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Finding Common Mistakes In Modelling With Mathematical Formalisms Using LLMs

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knowledge-representation reasoning

artificial intelligence, reasoning, knowledge representation

arXiv:2609.17111v1 [cs.CY] 15 Sep 2026

Finding Common Mistakes In Modelling With Mathematical Formalisms Using LLMs Lilian Killich

Marko Schmellenkamp

[email protected] Ruhr University Bochum Bochum, Germany

Fabian Vehlken

Thomas Zeume

[email protected] Ruhr University Bochum Bochum, Germany

Abstract Modelling with mathematical formalisms like logical formulas, mathematical equations, or regular expressions is an important yet challenging task for students of computer science and other STEM disciplines. Identifying common mistakes occurring in this context is an important step towards helping struggling students by providing targeted high-quality feedback, e.g. in interactive learning systems. We present a tool-supported workflow that allows to (1) identify candidates for common mistakes that explain many student mistakes in large educational data sets, (2) cluster candidates according to similarities, and (3) visualize resulting clusters for instructors and CS education researchers. The visualization is designed to help researchers to identify common modelling mistakes. The candidates for common mistakes are represented by bug fixing transformations that translate incorrect formalizations into correct formalizations; they are generated by an LLM and validated algorithmically. We show that this approach works well by reproducing common mistakes in propositional logic modelling that were identified by hand in the literature; showing that, unlike other algorithmic approaches, the LLM-based approach is suitable for very large sets of data; and applying it to multiple other formalisms to showcase it generalizes beyond propositional logic.

CCS Concepts • Theory of computation → Logic; • Social and professional topics → Computer science education.

Keywords modelling, logical formulas, mathematical formalisms, common mistakes, large language model

Introduction and Motivation

Learning to model scenarios with mathematical formalisms such as logical formulas, mathematical equations and terms, regular expressions, and many others, is an important aspect when introducing students to mathematical foundations in computer science and other STEM disciplines. A typical learning objective is that students can model scenarios given in natural language within the intended formalism. In this paper we address the question:

Suppose we are given a large data set of student modelling attempts. How can common modelling mistakes be identified? Identifying common mistakes is a preliminary step for understanding students’ struggles, exploring their invalid conceptions, and ultimately providing advice and feedback in modelling assignments. Knowing common mistakes also helps to craft specific advice and feedback in interactive learning systems. We aim for a workflow for identifying common modelling mistakes that supports CS education researchers and educators with adequate algorithmic tools, so that mistakes can be identified from large scale data sets. Our starting point for an algorithmic approach is the observation that the syntax of many mathematical formalisms is specified by a context-free grammar and therefore each formalization is associated with a syntax tree. For such formalisms, modelling mistakes are reflected in differences between the syntax trees of correct and incorrect formalizations. A key idea for answering the above question is that common mistakes can now be found by identifying “bug fixing transformations” that explain differences between syntax trees [8]. We provide examples of this idea for two different formalisms. Example 1.1 (Modelling with propositional logic formulas). A typical assignment when learning propositional logic is the translation of natural language statements such as “Anna goes to the cinema, but Bob only goes if Celine does.” into formulas. A correct propositional formula is 𝐴 ∧ (𝐵 → 𝐶), while a frequent mistake by students is to model the statement by 𝐴∧(𝐶 → 𝐵). The difference is reflected in the syntax trees: ∧ ∧ , and 𝐴

→

𝐴

→

𝐵 𝐶 𝐶 𝐵 A similar difference occurs for the statement “The user interface is not working properly only if the database or the back end are not working properly.” modelled by ¬𝑈 → (¬𝐷 ∨ ¬𝐵) with frequent mistake (¬𝐷 ∨ ¬𝐵) → ¬𝑈 and the following syntax trees: → → ∨

¬ 𝑈

∨

, and

𝐷

𝐵

𝐷

𝐵

𝑈

The differences in both examples are captured by a bug fixing transformation that captures the mistake and locally modifies the incorrect trees to correct trees: → →

approach to identify transformations from data via a SAT solver was presented, our initial experiments show that this approach does not scale to large educational data sets. Yet, typical such data looks very structured and it is not very hard – though time consuming – for humans to find reasonable bug fixing transformations. This motivates our approach to use large language models (LLMs) to find candidate transformations (see Section 3 for details). We then employ the resulting tool-supported workflow on educational data sets for logical modelling:

⇝ 𝑌1

𝑌2

𝑌1

This bug fixing transformation matches the two examples from above as follows: the pattern on the left hand side can be mapped into a subtree of the syntax tree of the incorrect formalizations, and rearranging the subtree according to the right hand side of the transformation yields the correct formalizations (see [8] for precise semantics). This transformation captures the common mistake that students model “only if” statements with an implication in the wrong direction. It can be used, for instance, to automatically identify this mistake in student attempts and provide feedback [5].

(a) On a data set for propositional logic with 6106 pairs of correct and incorrect formalizations, our workflow discovers 248 clusters of bug fixing transformations that can help to identify common mistakes. These clusters explain 84.44% of the pairs in total, whereas common mistakes identified in previous work [13] explain 71.57% of the pairs. Our approach exclusively explains 15.03% of the pairs, while 2.16% of the pairs are explained exclusively by previous work. When only considering clusters explaining at least 0.5% of the pairs, there are 21 clusters which still explain 66.34% of pairs; with 4.6% explained exclusively. (b) On data sets for modal logic with 12482 pairs of correct and incorrect formalizations, our workflow yields clusters of bug fixing transformations that explain 79.39%. When only considering clusters explaining at least 0.5% of the pairs, there are 38 clusters for modal logic which explain 40.36%. For computation tree logic (CTL) with 7210 pairs, 35.89% of the pairs are explained, and 19.78% of the pairs are explained by 16 clusters which together explain 0.5% of the pairs each.

Example 1.2 (Modelling with regular expressions). For the assignment “Provide a regular expression for the language of words over the alphabet {𝑎, 𝑏, 𝑐} in which 𝑎 appears as first and last letter, but nowhere else.” with correct formalization 𝑎(𝑏 ∗𝑐 ∗ ) ∗𝑎 and frequent mistake 𝑎(𝑏 ∗𝑐 ∗ )𝑎 with syntax trees ◦ ◦ 𝑎 𝑎 ∗ 𝑎 𝑎 ◦ and , ◦ ∗ ∗ ∗ ∗ 𝑐 𝑏 𝑐 𝑏 the difference between the trees can be described by ∗ ◦ ◦ ∗ ⇝ ∗ ∗ ∗ 𝑌2 𝑌1 𝑌2 𝑌1

Related Work

Research on mistakes is conducted for several formalisms. Early work focused on propositional logic [2, 7], with later refinements by Schmellenkamp et al. [13], who distinguish linguistic from logical operators to explain discrepancies in earlier findings. Analyses have also been conducted among others for regular expressions [9] and context-free grammars [11, 14]. A common representation of such mistakes uses bug fixing transformations that describe how incorrect attempts can be systematically corrected. In theoretical domains, this idea has been applied to feedback generation for propositional logic in tutoring systems [5] and extended by [13]. It has since been generalized to other formalisms, including context-free grammars [15], demonstrating applicability beyond tree-structured representations. Related ideas also appear in work on regular expressions [9], where feedback is derived from systematic differences between incorrect and correct solutions. The view that student errors can be explained by systematic rules has a long tradition in intelligent tutoring systems. Brown and Burton [4] introduced diagnostic models of procedural bugs in arithmetic, and Sleeman [19] studied how such rules can be inferred for computer-aided instruction. Our notion of mistake candidates as transformations follows this tradition by representing errors as rule-like explanations for families of incorrect formalizations. Repair-based approaches synthesize transformations from incorrect to correct artifacts, for example in programming education [1, 6, 12, 18, 20]. More recent work explores large language models

Contributions. We present a tool-supported workflow (see Figure 1) that allows to (1) identify candidates for common mistakes by finding bug fixing transformations that explain many student mistakes in large educational data sets, (2) cluster candidates according to similarities, and (3) visualize resulting clusters for instructors and CS education researchers. The visualization is designed to help researchers to identify common modelling mistakes. We note that while candidates identified in Step (1) explain frequent structural differences between syntax trees of correct and incorrect formalizations, it is not guaranteed that these differences correspond to actual conceptual mistakes. Therefore, structuring and visualizing the candidates in Steps (2) and (3) is a necessary follow-up step to support researchers and educators to check the candidates and select sensible common mistakes. Identifying candidates for common mistakes has so far been a time consuming process for CS education researchers, which required to go over data sets by hand. We provide algorithms that support Steps (1) – (3) from above, implement them, and thereby reduce the required effort considerably. The main technical challenge is with respect to Step (1). While in [8] a suitable specification language for tree transformations has been crafted and an algorithmic 2

Algorithm 1 Finding bug fixing transformations

for automated feedback [16, 17], showing their potential while also highlighting the need for validation due to possible unreliability or misalignment with instructional goals. We adopt the repair-based perspective while incorporating external algorithmic verification.

Input: A set D of pairs (𝑡, 𝑡 ∗ ) of syntax trees Output: A set T of bugfixing transformations b := D 1: D 2: T := ∅ 3: while termination criterion is not fulfilled do b into batches of size 𝑠 4: B1, . . . , B𝑏 := randomly split D 5: for all batches B𝑖 do 6: prompt LLM to find transformations 7: add those transformations to T which explain more than a 𝛿-fraction of D 8: end for b := pairs in D b not explained by any transformation in T 9: D 10: end while 11: return T

A Workflow for Finding Common Mistakes

In this section we describe our tool-supported workflow (see Figure 1) that allows to (1) identify candidates for common mistakes by finding bug fixing transformations that explain many student mistakes in large educational data sets, (2) cluster candidates according to similarities, and (3) visualize resulting clusters for instructors and CS education researchers. Step (1) uses an iterative process to generate candidates via an LLM and validates them algorithmically (see Section 3.1). For Step (2), we compute a correlation graph between candidate transformations (see Section 3.2). The workflow is applicable to various data sets. It can be used, for example, on educational data sets for modelling with logical formulas (propositional formulas, modal formulas, temporal formulas, . . . ), regular expressions, mathematical expressions etc. Another use case is the application to multiple data sets for the same formalism to discover more specific mistakes. As an example, a partitioned data set for propositional modelling into smaller data sets according to linguistic operators occurring in natural language statements has been presented by Schmellenkamp et al. [13]. Applying our workflow to each of these smaller data sets results in candidates for common mistakes for each linguistic operator.

3.1

for special cases, and a description of the task to find transformations. The prompt that was used can be found in the (anonymously published) supplementary material [3]. Via this prompt, the LLM provides a list of transformations for each one of the batches. The number of desired transformations for each batch is configurable (in our experiments, we use ⌊0.2·𝑠⌋). For each transformation found by the LLM, the algorithm tests which fraction of pairs (𝑡, 𝑡 ∗ ) it can explains, i.e. pairs for which 𝑡 can be transformed into 𝑡 ∗ by applying the transformation. If a transformation explains more than a 𝛿-fraction of the data, it is added to the output T .

