed corecursively on the codomain, \({\textsf {dtree}}\):

*T*is the same as its generated derivation tree \({\textsf {treeOf}}\;T\), and we can prove by induction that wellformedness of

*T*implies wellformedness of \({\textsf {treeOf}}\;T\). With Theorem 25, this immediately gives our desired soundness theorem for cyclic derivation trees.

### Theorem 26

The above theorem is a slight generalization of the result by Brotherston et al. [16]. It is more general in that it does not require the cyclic proof trees to be regular. If we wanted to model precisely the regular trees, we would need to further restrain \({\textsf {pointsTo}}\); however, such restrictions do not affect soundness.

## 6 Concrete Instances of Infinite-Proof Soundness

We consider instances of the abstract development establishing soundness of infinite proofs. We discuss in detail our running example, FOL with inductive definitions. Then we briefly mention other possible instances.

### 6.1 First-Order Logic with Inductive Predicates

Recall FOL\(_{\textsf {ind}}\) from Sect. 2.2 and its associated Gentzen system. The abstract structures (elements of the set \({\textsf {structure}}\)) are not instantiated by mere concrete structures, but rather by pairs \((\mathscr {S},\xi )\) where \(\mathscr {S}= \bigl (\textit{S},\, (F_{f})_{f \,\in \, {\textsf {fsym}}},\, (P_{p})_{p \,\in \, {\textsf {psym}}}\bigr )\) is a structure and \(\xi : {\textsf {var}}\rightarrow S\) is a variable valuation. We instantiate the satisfaction relation, \((\mathscr {S},\xi ) \models \varphi \), with \(\mathscr {S}\models _\xi \varphi \).

To instantiate Theorems 25 and 26, we must define the notion of marker (forming the set \({\textsf {marker}}\)), the ordinal \({{\textsf {ord}}}\), and the functions \({\textsf {mark}}: \{(s,r,s') \in {\textsf {sequent}}\times {\textsf {rule}}\times {\textsf {sequent}}\mid \exists { ss }.\;{\textsf {eff}}\;s\;r\;{ ss }\mathrel \wedge s' \in { ss }\} \rightarrow ({\textsf {marker}}\times {\textsf {bool}}\times {\textsf {marker}})\;\textsf {{set}}\) and \({\textsf {height}}: {\textsf {marker}}\times {\textsf {structure}}\rightarrow {{\textsf {ord}}}\) and verify the Descent property.

- Rule
*r*is not of the form \(\textsc {Subst}_{t,x}\) or \(p_{\mathrm{split}}\). Then, assuming \(s = (\Gamma \mathrel {{\mathrel {\rhd }}}\Delta )\), we set$$\begin{aligned} {\textsf {mark}}\;(s,r,s') =\{(M,\textsf {{False}},M) \mid M \in {\textsf {marker}}\mathrel \cap \Gamma \} \end{aligned}$$ - Rule
*r*has the form \(\textsc {Subst}_{t,x}\). Then*s*has the form \(\Gamma [t/x] \mathrel {{\mathrel {\rhd }}}\Delta [t/x]\), and we set$$\begin{aligned} {\textsf {mark}}\;(s,r,s') =\{(M[t/x],\textsf {{False}},M) \mid M \in {\textsf {marker}}\mathrel \cap \Gamma \} \end{aligned}$$ - Rule
*r*has the form \(p_{\mathrm{split}}\) for some \(p \in {\textsf {ipsym}}\). Then*s*has the form \(\Gamma {,}\;p(\overline{x}) \mathrel {{\mathrel {\rhd }}}\Delta \), and \(s'\) has the form \(\Gamma [\overline{t}/\overline{x}],\,{\textsf {prems}}(\chi ') \mathrel {{\mathrel {\rhd }}}\Delta [\overline{t}/\overline{x}]\) for some \(\chi \in {\textsf {ind}}_p\). We set

^{2}Let \(\mathscr {S}= \bigl (\textit{S},\, (F_{f})_{f \,\in \, {\textsf {fsym}}},\, (P_{p})_{p \,\in \, {\textsf {psym}}}\bigr )\) be a fixed inductive structure. We define the family of interpretations \((P_{p,n} : S^{{\textsf {ar}}\;p} \rightarrow S)_{p \in {\textsf {ipsym}},n \in {\textsf {nat}}}\) recursively as follows:

\(P_{p,0}\,\overline{a} \,\iff \, \textsf {{False}}\);

\(P_{p,n+1}\,\overline{a} \,\iff \, P_{p,n}\,\overline{a} \mathrel {\vee } (\exists \chi \in {\textsf {ind}}_p.\;\exists \xi .\;\overline{a} = \xi \;({\textsf {varsOf}}({\textsf {concl}}(\chi ))) \mathrel \wedge \mathscr {S}_n \models _\xi {\textsf {prems}}(\chi ))\), where \(\mathscr {S}_n = \bigl (\textit{S},\, (F_{f})_{f \,\in \, {\textsf {fsym}}},\, (P_{p,n})_{p \,\in \, {\textsf {psym}}}\bigr )\) and \({\textsf {varsOf}}({\textsf {concl}}(\chi ))\) is the tuple \(\overline{x}\) appearing in the conclusion \(p(\overline{x})\) of \(\chi \).

### Lemma 27

For all \(p \in {\textsf {ipsym}}\) and \(\overline{a} \in S^{{\textsf {ar}}\;p}\), we have \(P_p\,\overline{a}\) iff \(\exists n.\;P_{p,n}\,\overline{a}\).

Assume \(\mathscr {S}\models _\xi M\), and assume

*M*has the form \(p(\overline{t})\) for \(p \in {\textsf {ipsym}}\). Then \(P_p(\llbracket \overline{t}\rrbracket ^{\mathscr {S}}_{\xi })\). We let \({\textsf {height}}\;(M,(\mathscr {S},\xi ))\) be the smallest*n*such that \(P_{p,n}(\llbracket \overline{t}\rrbracket ^{\mathscr {S}}_{\xi })\) (which exists by Lemma 27).Assume \(\mathscr {S}\not \models _\xi M\). Then the value of \({\textsf {height}}\;(M,(\mathscr {S},\xi ))\) is irrelevant, and we can put anything here.

*s*be such that \(\mathscr {S}\not \models _\xi s\), and let

*r*and \({ ss }\) be such that \({\textsf {eff}}\;s\;r\;{ ss }\). We need to provide \(s' \in { ss }\) and \((\mathscr {S}',\xi ')\) such that \(\mathscr {S}' \not \models _{\xi '} s'\) and \({{\textsf {descent}}}\;(s,(\mathscr {S},\xi ))\;r\;(s',(\mathscr {S}',\xi '))\) holds. We will actually take \(\mathscr {S}'\) to be \(\mathscr {S}\), so we only need to provide \(s'\) and \(\xi '\).

Rule

*r*is not of the form \(\textsc {Subst}_{t,x}\) or \(p_{\mathrm{split}}\). We further distinguish some subcases:

*r*is Ax. This is impossible, since Ax is sound for all (inductive) structures, which contradicts the hypothesis \(\mathscr {S}\not \models _\xi s\).*r*is a single-premise rule involving no freshness side condition, i.e., is one of NegL, NegR, ConjR, AllL, and \(p_\chi \). Then we take \(s'\) to be the single premise and \(\xi ' = \xi \).*r*is conjL. Then we take \(\xi ' = \xi \) and \(s'\) to be one of the two premises, say, \(s_i\), such that \(\mathscr {S}\not \models _\xi s_i\) (which is known to exist by the soundness of ConjL).*r*is AllR. Then \(s = (\Gamma \mathrel {{\mathrel {\rhd }}}\Delta {,}\;{\textsf {All}}\;x\;\varphi )\), and we take \(s'\) to be the only premise of*r*, namely, \(s' = \Gamma \mathrel {{\mathrel {\rhd }}}\Delta {,}\;\varphi [y / x]\), where*y*is known to be fresh. Since \(\mathscr {S}\not \models _\xi {\textsf {All}}\;x\;\varphi \), we obtain \(a \in S\) such that \(\mathscr {S}\not \models _{\xi [x \leftarrow a]} \varphi \); by the freshness of*y*, this also implies \(\mathscr {S}\not \models _{\xi [y \leftarrow a]_{\phantom {]}}} \varphi [y/x]\). We let \(\xi ' = \xi [y \leftarrow a]\).