Finding Bug Fixing Transformations

3.2

The goal of the first step of our workflow is, given a data set containing pairs of incorrect and correct formalizations, to find transformations that explain (frequent) structural differences between the correct and incorrect formalizations. More formally, the input is a data set D which contains pairs (𝑡, 𝑡 ∗ ) of syntax trees 𝑡 and 𝑡 ∗ representing incorrect and correct formalizations. The goal is to identifiy a small set T of bug fixing transformations that explain the difference between as many pairs in D as possible. Here, roughly speaking, a bug fixing transformation is of the form 𝜎 ⇝ 𝜎 ∗ for two tree patterns 𝜎 and 𝜎 ∗ . Such a transformation explains a pair (𝑡, 𝑡 ∗ ) if the pattern 𝜎 can be found in 𝑡 and rearranging 𝑡 according to the shape of 𝜎 ∗ yields the tree 𝑡 ∗ (see Examples 1.1 and 1.2 as well as [8] for details). Unfortunately, finding a small set of bug fixing transformations that explains a large data set is NP-hard [8] and SAT-solving based approaches do not scale well. Our algorithmic approach (see Algorithm 1 for pseudo code) is to generate candidates for bug fixing transformations with an LLM, validate those candidates algorithmically, and collect transformations that explain a 𝛿-fraction of the initial data set in the output set T . This procedure is iterated until no new transformation has been found for 𝑘 iterations. In our experiments we used 𝛿 = 0.01 and 𝑘 = 10, see Section 4 for details. We provide some details. In each iteration, the data set at hand is split into equally sized randomized batches of size 𝑠 (in our experiments, we use 𝑠 ∼ 50), to accommodate limits of used LLMs. Each of the batches is put into a prompt that also contains a brief description of the transformation language, a few sample specifications

Clustering Bug Fixing Transformations

The goal of the second step of the workflow is to structure the list of bug fixing transformations resulting from the first step for CS education researchers and educators. The idea is to cluster transformations according to “similarity” and to sort those clusters by the number of explained data points. The idea is that transformations in the list may explain similar kinds of mistakes or one transformation may explain all mistakes also explained by another transformation. As an example, consider the transformations ¬ ¬ ↔ ⇝ ↔ 𝜌1: and 𝜌 2 : 𝑌1 ⇝ . 𝑌1 𝑌2 𝑌1 𝑌2 𝑌1 Both explain mistakes where a negation ¬ has been forgotten, but the first one is more specific: it only explains pairs in which the negation is missing in front of a bi-implication ↔. The first may explain mistakes that are specific when modelling natural language statements that involve negations and equivalences, while the second can explain natural language statements with negations more broadly. Our algorithmic approach for finding similar transformations is to compute, for each pair 𝜌 1 and 𝜌 2 of transformations, the fraction of data points explained by 𝜌 1 that are also explained by 𝜌 2 . More precisely, denote by 𝐸 𝜌 (D) the set of pairs in D that are explained by 𝜌. We compute a correlation graph GD for the transformations found for data set D that has a node for each transformation found in Step (1). For each pair 𝜌 1 and 𝜌 2 of transformations, the graph has 3

Finding Bug Fixing Transformations

Clustering

randomized batches data set

(𝐹 → 𝑆, 𝑆 → 𝐹 ), (𝐴 ∧ (𝐶 → 𝐵), 𝐴 ∧ (𝐵 → 𝐶 ) ), ...

(𝐴 ∧ (𝐶 → 𝐵), 𝐴 ∧ (𝐵 → 𝐶)), (¬𝑈 → (𝑆 ∨ 𝑇 ), (𝑆 ∨ 𝑇 ) → ¬𝑈 ),

(¬𝑈 → (𝑆 ∨ 𝑇 ), (𝑆 ∨ 𝑇 ) → ¬𝑈 ), ...

LLM Prompt

.. .

(𝐴 ↔ 𝐶, 𝐴 → 𝐶 ) ...

LLM Prompt

.. . (𝐴 ↔ 𝐶, 𝐴 → 𝐶)

data set

LLM Prompt

→

→ 𝑌2

𝑌1

(𝐹 → 𝑆, 𝑆 → 𝐹 ),

⇝

↔

termination criterion satisfied?

⇝

→ 𝑌2

𝑌1

⇝

, 𝑌1

𝑌2

⇝

, 𝑌1

⇝

𝑌2

.. . apply to input data set

output: set of collected transformations

⇝ ⇝

remaining data set with unexplained pairs

repeat

Visualization Hierarchical clusters table:

set of transformations

⇝

yes ⇝

Figure 1: Illustration of the workflow for finding bug fixing transformations, clustering them, and visualizing the clusters. |𝐸 𝜌 1 (D) ∩ 𝐸 𝜌 2 (D)| . Thus, the weight of |𝐸 𝜌 2 (D)| (𝜌 1, 𝜌 2 ) is the fraction of the data points explained by 𝜌 2 which are also explained by 𝜌 1 . In particular, a value of 1 means 𝜌 1 explains all pairs of D explained by 𝜌 2 (and possibly more) and a value of 0 means that it explains none. A cluster within a correlation graph is a set of nodes that are connected by a path consisting of edges of non-zero weight. Consider the following illustrative toy example of a correlation graph for a small hand-chosen data set.

transformation. Self-loops, edges with weight 0, and “transitive edges” (𝜌 1, 𝜌 4 ), (𝜌 4, 𝜌 1 ), (𝜌 1, 𝜌 5 ), (𝜌 5, 𝜌 1 ) are omitted for clarity.

an edge (𝜌 1, 𝜌 2 ) of weight

𝜌4

1 2 3

𝜌2

𝜌5

1 3

1 2

1 𝜌1

𝜌3

1 6

As the correlation graph indicates, 𝜌 1, . . . , 𝜌 5 form a cluster.

□

Example 3.1. Consider the following five transformations ∧ →

→

𝜌1:

, 𝜌2:

⇝ 𝑌2

𝑌1

→

⇝ 𝑌2

𝑌3

𝑌2

𝜌5:

, 𝜌4: 𝑌1

∧ →

∨ 𝑌1

𝑌2

𝑌3

𝑌1

→

∨

⇝ 𝑌4

𝑌2

𝑌4

𝑌3

∧ →

𝑌2

𝑌3

∧

∧ 𝑌1

→

⇝ 𝑌1

𝑌1

𝑌2

𝑌2 𝜌3:

We use correlation graphs to provide visualizations of how bug fixing transformations found in Step (1) relate to each other. In our experiments, all correlation graphs can be partitioned into clusters of a very simple shape: (a) single-transformation clusters consisting of a single transformation that only explains data points that no other transformation explains; (b) clusters with equivalent transformations consisting of transformations that all explain the same data points; and (c) hierarchical clusters whose transformations can be arranged in a tree, such that if a transformation explains data points D ′ ⊆ D then each of its children explains only data points D ′′ ⊆ D ′ . In particular, clusters of type (b) and (c) are helpful for CS education researchers to identify transformations representative for common mistakes and to distinguish “high-level” from “low-level” mistakes.

∧ →

𝑌1

→

𝑌1 ¬

⇝

𝑌3

𝑌2

𝑌3

and data set S consisting of the following (incorrect, correct)-pairs: 𝑡 1 = (𝐴 → 𝐵, 𝐵 → 𝐴), 𝑡 2 = (¬𝐴 → 𝐵, 𝐵 → ¬𝐴), 𝑡 3 = ((𝐴 ∨ ¬𝐷) ∧ (𝐵 → 𝐶), (𝐴 ∨ ¬𝐷) ∧ (𝐶 → 𝐵)),

𝑡 4 = (¬((𝐴 ∨ 𝐵) → (𝐶 → 𝐷)), ¬((𝐶 → 𝐷) → (𝐴 ∨ 𝐵))),

Evaluation

In this section we employ our tool-supported workflow exemplarily on educational data sets for logical modelling. In all experiments we use the workflow to find transformations, evaluate how well they cover data, and present a selection of transformations that explain large fractions of the data. Detailed tables on all experiments can be found in the supplementary material [3]. In a first set of experiments we use our workflow on a data set for modelling in propositional logic that has been used by Schmellenkamp et al. [13] to identify common mistakes by hand. Here we apply our workflow to a partition of the data set according to linguistic operators used in natural language statements as well as to the full data set. The objective is to see whether our workflow works as well as the by-hand-approach used in previous work. Then, in a second set of experiments, we apply our workflow to data sets for modelling with modal logic and computation tree

𝑡 5 = ((𝐴 ∨ 𝐵) ∧ (𝐶 → 𝐷), (𝐴 ∨ 𝐵) ∧ (𝐷 → 𝐶)), 𝑡 6 = (𝐴 ∧ (𝐵 → ¬𝐶), 𝐴 ∧ (¬𝐶 → 𝐵)) We first compute the sets 𝐸 𝜌𝑖 (S) of pairs explained by each transformation 𝜌𝑖 : 𝐸 𝜌 1 (S) = {𝑡 1, 𝑡 2, 𝑡 3, 𝑡 4, 𝑡 5, 𝑡 6 }, 𝐸 𝜌 2 (S) = {𝑡 3, 𝑡 5, 𝑡 6 }, 𝐸 𝜌 3 (S) = {𝑡 4 }, 𝐸 𝜌 4 (S) = {𝑡 3, 𝑡 5 }, 𝐸 𝜌 5 (S) = {𝑡 6 }. The resulting correlation graph is depicted below. The edges are labelled with their weight, i.e. the fraction of pairs explained by the target transformation that are also explained by the source 4

The transformation 𝜌 xor-nand represents the common mistake of using an exclusive or, expressed using a disjunction of conjunctions, instead of a negated conjunction which explains 114 pairs (6.57% of the pairs unexplained by the hand-picked mistakes). While a similar hand-picked mistake was found in [13], theirs only considers the case in which the exclusive or is expressed using a bi-implication, and thus does not explain the pairs explained by 𝜌 xor-nand . The transformation 𝜌 at-least-2 , which represents the common mistake of only using a disjunction to express that at least two literals are true, explains 28 pairs (1.61% of the pairs unexplained by the hand-picked mistakes).

logic (CTL). Here the objective is to see, exemplarily, whether the workflow scales to other formalisms. We first describe the experimental setup and then the results for the experiments sketched above. For finding bug fixing transformations in Step (1) of our workflow, we use the GPT-OSS-120B model [10]. Initial experiments showed that this model is powerful enough, but small enough to be locally deployable. As parameters we choose a batch size of 50, due to the token limitations of the model. In all experiments, the prompt asks for transformations that explain at least 20% of the pairs in a batch. For the propositional data set grouped by linguistic operators, found transformations additionally need to explain 𝛿 = 1% of all data points. The iterative process is terminated if no new transformation is chosen for 10 iterations.

4.1

4.2

Results for propositional logic

The propositional data set from [13] contains 6106 pairs with 1572 of them being distinct. This data set comes with a partition into 34 subdata sets for different linguistic operators. The median and average sizes of the subdata sets are 65.5 and 177.8, respectively. Results for linguistic operator data sets. Across the subdata sets, the found transformations explain at least as many pairs as the hand-picked mistakes in a substantial fraction of cases. For > 64% of the lingustic operator data sets, our workflow finds bug fixing transformations transformations that explain more data points than the mistakes identified in [13]. For > 14% of the subdata sets, the transformations explain exactly as many pairs as the hand-picked mistakes.

𝜌 1ml :

𝑌1

𝑌2

, and

𝑌1

𝑌2

∨ ∨

𝜌 at-least-2 :

⇝ 𝑌1

𝑌2

∧

𝑌3 𝑌1

𝑌2

𝑌1

𝑌3

𝑌2

□ 𝑌1

, and 𝜌 3ml :

∨

↔

⇝ 𝑌2

𝑌1

𝑌2

AG 𝑌1

⇝ 𝑌1

, and 𝜌 2ctl :

⇝ 𝑌1

𝑌1

Determining Usefulness of Transformations. To exemplarily determine the usefulness of clusters of transformations found by our workflow, a CS educator inspected all clusters explaining at least 0.5% of the pairs in each of the three data sets for propositional logic, modal logic, and CTL (see also Table 2). The focus was on whether the “main transformation”, i.e. the transformation forming the root of the cluster tree, is sufficient to provide targeted feedback, or if sub-transformations, those lower in the cluster’s hierarchy, would be required for this. In propositional logic, 19 clusters had main transformations suitable for feedback, though at least 4 required sub-transformations for more targeted feedback. In CTL, all 17 clusters had useful main transformations, with at least 6 requiring sub-transformations for sufficiently specific feedback. In modal logic, the main transformation of 18 clusters was suitable for feedback, but at least 7 required sub-transformations. In 16 clusters, the main transformations were too general to identify specific mistakes directly; for 5 of these, frequent sub-transformations revealed mistakes.

𝑌2

𝑌1

, 𝜌 2ml : 𝑌1 ⇝

explaining 2.98% and 2.48%, respectively.

¬ ∧

𝑌1

𝜌 1ctl :

∧

⇝

which explain 2.1%, 1.43%, and 3.37% of the data. While 𝜌 1ml and 𝜌 2ml change or add the modal operators ^ and □, 𝜌 3modal was also found for the propositional logic data set. It represents the common mistake of modelling an “exclusive or” as an “or”. For CTL, the transformations found by our workflow explain 35.89% of the pairs with the two transformations that explain the largest fractions of the data set being

∨ ∧

□

𝑌1

Results for the full propositional data set. We conducted two different experiments. First, we compared hand-picked mistakes to transformations found by our workflow as follows: For each handpicked mistake, we identified all transformations that explain at least one pair that is explained by the hand-picked mistake and clustered them as described in Section 3.2. As an example, Table 1 shows a hand-picked mistake that a disjunction was erroneously used instead of a conjunction, and the transformations corresponding to it. This particular hand-picked mistake occurred in 334 pairs (column #), which is 5.47% of the whole data set. Of these, 330 (98.80% of 334) were explained by bug fixing transformations. Across the whole data set, the bug fixing transformations found by our workflow explain 5156 of the pairs (84.44%), whereas the hand-picked mistakes only explain 4370 of the pairs (71.57%). 918 pairs (15.03%) are exclusively explained by the transformations, i.e. no hand-picked mistake explains any of these pairs; 132 pairs (2.16%) exclusively by the hand-picked mistakes. For instance, the following bug fixing transformations exclusively explain pairs that have not been explained by common mistakes in previous work

𝜌 xor-nand :

Results for Modal Logic and CTL

The data set for modal logic contains 12482 pairs with 3951 of them being distinct and is also from [13]. The data set for CTL contains 7210 pairs with 2695 distinct pairs and was collected by us using an interactive learning tool in two iterations of an introductory logic course (summer 2025 and 2026). The results for the modal logic and CTL data sets are summarized in Table 2. For modal logic, the transformations found by our workflow explain 79.39% of the pairs in the data set, which is comparable to the propositional logic case. Examples of transformations are

𝑌3 5

Table 1: Exemplary excerpt of the table of hierarchical bug-fixing transformation clusters that cover most of the pairs explained by the common mistake that a disjunction was erroneously used instead of a conjunction (as identified in [13]). For each transformation, a simple representative example is provided. For the mistake as a whole, each transformation, and the entire cluster, the number # and distinct number #dst of pairs are shown. The common mistake occurrs in 5.47% of all pairs in the full data set (3.88% distinct). Of these, the complete bug fixing transformation cluster found by our workflow explains 330 (98.8%). The fraction of the 334 pairs each transformation explains is also shown (columns % and %dst, respectively). A table with all bug fixing transformation clusters found by our workflow is presented in the (anonymously published) supplementary material [3]. Hand-picked mistake / transformations Mistake: disjunction instead of conjunction Transformations combined ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 ⌈ = (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2 (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3

#dst

%dst

334 330 201 153 150 128 118 35 35 17

5.47% 98.80% 60.18% 45.81% 44.91% 38.32% 35.33% 10.48% 10.48% 5.09%

61 58 23 16 15 34 29 9 9 6

3.88% 95.08% 37.70% 26.23% 24.59% 55.74% 47.54% 14.75% 14.75% 9.84%

Table 2: Overview of the data sets, number of clusters found by our workflow, and the percentage of pairs explained by clusters for the propositional logic (PL), modal logic (PL) and computation tree logic (CTL) data sets. For each data set, we provide the number of (incorrect, correct) formula pairs, modelling contexts (i.e. different assignments), and statements to be modelled.

# total formula pairs # modelling contexts # statements to be modelled # total clusters % total explained # clusters explaining ≥ 0.1% % explained by clusters explaining ≥ 0.1% # clusters explaining ≥ 0.5% % explained by clusters explaining ≥ 0.5% # clusters explaining ≥ 1% % explained by clusters explaining ≥ 1%

CTL

6106 11 58 248 84.44% 78 76.06% 21 66.34% 15 62.48%

12482 6 28 1585 79.39% 132 56.00% 38 40.36% 17 28.26%

7210 6 20 271 35.89% 62 29.76% 16 19.78% 8 13.86%

example of (incorrect, correct) formula pairs

¬(𝐴 ∧ 𝐵 ∧ 𝐶), ¬(𝐴 ∧ 𝐵), ¬(𝐴 ∧ 𝐵) → 𝐶, 𝐴 ∨ 𝐵 ∨ 𝐶, 𝐴 ∨ 𝐵, (¬𝐴 ∨ ¬𝐵) → 𝐶, (¬𝐴 ∨ ¬𝐵), (𝐴 ∨ 𝐵) → ¬𝐶,

¬𝐴 ∧ ¬𝐵 ∧ ¬𝐶 ¬𝐴 ∧ ¬𝐵 (¬𝐴 ∧ ¬𝐵) → 𝐶 𝐴 ∧ 𝐵 ∧𝐶 𝐴∧𝐵 (¬𝐴 ∧ ¬𝐵) → 𝐶 (¬𝐴 ∧ ¬𝐵) (𝐴 ∧ 𝐵) → ¬𝐶

applying the intended equivalence transformation to the previous correct formula, yielding the correct formula for that step. Overall, the results suggest that transformation-based mistake candidates can help scale the discovery of common mistakes in education settings. Future work includes applying the workflow to diverse data sets from different domains, and studying how the resulting candidates can be integrated into feedback systems of interactive learning tools.

Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 448468041.

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Discussion and Conclusion

We presented a tool-supported workflow for identifying candidates for common mistakes in modelling with mathematical formalisms as bug fixing transformations, clustering them, and presenting them to educators and CS education researchers. Our experiments show that the workflow recovers many known propositional logic mistakes, finds additional ones, and can also be applied to other formalisms, such as modal logic and CTL. Beyond modelling, the workflow is likely also applicable to tasks in which students transform logical formulas into normal forms. A difference in that setting is that one does not immediately have (incorrect, correct) pairs. Instead, one has an incorrect formula, a previous correct formula, and the equivalence transformation that was applied incorrectly. A suitable pair can then be constructed by 6

(LIPIcs, Vol. 328), Sudeepa Roy and Ahmet Kara (Eds.). Schloss Dagstuhl - LeibnizZentrum für Informatik, 24:1–24:20. doi:10.4230/LIPICS.ICDT.2025.24 [9] Olaperi Okuboyejo, Sigrid Ewert, and Ian Sanders. 2025. Automatic Feedback Generation for the Learning of Regular Expressions. ACM Trans. Comput. Educ. 25, 3, Article 36 (Aug. 2025), 26 pages. doi:10.1145/3743682 [10] OpenAI. 2025. gpt-oss-120b & gpt-oss-20b Model Card. arXiv:2508.10925 [cs.CL] https://arxiv.org/abs/2508.10925 [11] Nelishia Pillay. 2010. Learning Difficulties Experienced by Students in a Course on Formal Languages and Automata Theory. 41, 4 (2010), 48–52. doi:10.1145/ 1709424.1709444 [12] Kelly Rivers and Kenneth R Koedinger. 2017. Data-driven hint generation in vast solution spaces: a self-improving python programming tutor. International Journal of Artificial Intelligence in Education 27, 1 (2017), 37–64. [13] Marko Schmellenkamp, Alexandra Latys, and Thomas Zeume. 2023. Discovering and Quantifying Misconceptions in Formal Methods Using Intelligent Tutoring Systems. In Proceedings of the 54th ACM Technical Symposium on Computer Science Education, Volume 1, SIGCSE 2023, Toronto, ON, Canada, March 15-18, 2023, Maureen Doyle, Ben Stephenson, Brian Dorn, Leen-Kiat Soh, and Lina Battestilli (Eds.). ACM, 465–471. doi:10.1145/3545945.3569806 [14] Marko Schmellenkamp, Dennis Stanglmair, Tilman Michaeli, and Thomas Zeume. 2024. Exploring Error Types in Formal Languages Among Students of Upper Secondary Education. In Proceedings of the 24th Koli Calling International Conference on Computing Education Research, Koli Calling 2024, KoliFinland, November 12-17, 2024, Juho Leinonen and Andreas Mühling (Eds.). ACM, 20:1–20:8. doi:10.1145/3699538.3699540 [15] Marko Schmellenkamp, Thomas Zeume, Sven Argo, Sandra Kiefer, Cedric Siems, and Fynn Stebel. 2025. Detecting and Explaining (In-)equivalence of ContextFree Grammars. Proc. ACM Program. Lang. 9, OOPSLA2, Article 378 (Oct. 2025), 27 pages. doi:10.1145/3763156 [16] Niklas Scholz, Manh Hung Nguyen, Adish Singla, and Tomohiro Nagashima. 2026. Partnering with AI: A Pedagogical Feedback System for LLM Integration Into Programming Education. In Two Decades of TEL. From Lessons Learnt to Challenges Ahead, Kairit Tammets, Sergey Sosnovsky, Rafael Ferreira Mello, Gerti Pishtari, and Tanya Nazaretsky (Eds.). Springer Nature Switzerland, Cham, 243–248. [17] Priscylla Silva and Evandro Costa. 2025. Assessing Large Language Models for Automated Feedback Generation in Learning Programming Problem Solving. arXiv:2503.14630 [cs.SE] https://arxiv.org/abs/2503.14630 [18] Rishabh Singh, Sumit Gulwani, and Armando Solar-Lezama. 2013. Automated feedback generation for introductory programming assignments. SIGPLAN Not. 48, 6 (June 2013), 15–26. doi:10.1145/2499370.2462195 [19] Derek H. Sleeman. 1983. 16 - INFERRING STUDENT MODELS FOR INTELLIGENT COMPUTER-AIDED INSTRUCTION. In Machine Learning, Ryszard S. Michalski, Jaime G. Carbonell, and Tom M. Mitchell (Eds.). Morgan Kaufmann, San Francisco (CA), 483–510. doi:10.1016/B978-0-08-051054-5.50020-0 [20] Songwen Xu and Yam San Chee. 2003. Transformation-based diagnosis of student programs for programming tutoring systems. IEEE Transactions on Software Engineering 29, 4 (2003), 360–384. doi:10.1109/TSE.2003.1191799