Rule

*r*has the form \(\textsc {Subst}_{t,x}\). Then*s*has the form \(\Gamma [t/x] \mathrel {{\mathrel {\rhd }}}\Delta [t/x]\). We set \(\xi ' = \xi [x \leftarrow \llbracket t\rrbracket ^{\mathscr {S}}_{\xi }]\) and*s*to the only premise of*r*.- Rule
*r*has the form \(p_{\mathrm{split}}\) for some \(p \in {\textsf {ipsym}}\). Then*s*has the form \(\Gamma {,}\;p(\overline{x}) \mathrel {{\mathrel {\rhd }}}\Delta \). Since \(\mathscr {S}\models _\xi p(\overline{x})\), we have \(P_p\,(\xi \,\overline{x})\), and let*m*be the least number such that \(P_{p,m}\,(\xi \,\overline{x})\). By the definition of \(P_{p,\_}\), \(m > 0\) and there exist \(\chi \in {\textsf {ind}}_p\) and the valuation \(\xi ''\) such that \(\xi \,\overline{x} = \xi ''\,\overline{x}\) and \(\mathscr {S}_{m-1} \models _{\xi ''} {\textsf {prems}}(\chi ')\).^{3}We take \(s'\) to be the premise of*r*corresponding to \(\chi \), namely \(s' = (\Gamma [\overline{t}/\overline{x}],\,{\textsf {prems}}(\chi ') \mathrel {{\mathrel {\rhd }}}\Delta [\overline{t}/\overline{x}])\). Finally, we define \(\xi '\) as follows:$$\begin{aligned} \xi '\;z = \left\{ \begin{array}{ll} \xi \;z &{} \text{ if } \text{ z } \text{ appears } \text{ free } \text{ in } \Gamma \,\cup \, \Delta \\ \xi ''\;z &{} \text{ if } \text{ z } \text{ appears } \text{ free } \text{ in } {\textsf {prems}}(\chi ')\\ \text{ anything } &{} \text{ otherwise } \end{array} \right. \end{aligned}$$

For all \(M \in {\textsf {marker}}\mathrel \cap \Gamma \), we have \(\llbracket M[\overline{t}/\overline{x}]\rrbracket ^{\mathscr {S}}_{\xi '} = \llbracket M\rrbracket ^{\mathscr {S}}_{\xi }\).

For all \(q\,(\overline{t}) \in {\textsf {marker}}\mathrel \cap {\textsf {prems}}(\chi ')\),

*n*is the smallest number such that \(P_{q,n}(\llbracket \overline{t}\rrbracket ^{\mathscr {S}}_{\xi '})\).

### Corollary 28

Assume that, in the Genzten system of FOL\(_{\textsf {ind}}\), a sequent \(\Gamma \mathrel {{\mathrel {\rhd }}}\Delta \) either has an iproof or has a cyclic proof and \({\textsf {pointsTo}}\) fulfills Good Links. Then \(\Gamma \mathrel {{\mathrel {\rhd }}}\Delta \) is satisfied by all pairs \((\mathscr {S},\xi )\), i.e., \(\mathscr {S}\models _\xi \Gamma \mathrel {{\mathrel {\rhd }}}\Delta \). Hence, \(\mathscr {S}\models \Gamma \mathrel {{\mathrel {\rhd }}}\Delta \).

### Example 29

*T*of Fig. 5 is an iproof, i.e., is well formed and satisfies the goodness predicate. To see the latter, observe that the only infinite path in this tree is \({\sigma }= ((s_1,r_1) \cdot (s_2,r_2) \cdot (s_3,r_3))^\omega \), where

\(s_1 = ({{\textsf {even}}}(x) \mathrel {{\mathrel {\rhd }}}{{\textsf {odd}}}({{\textsf {Suc}}}(x)))\), \(r_1 = \textsc {Even}_{\mathrm{split}}\);

\(s_2 = ({{\textsf {even}}}(y) \mathrel {{\mathrel {\rhd }}}{{\textsf {odd}}}({{\textsf {Suc}}}({{\textsf {Suc}}}({{\textsf {Suc}}}(y)))))\), \(r_2 = \textsc {Odd}{}_{{\textsf {Suc}}}\);

\(s_3 = ({{\textsf {even}}}(y) \mathrel {{\mathrel {\rhd }}}{{\textsf {odd}}}({{\textsf {Suc}}}(y)))\), \(r_3 = \textsc {Subst}_{y,x}\).