Prompts for Finding Bug Fixing Transformations

The prompts for all propositional logic, modal logic, and CTL share large parts; they mainly differ in the symbols the transformations are allowed to use. The following prompt represents all three prompts, with the only differing sentence being prefixed by the logic the prompt is used for.

Start of prompt: Instruction. Find structural differences between pairs of propositional logic formulas and express them by transformations of a transformation language. A transformation of our transformation language consists of two tree patterns: • body: the pattern that matches a subtree in the input tree (left side of the transformation). • head: the pattern that replaces the matched subtree (right side of the transformation). • Transformations are written in the format: body —> head Patterns are trees that follow the following rules: • propositional: Inner nodes are labeled by the operators of propositional logic: “or”, “and”, “impl”, “not”, “eq”. modal: Inner nodes are labeled by the operators of modal logic: “or”, “and”, “impl”, “not”, “eq”, “box” (for □), “dia” (for ^). CTL: Inner nodes are labeled by the operators of computation tree logic: “or”, “and”, “impl”, “not”, “eq”, “ax”, “ex”, “af”, “ef”, “ag”, “eg”, “au”, “eu”. • They are written as: operator[child1 child2 ...] with children separated by spaces. • Leaf nodes are labeled by tree variables, which are placeholders for subtrees. Write them as Y1, Y2, ... • Variadic patterns: “and” and “or” may have arbitrary arity. Use Y1 .. Yn to denote an ordered sequence of two or more child subtrees. Example: and[Y1 .. Yn] matches and[A B], and[A B C], and[A B C D], etc.; or[Y1 .. Yn] matches or[A B], or[A B C], or[A B C D], etc. • The sequence Y1 .. Yn must be preserved unchanged: do not reorder, duplicate, or split it. Do not remove or add single elements. • Use Y1 .. Yn only as the complete child list of an and/or node. Example: and[Y1 .. Yn] • Every tree variable that appears in the head must also appear in the body. • Propositional variables of the formulas may not appear in the transformations. Use only tree variables and the operators as node labels. Matching: • A match for the body may occur anywhere in the tree, not only at the root. • Matching is injective: every body-node is mapped to a unique node in the input tree. • If tree variables appear multiple times in the same pattern, the occurrences have to be matched by isomorphic subtrees. Application: When a match is found, the corresponding subtree is replaced by the head, with variables substituted by the subtrees/labels found in the match. Your Task: Find <n_transformations> transformation(s) that can transform as many of the formula pairs as possible in one step. Return the answer exclusively in the following JSON format. Write nothing outside this JSON. Pay attention to correct quotation marks, no comments, no headings. Replace TRANSFORMATION by the respective transformation. { "transformations": [ { "name": "Transformation_1", "transformation": "TRANSFORMATION" } ] } Here is the dataset of formula pairs: Input: <error_formula_1> Target: <correct_formula_1> Input: <error_formula_2> Target: <correct_formula_2> .. .

B Detailed Evaluation Data B.1 Propositional Logic Data Set Grouped by Linguistic Operator The hand-picked mistakes are from [13]. Table 3: Hand-picked errors and transformations for linguistic operator(s) AND_AND_NONE_OF Hand-picked mistake / Transformations

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Unexplained by ALL hand errors Transformations combined T1: 𝑌1 ∧ 𝑌2 ∧ ¬ (𝑌3 ∧ 𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T2: (𝑌1 ∧ 𝑌2 ) ↔ ¬ (𝑌3 ∧ 𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T3: (𝑌1 ∧ 𝑌2 ) → (¬𝑌3 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T4: (𝑌1 ∧ 𝑌2 ) ↔ (¬𝑌3 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T5: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T6: (¬𝑌1 ∧ ¬𝑌2 ) → (𝑌3 ∧ 𝑌4 ) ⇝ 𝑌3 ∧ 𝑌4 ∧ ¬𝑌1 ∧ ¬𝑌2 T7: (𝑌1 ∧ 𝑌2 ) ∨ (¬𝑌3 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T8: 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 ⇝ 𝑌3 ∧ 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌4 T9: 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 ⇝ 𝑌1 ∧ 𝑌3 ∧ ¬𝑌4 ∧ ¬𝑌2 T10: 𝑌1 ∧ 𝑌2 ∧ ¬ (𝑌3 ∧ 𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4

48 13 4 1 1 1 1 1 1 1 1 1

100.00% 27.08% 8.33% 2.08% 2.08% 2.08% 2.08% 2.08% 2.08% 2.08% 2.08% 2.08%

26 12 3 1 1 1 1 1 1 1 1 1

100.00% 46.15% 11.54% 3.85% 3.85% 3.85% 3.85% 3.85% 3.85% 3.85% 3.85% 3.85%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

48 0 13 0 13

100.00% 0.00% 27.08% 0.00% 27.08%

26 0 12 0 12

100.00% 0.00% 46.15% 0.00% 46.15%

Table 4: Hand-picked errors and transformations for linguistic operator(s) AND_AND Hand-picked mistake / Transformations

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BUGGY_ADD_OPERAND_TO_CONJUNCTION Transformations combined

19 0

65.52% 0.00%

4 0

30.77% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T1: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 T4: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 ∧ 𝑌1 ∧ 𝑌2

2 2 1 1

6.90% 100.00% 50.00% 50.00%

2 2 1 1

15.38% 100.00% 50.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T5: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ 𝑌3

2 2 2

6.90% 100.00% 100.00%

1 1 1

7.69% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T6: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ 𝑌3

1 1 1

3.45% 100.00% 100.00%

1 1 1

7.69% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T3: ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛

1 1 1

3.45% 100.00% 100.00%

1 1 1

7.69% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T2: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛

1 1 1

3.45% 100.00% 100.00%

1 1 1

7.69% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

3 0

10.34% 0.00%

3 0

23.08% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

29 26 7 19 0

100.00% 89.66% 24.14% 65.52% 0.00%

13 10 6 4 0

100.00% 76.92% 46.15% 30.77% 0.00%

Table 5: Hand-picked errors and transformations for linguistic operator(s) AND_IF Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T2: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3

12 12 12

8.89% 100.00% 100.00%

1 1 1

1.69% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T4: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1

10 8 8

7.41% 80.00% 80.00%

3 1 1

5.08% 33.33% 33.33%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T3: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3

9 9 9

6.67% 100.00% 100.00%

1 1 1

1.69% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined ⌈ = T5: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⌊ = T6: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛

7 7 7 7

5.19% 100.00% 100.00% 100.00%

1 1 1 1

1.69% 100.00% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

BUGGY_REPLACE_XOR_BY_CONJUNCTION Transformations combined

1 0

0.74% 0.00%

1 0

1.69% 0.00%

Unexplained by ALL hand errors Transformations combined T1: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T7: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T8: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T9: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T10: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T11: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 T13: 𝑌1 → ¬ (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T14: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 → 𝑌2 ) ∧ 𝑌1 T12: ¬𝑌1 → (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T15: 𝑌1 → ¬ (¬𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1

90 47 22 3 3 3 3 3 3 3 2 2

66.67% 52.22% 24.44% 3.33% 3.33% 3.33% 3.33% 3.33% 3.33% 3.33% 2.22% 2.22%

46 11 2 1 1 1 1 1 1 1 1 1

77.97% 23.91% 4.35% 2.17% 2.17% 2.17% 2.17% 2.17% 2.17% 2.17% 2.17% 2.17%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

135 45 83 9 47

100.00% 33.33% 61.48% 6.67% 34.81%

59 13 15 9 11

100.00% 22.03% 25.42% 15.25% 18.64%

Table 6: Hand-picked errors and transformations for linguistic operator(s) AND_NOT_NOT Hand-picked mistake / Transformations

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BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T1: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

1 1 1

100.00% 100.00% 100.00%

1 1 1

100.00% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

0 0

0.00% 0.00%

0 0

0.00% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

1 1 1 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

1 1 1 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

Table 7: Hand-picked errors and transformations for linguistic operator(s) AND_NOT Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T1: 𝑌1 ∨ ¬𝑌2 ⇝ 𝑌1 ∧ ¬𝑌2

4 4 4

80.00% 100.00% 100.00%

1 1 1

50.00% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T2: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

1 1 1

20.00% 100.00% 100.00%

1 1 1

50.00% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

0 0

0.00% 0.00%

0 0

0.00% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

5 5 5 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

2 2 2 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

Table 8: Hand-picked errors and transformations for linguistic operator(s) AND Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined ⌈ = T1: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⌊ = T2: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

50 50 50 50

78.12% 100.00% 100.00% 100.00%

6 6 6 6

40.00% 100.00% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T3: 𝑌1 → 𝑌2 ⇝ 𝑌2 ∧ 𝑌1 T5: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

7 7 6 1

10.94% 100.00% 85.71% 14.29%

3 3 2 1

20.00% 100.00% 66.67% 33.33%

BUGGY_ADD_OPERAND_TO_CONJUNCTION Transformations combined

2 0

3.12% 0.00%

2 0

13.33% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T4: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ∧ 𝑌1

2 2 2

3.12% 100.00% 100.00%

1 1 1

6.67% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined T6: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 ∧ 𝑌1

1 1 1

1.56% 100.00% 100.00%

1 1 1

6.67% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

2 0

3.12% 0.00%

2 0

13.33% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

64 62 60 2 0

100.00% 96.88% 93.75% 3.12% 0.00%

15 13 11 2 0

100.00% 86.67% 73.33% 13.33% 0.00%

Table 9: Hand-picked errors and transformations for linguistic operator(s) AT_LEAST_NEUTRAL Hand-picked mistake / Transformations

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BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined T1: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ 𝑌4 ) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4

2 2 2

40.00% 100.00% 100.00%

1 1 1

25.00% 100.00% 100.00%

BUGGY_ADD_OPERAND_TO_DISJUNCTION Transformations combined

1 0

20.00% 0.00%

1 0

25.00% 0.00%

Unexplained by ALL hand errors Transformations combined (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 ) (𝑌2 ∧ ¬𝑌1 ∧ ¬𝑌3 ∧ ¬𝑌4 ) T2: ∨ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌4 ) ∨ (𝑌4 ∧ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4 (𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ∨ 𝑌4 )) (𝑌2 ↔ ¬ (𝑌1 ∨ 𝑌3 ∨ 𝑌4 )) T3: ∨ (𝑌3 ↔ ¬ (𝑌1 ∨ 𝑌2 ∨ 𝑌4 )) ∨ (𝑌4 ↔ ¬ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 )) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4

∨

2 2 1

40.00% 100.00% 50.00%

2 2 1

50.00% 100.00% 50.00%

∨

50.00%

5 3 4 1 2

100.00% 60.00% 80.00% 20.00% 40.00%

4 2 3 1 2

100.00% 50.00% 75.00% 25.00% 50.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

Table 10: Hand-picked errors and transformations for linguistic operator(s) AT_LEAST Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T10: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ 𝑌2 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

3 3 3

1.46% 100.00% 100.00%

2 2 2

1.61% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined

2 0

0.97% 0.00%

2 0

1.61% 0.00%

BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined

2 0

0.97% 0.00%

2 0

1.61% 0.00%

Unexplained by ALL hand errors Transformations combined T1: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ∧ ¬𝑌1 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T2: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T3: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T5: (𝑌1 ∨ 𝑌2 ∨ ¬𝑌3 ) ∧ (𝑌1 ∨ 𝑌3 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ 𝑌3 ∨ ¬𝑌1 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T4: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ∨ (𝑌1 ∧ 𝑌3 ) ∨ 𝑌2 ∨ (𝑌2 ∧ 𝑌3 ) ∨ 𝑌1 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T6: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T7: ¬ (¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T8: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T9: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

199 101 31 28 10 7 7 6 5 4 3

96.60% 50.75% 15.58% 14.07% 5.03% 3.52% 3.52% 3.02% 2.51% 2.01% 1.51%

118 30 9 6 1 4 3 2 2 1 2

95.16% 25.42% 7.63% 5.08% 0.85% 3.39% 2.54% 1.69% 1.69% 0.85% 1.69%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

206 7 104 4 101

100.00% 3.40% 50.49% 1.94% 49.03%

124 6 32 4 30

100.00% 4.84% 25.81% 3.23% 24.19%

Table 11: Hand-picked errors and transformations for linguistic operator(s) EITHER_OR Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_XOR Transformations combined T1: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

91 91 91

50.00% 100.00% 100.00%

2 2 2

6.25% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_XOR Transformations combined T2: 𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

21 21 21

11.54% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_XOR Transformations combined T3: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T15: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

12 11 9 2

6.59% 91.67% 75.00% 16.67%

3 2 1 1

9.38% 66.67% 33.33% 33.33%

BUGGY_REPLACE_NOR_BY_XOR Transformations combined T4: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T9: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

10 10 6 4

5.49% 100.00% 60.00% 40.00%

2 2 1 1

6.25% 100.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_XOR Transformations combined T8: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) T10: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

6 6 3 3

3.30% 100.00% 50.00% 50.00%

2 2 1 1

6.25% 100.00% 50.00% 50.00%

BUGGY_REPLACE_CONJUNCTION_BY_XOR Transformations combined T7: 𝑌1 ∧ 𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 )

5 5 5

2.75% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined T11: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

4 3 3

2.20% 75.00% 75.00%

2 1 1

6.25% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_EQUIVALENCE Transformations combined T12: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

3 2 2

1.65% 66.67% 66.67%

2 1 1

6.25% 50.00% 50.00%

BUGGY_REPLACE_NAND_BY_EQUIVALENCE Transformations combined

1 0

0.55% 0.00%

1 0

3.12% 0.00%

BUGGY_REPLACE_NOR_BY_EQUIVALENCE Transformations combined

1 0

0.55% 0.00%

1 0

3.12% 0.00%

Unexplained by ALL hand errors Transformations combined T5: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T6: 𝑌1 → ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T14: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T13: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T8: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 )

28 19 6 5 4 2 2

15.38% 67.86% 21.43% 17.86% 14.29% 7.14% 7.14%

15 6 1 1 1 2 1

46.88% 40.00% 6.67% 6.67% 6.67% 13.33% 6.67%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

182 154 168 5 19

100.00% 84.62% 92.31% 2.75% 10.44%

32 17 18 5 6

100.00% 53.12% 56.25% 15.62% 18.75%

Table 12: Hand-picked errors and transformations for linguistic operator(s) IF_AND_NEUTRAL Hand-picked mistake / Transformations

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BUGGY_REMOVE_NEGATION Transformations combined T4: 𝑌1 → ¬𝑌2 ⇝ 𝑌1 → 𝑌2

9 9 9

37.50% 100.00% 100.00%

3 3 3

21.43% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

4 4 4

16.67% 100.00% 100.00%

2 2 2

14.29% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T3: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3

2 2 2

8.33% 100.00% 100.00%

1 1 1

7.14% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_NAND Transformations combined T2: ¬𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

1 1 1

4.17% 100.00% 100.00%

1 1 1

7.14% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T5: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

1 1 1

4.17% 100.00% 100.00%

1 1 1

7.14% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

7 3 3

29.17% 42.86% 42.86%

6 2 2

42.86% 33.33% 33.33%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

24 17 20 0 3

100.00% 70.83% 83.33% 0.00% 12.50%

14 8 10 0 2

100.00% 57.14% 71.43% 0.00% 14.29%

Table 13: Hand-picked errors and transformations for linguistic operator(s) IF_AND_NOT Hand-picked mistake / Transformations

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BUGGY_ADD_NEGATION Transformations combined T1: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3

24 24 24

33.80% 100.00% 100.00%

6 6 6

15.00% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T2: ¬𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1

12 12 12

16.90% 100.00% 100.00%

5 5 5

12.50% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined ⌈ = T3: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⌊ = T18: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

9 9 9 9

12.68% 100.00% 100.00% 100.00%

5 5 5 5

12.50% 100.00% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T4: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1

5 5 5

7.04% 100.00% 100.00%

3 3 3

7.50% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T7: ¬𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 T16: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1

2 2 1 1

2.82% 100.00% 50.00% 50.00%

2 2 1 1

5.00% 100.00% 50.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T11: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3

1 1 1

1.41% 100.00% 100.00%

1 1 1

2.50% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T9: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1

1 1 1

1.41% 100.00% 100.00%

1 1 1

2.50% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T8: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 T5: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T6: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 T10: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T12: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T13: 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T14: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ¬𝑌2 T15: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T17: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1

17 13 3 2 2 1 1 1 1 1 1

23.94% 76.47% 17.65% 11.76% 11.76% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88%

17 13 3 2 2 1 1 1 1 1 1

42.50% 76.47% 17.65% 11.76% 11.76% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

71 54 67 0 13

100.00% 76.06% 94.37% 0.00% 18.31%

40 23 36 0 13

100.00% 57.50% 90.00% 0.00% 32.50%

Table 14: Hand-picked errors and transformations for linguistic operator(s) IF_AND_ONLY_IF Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_EQUIVALENCE Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 T2: 𝑌1 → 𝑌2 ⇝ 𝑌1 ↔ 𝑌2

628 626 344 282

88.20% 99.68% 54.78% 44.90%

39 38 20 18

52.00% 97.44% 51.28% 46.15%

BUGGY_REPLACE_CONJUNCTION_BY_EQUIVALENCE Transformations combined ⌈ = T3: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 ⌊ = T4: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ↔ 𝑌2 ⌈ = T5: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 ⌊ = T6: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌2 ↔ 𝑌1

35 35 18 18 17 17

4.92% 100.00% 51.43% 51.43% 48.57% 48.57%

10 10 4 4 6 6

13.33% 100.00% 40.00% 40.00% 60.00% 60.00%

BUGGY_ADD_EQUIVALENCE Transformations combined

9 0

1.26% 0.00%

3 0

4.00% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

8 0

1.12% 0.00%

7 0

9.33% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_EQUIVALENCE Transformations combined

2 0

0.28% 0.00%

2 0

2.67% 0.00%

BUGGY_REPLACE_IMPLICATION_BY_XOR Transformations combined

1 0

0.14% 0.00%

1 0

1.33% 0.00%

Unexplained by ALL hand errors Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 T2: 𝑌1 → 𝑌2 ⇝ 𝑌1 ↔ 𝑌2

29 18 9 9

4.07% 62.07% 31.03% 31.03%

13 4 2 2

17.33% 30.77% 15.38% 15.38%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

712 683 679 22 18

100.00% 95.93% 95.37% 3.09% 2.53%

75 62 52 14 4

100.00% 82.67% 69.33% 18.67% 5.33%

Table 15: Hand-picked errors and transformations for linguistic operator(s) IF_AND Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

6 6 6

25.00% 100.00% 100.00%

3 3 3

17.65% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T2: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

2 2 2

8.33% 100.00% 100.00%

2 2 2

11.76% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T3: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3

1 1 1

4.17% 100.00% 100.00%

1 1 1

5.88% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T5: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

1 1 1

4.17% 100.00% 100.00%

1 1 1

5.88% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T7: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ (𝑌2 ∧ 𝑌3 ) → 𝑌1 T4: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌3 ∧ 𝑌2 ) → 𝑌1 T6: (𝑌1 → 𝑌2 ) ∧ (𝑌1 → 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌3 ) → 𝑌2

14 7 3 2 1 1

58.33% 50.00% 21.43% 14.29% 7.14% 7.14%

10 6 2 2 1 1

58.82% 60.00% 20.00% 20.00% 10.00% 10.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

24 10 17 0 7

100.00% 41.67% 70.83% 0.00% 29.17%

17 7 13 0 6

100.00% 41.18% 76.47% 0.00% 35.29%

Table 16: Hand-picked errors and transformations for linguistic operator(s) IF_IF Hand-picked mistake / Transformations

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BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T1: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 T16: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 )

40 40 40 13

29.63% 100.00% 100.00% 32.50%

2 2 2 1

4.00% 100.00% 100.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined ⌈ = T8: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) ⌊ = T17: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

10 9 9 9

7.41% 90.00% 90.00% 90.00%

2 1 1 1

4.00% 50.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T6: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