For accepting a cyclic or infinite proof in FOL\(_{\textsf {ind}}\) as a good (and hence valid) proof, any infinite path should infinitely often apply the split rule to the (persistent) instance of the same inductive predicate *p*.^{4} This can be seen from the fact that a triple \((\_,\textsf {{True}},\_)\), meaning “decrease,” only appears in the definition of \({\textsf {mark}}\) for the split rule. In the above example, the predicate *p* that ensures goodness along the tree’s unique infinite path is \({{\textsf {even}}}\).

### 6.2 Other Instances

Variations of Gentzen systems for FOL, as discussed in Sect. 4.2, can in principle be extended to FOL\(_{\textsf {ind}}\), and the abstract infinite-proof soundness result would apply. Other instances include extensions of modal logic with inductive definitions. In such logics, the structures can be viewed as tuples \(\mathscr {S}= \bigl ((\textit{S}_w)_{w \in W},\, (F_{f,w})_{f \,\in \, {\textsf {fsym}}, w \in W},\, (P_{p,w})_{p \,\in \, {\textsf {psym}}, w \in W}\bigr )\), where *W* is a set of “worlds,” perhaps endowed with additional structure (algebraic, order-theoretic, etc.). The inductiveness condition for predicates can be stated similarly to that of FOL\(_{\textsf {ind}}\), but can also spread across different worlds. (The split rule can be adapted accordingly.)

Separation logic can be regarded as a variation of the above, where the structure carrier \(\textit{S}_w\) is fixed to some \(\textit{S}\) and the worlds are heaps, i.e., partial functions from a fixed set of locations to \(\textit{S}\). (In separation logic terminology, the worlds are the heaps, whereas the valuations \(\xi \) are the stacks.) Two such instances are described by Brotherston et al. [16], one for entailment [15] and one for termination proofs [14].

## 7 Formalization and Implementation

The definitions, lemmas, and theorems presented in Sects. 3 and 5, pertaining to the presented abstract soundness and completeness results, have been formalized in the proof assistant Isabelle/HOL. The instantiation step of Sect. 4.1 is formalized for a richer version of FOL, with sorts and interpreted equality, as required by our motivating application (efficient encodings of sorts in unsorted FOL [7]). The formal development is publicly available [8, 10].

The necessary codatatypes and corecursive definitions are realized using a recently introduced definitional package for (co)datatypes [6] with support for mutual and nested (co)recursion [50] and mixed recursive–corecursive function definitions [11]. The \({\textsf {tree}}\) codatatype illustrates the support for corecursion through permutative data structures (with non-free constructors) such as finite sets, a feature that is not available in any other proof assistant. The formalization is close to this article’s presentation, with a few differences originating from Isabelle’s lack of support for dependent types.

^{5}From this function, we build the relational \({\textsf {eff}}\) as the partial function’s graph. Isabelle’s code generator [25] can then produce Haskell code for the computable part of our completeness proof—the abstract prover \({\textsf {mkTree}}\), defined corecursively in the proof of Theorem 15: Finite sets are represented as lists. The functions \({\textsf {isJust}} : \alpha \;{\textsf {option}}\rightarrow {\textsf {bool}}\) and \({\textsf {fromJust}} : \alpha \;{\textsf {option}}\rightarrow \alpha \) are the Haskell-style discriminator and selector for \({\textsf {option}}\). Since the Isabelle formalization is parametric over rule systems \(({\textsf {sequent}},{\textsf {rule}},{\textsf {eff}})\), the code for \({\textsf {mkTree}}\) explicitly takes \({\textsf {eff}}\) as a parameter.

Although Isabelle’s code generator was not designed with codatatypes in mind, it is general enough to handle them. Internally, it reduces Isabelle specifications to higher-order rewrite systems [35] and generates functional code in Haskell, OCaml, Scala, or Standard ML. Partial correctness is guaranteed regardless of the target language’s evaluation strategy. However, for the guarantee to be non-vacuous for corecursive definitions, one needs a language with a lazy evaluation strategy, such as Haskell.

The verified contract of the program reads as follows: Given an available and persistent rule system \(({\textsf {sequent}},{\textsf {rule}},{\textsf {eff}})\), a fair rule enumeration \(\rho \), and a sequent *s* representing the formula to prove, \({\textsf {mkTree}}\;{\textsf {eff}}\;\rho \;s\) either yields a finite derivation tree of *s* or produces an infinite fair derivation tree whose infinite paths are all countermodel paths. These guarantees involve only partial correctness of ground term evaluation.