3 3 3

2.22% 100.00% 100.00%

2 2 2

4.00% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1

1 1 1

0.74% 100.00% 100.00%

1 1 1

2.00% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

2.00% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.74% 0.00%

1 0

2.00% 0.00%

Unexplained by ALL hand errors Transformations combined T2: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T3: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T4: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T5: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T13: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T9: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T10: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T7: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T11: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T12: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌3 → 𝑌2 ) T14: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ 𝑌2 → (𝑌3 → 𝑌1 ) T15: (𝑌1 → 𝑌2 ) ∧ (𝑌3 → 𝑌3 ) ⇝ 𝑌3 → (𝑌1 → 𝑌2 )

79 50 16 5 5 4 4 3 3 2 2 2 2 2

58.52% 63.29% 20.25% 6.33% 6.33% 5.06% 5.06% 3.80% 3.80% 2.53% 2.53% 2.53% 2.53% 2.53%

41 12 1 1 1 1 1 1 1 1 1 1 1 1

82.00% 29.27% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44% 2.44%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

135 56 103 3 50

100.00% 41.48% 76.30% 2.22% 37.04%

50 9 18 3 12

100.00% 18.00% 36.00% 6.00% 24.00%

Table 17: Hand-picked errors and transformations for linguistic operator(s) IF_NEITHER_NOR_NOT Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined ⌈ = T1: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 | = T2: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 | = T3: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T28: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T20: (¬𝑌1 ∨ ¬𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

73 73 65 65 65 65 8

20.05% 100.00% 89.04% 89.04% 89.04% 89.04% 10.96%

3 3 2 2 2 2 1

3.16% 100.00% 66.67% 66.67% 66.67% 66.67% 33.33%

BUGGY_ADD_NEGATION Transformations combined ⌈ = T4: (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → ¬𝑌3 ⌊ = T5: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⌈ = T8: ¬ (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → ¬𝑌3 ⌊ = T9: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

61 58 43 43 15 15

16.76% 95.08% 70.49% 70.49% 24.59% 24.59%

4 2 1 1 1 1

4.21% 50.00% 25.00% 25.00% 25.00% 25.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T6: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 T13: ¬𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1

28 28 21 7

7.69% 100.00% 75.00% 25.00%

3 3 2 1

3.16% 100.00% 66.67% 33.33%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T19: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

10 8 8

2.75% 80.00% 80.00%

2 1 1

2.11% 50.00% 50.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined ⌈ = T23: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⌊ = T32: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T27: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1

8 8 4 4 4

2.20% 100.00% 50.00% 50.00% 50.00%

2 2 1 1 1

2.11% 100.00% 50.00% 50.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T15: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1

7 6 6

1.92% 85.71% 85.71%

2 1 1

2.11% 50.00% 50.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined ⌈ = T14: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T21: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

6 6 6 6

1.65% 100.00% 100.00% 100.00%

1 1 1 1

1.05% 100.00% 100.00% 100.00%

BUGGY_REPLACE_XOR_BY_NOR Transformations combined ⌈ = T25: ¬ (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 | = T29: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 ⌊ = T31: ¬ (𝑌1 ↔ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

5 5 5 5 5

1.37% 100.00% 100.00% 100.00% 100.00%

1 1 1 1 1

1.05% 100.00% 100.00% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined

2 0

0.55% 0.00%

2 0

2.11% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.27% 0.00%

1 0

1.05% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.27% 0.00%

1 0

1.05% 0.00%

162 73 18 13

44.51% 45.06% 11.11% 8.02%

73 10 1 1

76.84% 13.70% 1.37% 1.37%

Unexplained by ALL hand errors Transformations combined T7: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T18: ¬𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1

Continued on next page 23

Hand-picked mistake / Transformations T10: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⌈ = T11: (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → ¬𝑌3 ⌊ = T12: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T24: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T30: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T16: ¬ (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T17: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T22: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T26: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

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9 7 7 5 5 4 4 4 4

5.56% 4.32% 4.32% 3.09% 3.09% 2.47% 2.47% 2.47% 2.47%

1 1 1 1 1 1 1 1 1

1.37% 1.37% 1.37% 1.37% 1.37% 1.37% 1.37% 1.37% 1.37%

364 202 265 10 73

100.00% 55.49% 72.80% 2.75% 20.05%

95 22 24 8 10

100.00% 23.16% 25.26% 8.42% 10.53%

Table 18: Hand-picked errors and transformations for linguistic operator(s) IF_NEITHER_NOR Hand-picked mistake / Transformations

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BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T1: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T2: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

37 37 28 9

45.12% 100.00% 75.68% 24.32%

2 2 1 1

6.25% 100.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T4: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T14: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1

5 5 4 1

6.10% 100.00% 80.00% 20.00%

2 2 1 1

6.25% 100.00% 50.00% 50.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T3: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

4 4 4

4.88% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T6: (¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T12: ¬ (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

3 3 2 1

3.66% 100.00% 66.67% 33.33%

2 2 1 1

6.25% 100.00% 50.00% 50.00%

BUGGY_REPLACE_XOR_BY_NOR Transformations combined T5: ¬ (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

3 3 3

3.66% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_NOR Transformations combined T15: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

2 2 2

2.44% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_ADD_CONCLUSION_FOR_IMPLICATION Transformations combined

1 0

1.22% 0.00%

1 0

3.12% 0.00%

BUGGY_ADD_NEGATION Transformations combined T28: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ 𝑌2 ) → 𝑌3

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_MOVE_PARENTHESIS2 Transformations combined T22: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined T25: ¬ (𝑌1 ∨ ¬𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T9: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T24: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T8: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

1 1 1

1.22% 100.00% 100.00%

1 1 1

3.12% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T10: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T7: ¬ (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T11: (𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

21 18 3 2 2

25.61% 85.71% 14.29% 9.52% 9.52%

16 13 1 1 1

50.00% 81.25% 6.25% 6.25% 6.25%

Continued on next page 25

Hand-picked mistake / Transformations T17: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T13: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T16: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T18: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T19: (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T20: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T21: (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T23: (¬𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T26: 𝑌1 ∧ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T27: ¬ (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

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2 1 1 1 1 1 1 1 1 1

9.52% 4.76% 4.76% 4.76% 4.76% 4.76% 4.76% 4.76% 4.76% 4.76%

1 1 1 1 1 1 1 1 1 1

6.25% 6.25% 6.25% 6.25% 6.25% 6.25% 6.25% 6.25% 6.25% 6.25%

82 61 78 1 18

100.00% 74.39% 95.12% 1.22% 21.95%

32 16 28 1 13

100.00% 50.00% 87.50% 3.12% 40.62%

Table 19: Hand-picked errors and transformations for linguistic operator(s) IF_NOT_AND Hand-picked mistake / Transformations

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BUGGY_ADD_NEGATION Transformations combined T1: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T11: 𝑌1 ∨ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌2 ) → 𝑌1

16 16 15 1

39.02% 100.00% 93.75% 6.25%

2 2 1 1

12.50% 100.00% 50.00% 50.00%

BUGGY_REMOVE_NEGATION Transformations combined T2: (¬𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3 T7: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 ∧ ¬𝑌2

6 6 5 1

14.63% 100.00% 83.33% 16.67%

2 2 1 1

12.50% 100.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T3: 𝑌1 → (𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌2 ∧ ¬𝑌3 ) → 𝑌1

4 4 4

9.76% 100.00% 100.00%

2 2 2

12.50% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T4: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3

2 2 2

4.88% 100.00% 100.00%

1 1 1

6.25% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T13: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2

1 1 1

2.44% 100.00% 100.00%

1 1 1

6.25% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T5: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌2 ∧ ¬𝑌1 T6: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T10: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌2 ) → 𝑌1 T8: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌2 ) → 𝑌1 T9: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌1 ) → 𝑌2 T12: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌1 ) → 𝑌2

12 9 2 2 2 1 1 1

29.27% 75.00% 16.67% 16.67% 16.67% 8.33% 8.33% 8.33%

8 6 1 1 1 1 1 1

50.00% 75.00% 12.50% 12.50% 12.50% 12.50% 12.50% 12.50%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

41 29 38 0 9

100.00% 70.73% 92.68% 0.00% 21.95%

16 8 14 0 6

100.00% 50.00% 87.50% 0.00% 37.50%

Table 20: Hand-picked errors and transformations for linguistic operator(s) IF_NOT Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: ¬𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1

137 137 137

47.24% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T2: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1 T8: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1

54 54 49 5

18.62% 100.00% 90.74% 9.26%

2 2 1 1

8.70% 100.00% 50.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T5: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2 T9: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2

20 20 15 5

6.90% 100.00% 75.00% 25.00%

2 2 1 1

8.70% 100.00% 50.00% 50.00%

BUGGY_ADD_NEGATION Transformations combined T4: 𝑌1 → 𝑌2 ⇝ 𝑌1 → ¬𝑌2

20 20 20

6.90% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T3: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2

14 14 14

4.83% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined T6: 𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1

10 10 10

3.45% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T3: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2

8 8 8

2.76% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T7: 𝑌1 ∨ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2

4 4 4

1.38% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined T7: 𝑌1 ∨ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2

2 2 2

0.69% 100.00% 100.00%

1 1 1

4.35% 100.00% 100.00%

BUGGY_EXPLICIFY_IMPLICATION_1 Transformations combined

1 0

0.34% 0.00%

1 0

4.35% 0.00%

BUGGY_REPLACE_NOR_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.34% 0.00%

1 0

4.35% 0.00%

Unexplained by ALL hand errors Transformations combined T11: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → ¬𝑌2 T10: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1

19 10 7 3

6.55% 52.63% 36.84% 15.79%

10 2 1 1

43.48% 20.00% 10.00% 10.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

290 271 279 2 10

100.00% 93.45% 96.21% 0.69% 3.45%

23 13 13 2 2

100.00% 56.52% 56.52% 8.70% 8.70%

Table 21: Hand-picked errors and transformations for linguistic operator(s) IF_OR_NEUTRAL Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T1: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

6 6 6

17.65% 100.00% 100.00%

2 2 2

9.09% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined T2: 𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T5: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → 𝑌2

5 5 4 1

14.71% 100.00% 80.00% 20.00%

2 2 1 1

9.09% 100.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T3: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

4 4 4

11.76% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_DISJUNCTION Transformations combined T4: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T12: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

3 3 2 1

8.82% 100.00% 66.67% 33.33%

2 2 1 1

9.09% 100.00% 50.00% 50.00%

BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined T8: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∨ 𝑌2 T13: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

2 2 1 1

5.88% 100.00% 50.00% 50.00%

2 2 1 1

9.09% 100.00% 50.00% 50.00%

BUGGY_REMOVE_FIRST_OPERAND_FROM_CONJUNCTION Transformations combined T9: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∨ 𝑌2

1 1 1

2.94% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T7: 𝑌1 ↔ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

1 1 1

2.94% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T10: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

1 1 1

2.94% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T6: ¬𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T11: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T14: ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T15: 𝑌1 ↔ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T16: 𝑌1 ↔ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T17: 𝑌1 ↔ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T18: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