The generated code is a generic countermodel-producing semidecision procedure parameterized by the proof system. Moreover, the fair rule enumeration parameter \(\rho \) can be instantiated to various choices that may perform better than the simple scheme described in Sect. 3. However, more research is needed to understand how our framework or a refinement of it can accommodate state-of-the-art proof search procedures, such as conflict-driven clause learning [5]. For example, one would rather want to specify \(\rho \) lazily (using unification) during, and not before, the evaluation of \({\textsf {mkTree}}\).

## 8 Related Work

This article joins a series of pearls aimed at reclaiming mathematical concepts and results for coinductive methods, including streams [43, 47], regular expressions [44, 46], and automata [45]. Some developments pass the ultimate test of formalization, usually in Agda and Coq, the codatatype-aware proof assistants par excellence: the sieve of Eratosthenes [3], real number basics [18], and temporal logic for red–blue trees [36].

So why write yet another formalized manifesto for coinduction and corecursion? First, because we finally could—with the new codatatype package, Isabelle has caught up with its rivals in this area, and has even superseded them in some respects; for example, the mixture of recursion and corecursion allowed in function definitions is unique to Isabelle. Second, because although codatatypes are a good match for the completeness and the infinite-proof soundness theorems (as we hope to have shown), there seems to be no proof in the literature that takes advantage of this.

There are many accounts of the completeness theorem for FOL and related logics, including Petria’s very abstract account[40] within institution-independent model theory [19]. However, most of these accounts favor the more mathematical Henkin style, which obfuscates the rich structure of proof and failure. This preference has a long history. It is positively motivated by the ability to support uncountable languages. More crucially, it is negatively motivated by the lack of rigor perceived in the alternative: “geometric” reasoning about infinite trees. Negri [37] gives a revealing account in the context of modal logic, quoting reviews that were favorable to Kripke’s completeness result [32] but critical of his informal argument based on infinite tableau trees.^{6} Kaplan [30] remarks that “although the author extracts a great deal of information from his tableau constructions, a completely rigorous development along these lines would be extremely tedious.”

A few textbooks venture in a proof-theoretic presentation of completeness, notably Gallier’s [22]. Such a treatment highlights not only the structure, but also the algorithmic content of the proofs. The price is usually a lack of rigor, in particular a gap between the definition of derivation trees and its use in the completeness argument. This lack of rigor should not be taken lightly, as it may lead to serious ambiguities or errors: In the context of a tableau completeness proof development, Hähnle [26] first performs an implicit transition from finite to possibly infinite tableaux, and then claims that tableau chain suprema exist by wrongly invoking Zorn’s lemma [26, Definition 3.16].^{7}

Orthogonally, we wanted to isolate and reuse the abstract core of the argument involving potentially infinite derivation trees and countermodel paths. Except for syntactic details, the different accounts are after the same goal, and they reach it in a variety of more or less colorful, if not noisy, ways; most of them do acknowledge that their approach is similar to previous ones, but cannot refer to a given abstract result that addresses this goal. Consequently, they have to repeat a variation of the same argument. For example, Gallier’s monograph [22] repeats the argument four times, for logics of increasing complexity: propositional logic, FOL with no function symbols or equality, FOL with function symbols but no equality, and finally full FOL; Bell and Machover [1] employ a different fair tree generation strategy, to the same effect; for a world-instrumented system for modal logic, Negri [37] employs yet another strategy.

Regrettably, one of the completeness results not covered by our abstract completeness is connected to our other case study: logics admitting infinite proofs. Brotherston and Simpson [17] show that completeness for FOL\(_{\textsf {ind}}\) (with equality) can be achieved by allowing infinite proofs satisfying the Goodness condition from Sect. 5.1. This nonstandard completeness result is not captured by our framework from Sect. 3, which requires finite proofs.

Unlike the infinite-proof soundness theorem (which represents a newer line of research), the completeness theorem has been mechanized before in proof assistants. Schlöder and Koepke, in Mizar [49], formalize a Henkin-style argument for possibly uncountable languages. Building on an early insight by Krivine [33] concerning the expressibility of the completeness proof in intuitionistic second-order logic, Ilik [28] analyzes Henkin-style arguments for classical and intuitionistic logic with respect to standard and Kripke models and formalizes them in Coq (without employing codatatypes).

At least four proofs were developed using HOL-based systems. Harrison [27], in HOL Light, and Berghofer [2], in Isabelle, opted for Henkin-style arguments. Berghofer’s work was recently extended by Schlichtkrull [48] to prove the completeness of first-order resolution. Ridge and Margetson [34, 42], also in Isabelle, employ proof trees constructed as graphs of nodes that carry their levels as natural numbers. This last work has the merits of analyzing the computational content of proofs in the style of Gallier [22] and discussing an OCaml implementation. Our formalization relates to this work in a similar way to which our presentation relates to Gallier’s: The newly introduced support for codatatypes and corecursion in Isabelle provides suitable abstraction mechanisms for reasoning about infinite trees, avoiding boilerplate for tree manipulation based on numeric indexing. Moreover, codatatypes are mapped naturally to Haskell types, allowing Isabelle’s code generator to produce certified Haskell code. Finally, our proof is abstract and applies to a wide range of FOL variants and beyond.

## 9 Conclusion

The completeness theorem is a fundamental result about classical logic. Its proof is presented in many variants in the literature. Few of these presentations emphasize the algorithmic content, and none of them uses codatatypes. Despite the variety of approaches proposed in textbooks and formalizations, we found them lacking in rigor or readability. Gallier’s pseudo-Pascal code is inspiring, but we prefer “pseudo-Haskell,” i.e., Isabelle/HOL with codatatypes, to combine computational intuition and full mathematical rigor.

In our view, coinduction is the key to formulate an account that is both mathematically rigorous and abundant in algorithmic content. This applies to both of our case studies: classic completeness and infinite-proof soundness. The definition of the abstract prover \({\textsf {mkTree}}\) is stated rigorously, is accessible to functional programmers, and replaces pages of verbose descriptions.

The advantages of machine-checked metatheory are well known from programming language research, where new results are often formalized and proof assistants are used in the classroom. This article reported on some steps we have taken to apply the same methods to formal logic and automated reasoning.

## Footnotes

- 1.
Given formulas \(\psi _1,\ldots ,\psi _k\), we let \({\textsf {Conj}}\;\psi _1\;\ldots \;\psi _k\) denote \({\textsf {Conj}}\;\psi _1\;({\textsf {Conj}}\;\psi _2\;(\ldots \psi _n)\ldots )\). In particular, when \(k = 0\) it denotes the “true” formula \(\top \), defined in a standard way, e.g., as \({\textsf {Imp}}\;a\;a\) for some atom

*a*. - 2.
This is acceptable here, since we employ finitary Horn clauses and the language is countable. Different assumptions may require larger ordinals.

- 3.
The definition of \(P_{p,\_}\) works with the original clauses \(\chi \in {\textsf {ind}}_p\), whereas here we apply it to the “copies” \(\chi '\) of \(\chi \) guaranteed to have their variables fresh for \(\Gamma \) and \(\Delta \), as stipulated in the \(p_{\mathrm{split}}\) rule. This is unproblematic, since it is easy to verify that the definition of \(P_{p,\_}\) is invariant under bijective renaming of variables in the clauses \(\chi \).

- 4.
Goodness is decidable for cyclic trees in logics where rule application is decidable, such as FOL\(_{\textsf {ind}}\) [16].

- 5.
In the proof system from Example 2, \({\textsf {eff}}\) is not deterministic due to the rule AllR. It can be made deterministic by refining the rule with a systematic choice of the fresh variable

*y*. - 6.
And Kripke’s degree of rigor in this early article is not far from today’s state of the art in proof theory; see, e.g., Troelstra and Schwichtenberg [51].

- 7.
This is the only error we found in this otherwise excellent chapter on tableaux.

## Notes

### Acknowledgments

Tobias Nipkow made this work possible. Mark Summerfield and the anonymous reviewers suggested many textual improvements to earlier versions of this article. The reviewers read the submitted paper carefully and made useful and insightful comments and suggestions. Blanchette was partially supported by the Deutsche Forschungsgemeinschaft (DFG) project Hardening the Hammer (Grant NI 491/14-1). Popescu was partially supported by the EPSRC project Verification of Web-based Systems (VOWS, Grant EP/N019547/1) and by the DFG project Security Type Systems and Deduction (Grant NI 491/13-3). Traytel was supported by the DFG program Program and Model Analysis (PUMA, Doctorate Program 1480). The authors are listed alphabetically.

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