11 7 1 1 1 1 1 1 1

32.35% 63.64% 9.09% 9.09% 9.09% 9.09% 9.09% 9.09% 9.09%

10 7 1 1 1 1 1 1 1

45.45% 70.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

34 23 30 0 7

100.00% 67.65% 88.24% 0.00% 20.59%

22 12 19 0 7

100.00% 54.55% 86.36% 0.00% 31.82%

Table 22: Hand-picked errors and transformations for linguistic operator(s) IF_OR Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined ⌈ = T1: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) ⌊ = T9: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

18 18 18 18

52.94% 100.00% 100.00% 100.00%

4 4 4 4

23.53% 100.00% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T3: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

3 3 3

8.82% 100.00% 100.00%

2 2 2

11.76% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T2: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

3 3 3

8.82% 100.00% 100.00%

2 2 2

11.76% 100.00% 100.00%

BUGGY_PULL_NEGATION_DISJUNCTION Transformations combined T6: (¬𝑌1 ∨ ¬𝑌2 ) → ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

1 1 1

2.94% 100.00% 100.00%

1 1 1

5.88% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T7: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

1 1 1

2.94% 100.00% 100.00%

1 1 1

5.88% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T8: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

1 1 1

2.94% 100.00% 100.00%

1 1 1

5.88% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T4: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T5: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

7 2 1 1

20.59% 28.57% 14.29% 14.29%

6 2 1 1

35.29% 33.33% 16.67% 16.67%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

34 27 29 0 2

100.00% 79.41% 85.29% 0.00% 5.88%

17 11 13 0 2

100.00% 64.71% 76.47% 0.00% 11.76%

Table 23: Hand-picked errors and transformations for linguistic operator(s) IF Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

61 61 61

58.65% 100.00% 100.00%

4 4 4

19.05% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T2: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

20 20 20

19.23% 100.00% 100.00%

4 4 4

19.05% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T4: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 T3: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1

10 10 6 4

9.62% 100.00% 60.00% 40.00%

4 4 1 3

19.05% 100.00% 25.00% 75.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T3: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1

3 3 3

2.88% 100.00% 100.00%

2 2 2

9.52% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T5: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

3 3 3

2.88% 100.00% 100.00%

1 1 1

4.76% 100.00% 100.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.96% 0.00%

1 0

4.76% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined

1 0

0.96% 0.00%

1 0

4.76% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined

1 0

0.96% 0.00%

1 0

4.76% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.96% 0.00%

1 0

4.76% 0.00%

BUGGY_REPLACE_XOR_BY_IMPLICATION Transformations combined

1 0

0.96% 0.00%

1 0

4.76% 0.00%

Unexplained by ALL hand errors Transformations combined

2 0

1.92% 0.00%

1 0

4.76% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

104 102 97 5 0

100.00% 98.08% 93.27% 4.81% 0.00%

21 20 15 5 0

100.00% 95.24% 71.43% 23.81% 0.00%

Table 24: Hand-picked errors and transformations for linguistic operator(s) NEITHER_NOR Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T1: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⌈ = T2: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⌊ = T7: ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T11: ¬𝑌1 ∨ ¬𝑌2 ∨ ¬𝑌3 ⇝ ¬𝑌1 ∧ ¬𝑌3 ∧ ¬𝑌2 ⌈ = T4: ¬𝑌1 ∨ ¬𝑌2 ∨ ¬𝑌3 ⇝ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⌊ = T9: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ∨ 𝑌3 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 ∧ 𝑌3 T14: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 ∧ 𝑌2 ∧ 𝑌3

25 25 14 7 7 2 1 1 1

37.31% 100.00% 56.00% 28.00% 28.00% 8.00% 4.00% 4.00% 4.00%

7 7 2 2 2 1 1 1 1

23.33% 100.00% 28.57% 28.57% 28.57% 14.29% 14.29% 14.29% 14.29%

BUGGY_ADD_OPERAND_TO_DISJUNCTION Transformations combined

4 0

5.97% 0.00%

2 0

6.67% 0.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T3: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

3 3 3

4.48% 100.00% 100.00%

1 1 1

3.33% 100.00% 100.00%

BUGGY_ADD_NEGATION Transformations combined T10: 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ ¬𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌1 T12: 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3

2 2 1 1

2.99% 100.00% 50.00% 50.00%

2 2 1 1

6.67% 100.00% 50.00% 50.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

1.49% 0.00%

1 0

3.33% 0.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T6: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

1 1 1

1.49% 100.00% 100.00%

1 1 1

3.33% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T5: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌1 T8: 𝑌1 ↔ (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌3 ∧ 𝑌2 ∧ 𝑌1 T13: (𝑌1 ↔ ¬𝑌2 ) ∧ 𝑌3 ⇝ ¬𝑌2 ∧ ¬𝑌1 ∧ ¬𝑌3 T15: ¬ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ⇝ ¬𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 T16: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 ∧ ¬𝑌3 ∧ ¬𝑌2

31 5 1 1 1 1 1

46.27% 16.13% 3.23% 3.23% 3.23% 3.23% 3.23%

16 5 1 1 1 1 1

53.33% 31.25% 6.25% 6.25% 6.25% 6.25% 6.25%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

67 36 36 5 5

100.00% 53.73% 53.73% 7.46% 7.46%

30 14 16 3 5

100.00% 46.67% 53.33% 10.00% 16.67%

Table 25: Hand-picked errors and transformations for linguistic operator(s) NEUTRAL Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REMOVE_NEGATION Transformations combined T1: ¬𝑌1 ⇝ 𝑌1

52 52 52

100.00% 100.00% 100.00%

2 2 2

100.00% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

0 0

0.00% 0.00%

0 0

0.00% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

52 52 52 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

2 2 2 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

Table 26: Hand-picked errors and transformations for linguistic operator(s) NONE_OF Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T1: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T7: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛

36 36 34 2

78.26% 100.00% 94.44% 5.56%

7 7 5 2

46.67% 100.00% 71.43% 28.57%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T2: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

4 4 4

8.70% 100.00% 100.00%

2 2 2

13.33% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T3: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T6: ¬ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

2 2 1 1

4.35% 100.00% 50.00% 50.00%

2 2 1 1

13.33% 100.00% 50.00% 50.00%

Unexplained by ALL hand errors Transformations combined T4: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 T5: ¬ (𝑌1 ∧ 𝑌2 ) ∧ ¬𝑌3 ⇝ ¬𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2

4 2 1 1

8.70% 50.00% 25.00% 25.00%

4 2 1 1

26.67% 50.00% 25.00% 25.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

46 42 44 0 2

100.00% 91.30% 95.65% 0.00% 4.35%

15 11 13 0 2

100.00% 73.33% 86.67% 0.00% 13.33%

Table 27: Hand-picked errors and transformations for linguistic operator(s) NOT_ALL_OF Hand-picked mistake / Transformations

#dst

%dst

BUGGY_PULL_NEGATION_CONJUNCTION Transformations combined ⌈ = T3: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T4: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T17: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 )

67 67 61 61 6

12.48% 100.00% 91.04% 91.04% 8.96%

13 13 12 12 1

7.78% 100.00% 92.31% 92.31% 7.69%

BUGGY_REPLACE_XOR_BY_NAND Transformations combined T7: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T11: ¬𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

44 43 24 19

8.19% 97.73% 54.55% 43.18%

5 4 2 2

2.99% 80.00% 40.00% 40.00%

BUGGY_REPLACE_DISJUNCTION_BY_NAND Transformations combined ⌈ = T5: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T6: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 )

39 35 35 35

7.26% 89.74% 89.74% 89.74%

8 6 6 6

4.79% 75.00% 75.00% 75.00%

BUGGY_REPLACE_NOR_BY_NAND Transformations combined ⌈ = T8: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T9: ¬ (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 )

26 22 22 22

4.84% 84.62% 84.62% 84.62%

7 6 6 6

4.19% 85.71% 85.71% 85.71%

BUGGY_REPLACE_IMPLICATION_BY_NAND Transformations combined T15: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T21: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 )

18 15 8 7

3.35% 83.33% 44.44% 38.89%

5 3 2 1

2.99% 60.00% 40.00% 20.00%

BUGGY_REPLACE_CONJUNCTION_BY_NAND Transformations combined ⌈ = T12: 𝑌1 ∧ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T24: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 )

14 13 13 13

2.61% 92.86% 92.86% 92.86%

4 3 3 3

2.40% 75.00% 75.00% 75.00%

BUGGY_REPLACE_EQUIVALENCE_BY_NAND Transformations combined T13: 𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

11 10 10

2.05% 90.91% 90.91%

5 4 4

2.99% 80.00% 80.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined

5 0

0.93% 0.00%

4 0

2.40% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_DISJUNCTION Transformations combined

4 0

0.74% 0.00%

3 0

1.80% 0.00%

BUGGY_MOVE_PARENTHESIS1 Transformations combined

4 0

0.74% 0.00%

2 0

1.20% 0.00%

BUGGY_ADD_NEGATION Transformations combined

4 0

0.74% 0.00%

1 0

0.60% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.19% 0.00%

1 0

0.60% 0.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined

1 0

0.19% 0.00%

1 0

0.60% 0.00%

BUGGY_REPLACE_XOR_BY_DISJUNCTION Transformations combined

1 0

0.19% 0.00%

1 0

0.60% 0.00%

298 245 114

55.49% 82.21% 38.26%

107 62 20

64.07% 57.94% 18.69%

Unexplained by ALL hand errors Transformations combined T1: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 )

Continued on next page 35

Hand-picked mistake / Transformations T2: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T10: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T23: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌈ = T19: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T20: 𝑌1 ∧ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬ (𝑌2 ∧ 𝑌3 ) T14: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T16: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T18: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T22: 𝑌1 ∧ ¬𝑌2 ∧ 𝑌2 ∧ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

#dst

%dst

66 22 15 6 6 9 7 6 6

22.15% 7.38% 5.03% 2.01% 2.01% 3.02% 2.35% 2.01% 2.01%

6 7 12 4 4 6 3 4 4

5.61% 6.54% 11.21% 3.74% 3.74% 5.61% 2.80% 3.74% 3.74%

537 239 450 34 245

100.00% 44.51% 83.80% 6.33% 45.62%

167 60 101 21 62

100.00% 35.93% 60.48% 12.57% 37.13%

Table 28: Hand-picked errors and transformations for linguistic operator(s) NOT Hand-picked mistake / Transformations

#dst

%dst

BUGGY_ADD_NEGATION Transformations combined T1: 𝑌1 ⇝ ¬𝑌1

1 1 1

100.00% 100.00% 100.00%

1 1 1

100.00% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined

0 0

0.00% 0.00%

0 0

0.00% 0.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

1 1 1 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

1 1 1 0 0

100.00% 100.00% 100.00% 0.00% 0.00%

Table 29: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF_AND Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined ⌈ = T1: (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) | = T2: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) ⌊ = T3: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

446 443 443 443 443

50.06% 99.33% 99.33% 99.33% 99.33%

5 3 3 3 3

5.75% 60.00% 60.00% 60.00% 60.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined ⌈ = T4: (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) | = T5: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) ⌊ = T6: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

202 201 201 201 201

22.67% 99.50% 99.50% 99.50% 99.50%

4 3 3 3 3

4.60% 75.00% 75.00% 75.00% 75.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined ⌈ = T7: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∧ 𝑌3 ) | = T8: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T9: 𝑌3 ↔ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

18 18 18 18 18

2.02% 100.00% 100.00% 100.00% 100.00%

3 3 3 3 3

3.45% 100.00% 100.00% 100.00% 100.00%

BUGGY_ADD_PREMISE_FOR_IMPLICATION Transformations combined

12 0

1.35% 0.00%

2 0

2.30% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T10: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

10 10 10

1.12% 100.00% 100.00%

2 2 2

2.30% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_NAND Transformations combined

5 0

0.56% 0.00%

1 0

1.15% 0.00%

BUGGY_ADD_NEGATION Transformations combined

4 0

0.45% 0.00%

1 0

1.15% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

4 0

0.45% 0.00%

1 0

1.15% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined

3 0

0.34% 0.00%

2 0

2.30% 0.00%

BUGGY_REPLACE_XOR_BY_REVERSED_IMPLICATION Transformations combined

3 0

0.34% 0.00%

1 0

1.15% 0.00%

BUGGY_ADD_OPERAND_TO_CONJUNCTION Transformations combined

2 0

0.22% 0.00%

1 0

1.15% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined

2 0

0.22% 0.00%

1 0

1.15% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.11% 0.00%

1 0

1.15% 0.00%

Unexplained by ALL hand errors Transformations combined ⌈ = T11: (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ↔ ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⌊ = T12: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

179 9 9 9

20.09% 5.03% 5.03% 5.03%

62 1 1 1

71.26% 1.61% 1.61% 1.61%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

891 712 681 40 9

100.00% 79.91% 76.43% 4.49% 1.01%

87 25 12 14 1

100.00% 28.74% 13.79% 16.09% 1.15%

Table 30: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF_AT_LEAST Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T3: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T7: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) → 𝑌4 ⇝ 𝑌4 → (𝑌1 ∨ 𝑌2 ∨ 𝑌3 )

277 274 239 35

45.26% 98.92% 86.28% 12.64%

20 18 4 14

10.10% 90.00% 20.00% 70.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

33 33 33

5.39% 100.00% 100.00%

4 4 4

2.02% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T6: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1

29 23 23

4.74% 79.31% 79.31%

17 12 12

8.59% 70.59% 70.59%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T4: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

29 29 29

4.74% 100.00% 100.00%

6 6 6

3.03% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T5: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛

12 11 11

1.96% 91.67% 91.67%

3 2 2

1.52% 66.67% 66.67%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T6: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1

8 6 6

1.31% 75.00% 75.00%

4 2 2

2.02% 50.00% 50.00%

BUGGY_REPLACE_XOR_BY_DISJUNCTION Transformations combined

6 0

0.98% 0.00%

3 0

1.52% 0.00%

BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined

5 0

0.82% 0.00%

2 0

1.01% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_DISJUNCTION Transformations combined

4 0

0.65% 0.00%

1 0

0.51% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

3 0

0.49% 0.00%

3 0

1.52% 0.00%

BUGGY_ADD_OPERAND_TO_DISJUNCTION Transformations combined

2 0

0.33% 0.00%

2 0

1.01% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_NOR Transformations combined

2 0

0.33% 0.00%

2 0

1.01% 0.00%

BUGGY_ADD_NEGATION Transformations combined

2 0

0.33% 0.00%

1 0

0.51% 0.00%

BUGGY_ADD_PREMISE_FOR_IMPLICATION Transformations combined

1 0

0.16% 0.00%

1 0

0.51% 0.00%

BUGGY_MOVE_PARENTHESIS2 Transformations combined

1 0

0.16% 0.00%

1 0

0.51% 0.00%

BUGGY_REMOVE_FIRST_OPERAND_FROM_CONJUNCTION Transformations combined

1 0

0.16% 0.00%

1 0

0.51% 0.00%

BUGGY_REMOVE_FIRST_OPERAND_FROM_DISJUNCTION Transformations combined

1 0

0.16% 0.00%

1 0

0.51% 0.00%

BUGGY_REMOVE_LATER_OPERAND_FROM_DISJUNCTION Transformations combined

1 0

0.16% 0.00%

1 0

0.51% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION

0.16%

0.51%

Continued on next page 39

Hand-picked mistake / Transformations

#dst

%dst

0.00%

1 0

0.16% 0.00%

1 0

0.51% 0.00%

Unexplained by ALL hand errors Transformations combined T2: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

193 10 10

31.54% 5.18% 5.18%

123 1 1

62.12% 0.81% 0.81%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

612 419 386 43 10

100.00% 68.46% 63.07% 7.03% 1.63%

198 75 45 31 1

100.00% 37.88% 22.73% 15.66% 0.51%

Transformations combined BUGGY_REPLACE_NOR_BY_DISJUNCTION Transformations combined

Table 31: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF_NEITHER_NOR Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T4: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1

138 138 117 21

28.34% 100.00% 84.78% 15.22%

33 33 20 13

15.21% 100.00% 60.61% 39.39%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined ⌈ = T2: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T3: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T14: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

47 47 29 29 18

9.65% 100.00% 61.70% 61.70% 38.30%

17 17 10 10 7

7.83% 100.00% 58.82% 58.82% 41.18%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T7: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T17: 𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1

25 25 19 6

5.13% 100.00% 76.00% 24.00%

15 15 9 6

6.91% 100.00% 60.00% 40.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined ⌈ = T5: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T6: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

21 21 21 21

4.31% 100.00% 100.00% 100.00%

7 7 7 7

3.23% 100.00% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined ⌈ = T8: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T10: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

17 17 17 17

3.49% 100.00% 100.00% 100.00%

8 8 8 8

3.69% 100.00% 100.00% 100.00%

BUGGY_ADD_NEGATION Transformations combined T18: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T16: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

9 9 5 4

1.85% 100.00% 55.56% 44.44%

5 5 3 2

2.30% 100.00% 60.00% 40.00%

BUGGY_REMOVE_NEGATION Transformations combined

6 0

1.23% 0.00%

4 0

1.84% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

4 0

0.82% 0.00%

4 0

1.84% 0.00%

BUGGY_MOVE_PARENTHESIS2 Transformations combined

3 0

0.62% 0.00%

2 0

0.92% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined

2 0

0.41% 0.00%

2 0

0.92% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined

2 0

0.41% 0.00%

2 0

0.92% 0.00%

BUGGY_REPLACE_EQUIVALENCE_BY_NOR Transformations combined

2 0

0.41% 0.00%

2 0

0.92% 0.00%

BUGGY_REMOVE_FIRST_FROM_EQUIVALENCE Transformations combined

1 0

0.21% 0.00%

1 0

0.46% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_NAND Transformations combined

1 0

0.21% 0.00%

1 0

0.46% 0.00%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

1 0

0.21% 0.00%

1 0

0.46% 0.00%

BUGGY_REPLACE_XOR_BY_NOR Transformations combined

1 0

0.21% 0.00%

1 0

0.46% 0.00%

Continued on next page 41

Hand-picked mistake / Transformations

#dst

%dst

Unexplained by ALL hand errors Transformations combined T11: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T12: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T9: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 ⌈ = T2: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T3: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T4: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T15: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T13: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T16: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3

207 132 40 19 17 15 15 14 13 9 5

42.51% 63.77% 19.32% 9.18% 8.21% 7.25% 7.25% 6.76% 6.28% 4.35% 2.42%

112 46 14 7 7 2 2 2 6 6 2

51.61% 41.07% 12.50% 6.25% 6.25% 1.79% 1.79% 1.79% 5.36% 5.36% 1.79%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

487 280 389 23 132

100.00% 57.49% 79.88% 4.72% 27.10%

217 105 131 20 46

100.00% 48.39% 60.37% 9.22% 21.20%

Table 32: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF_NOT_AND_NOT_NOT Hand-picked mistake / Transformations

#dst

%dst

BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T7: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 )

19 19 11 8

32.76% 100.00% 57.89% 42.11%

2 2 1 1

11.11% 100.00% 50.00% 50.00%

BUGGY_ADD_NEGATION Transformations combined T10: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) T16: 𝑌1 → 𝑌2 ⇝ ¬𝑌1 → 𝑌2

3 3 2 1

5.17% 100.00% 66.67% 33.33%

2 2 1 1

11.11% 100.00% 50.00% 50.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T3: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

3 3 3

5.17% 100.00% 100.00%

1 1 1

5.56% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T4: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

2 2 2

3.45% 100.00% 100.00%

1 1 1

5.56% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T2: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

1 1 1

1.72% 100.00% 100.00%

1 1 1

5.56% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T12: ¬ (𝑌1 ∨ 𝑌2 ) ↔ ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 )

1 1 1

1.72% 100.00% 100.00%

1 1 1

5.56% 100.00% 100.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T5: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

1 1 1

1.72% 100.00% 100.00%

1 1 1

5.56% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T6: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T8: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) T9: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T11: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T14: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T15: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T13: ¬𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) T17: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 )

28 27 8 7 4 2 2 2 1 1

48.28% 96.43% 28.57% 25.00% 14.29% 7.14% 7.14% 7.14% 3.57% 3.57%

9 8 1 1 1 1 1 1 1 1

50.00% 88.89% 11.11% 11.11% 11.11% 11.11% 11.11% 11.11% 11.11% 11.11%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

58 30 57 0 27

100.00% 51.72% 98.28% 0.00% 46.55%

18 9 17 0 8

100.00% 50.00% 94.44% 0.00% 44.44%

Table 33: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF_NOT_OR_NOT_NOT Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T11: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 )

50 50 44 6

30.12% 100.00% 88.00% 12.00%

10 10 7 3

12.05% 100.00% 70.00% 30.00%

BUGGY_ADD_NEGATION Transformations combined T2: 𝑌1 → 𝑌2 ⇝ ¬𝑌1 → 𝑌2 T16: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 ∨ ¬𝑌1 T14: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 )

21 21 16 3 2

12.65% 100.00% 76.19% 14.29% 9.52%

10 10 6 2 2

12.05% 100.00% 60.00% 20.00% 20.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T6: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

14 13 13

8.43% 92.86% 92.86%

4 3 3

4.82% 75.00% 75.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T3: ¬ (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ⇝ ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 ⌈ = T4: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬𝑌1 ∨ ¬𝑌2 ⌊ = T5: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛

12 12 8 4 4

7.23% 100.00% 66.67% 33.33% 33.33%

6 6 4 2 2

7.23% 100.00% 66.67% 33.33% 33.33%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T7: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

10 9 9

6.02% 90.00% 90.00%

5 4 4

6.02% 80.00% 80.00%

BUGGY_REPLACE_NOR_BY_DISJUNCTION Transformations combined T10: 𝑌1 → 𝑌2 ⇝ ¬𝑌2 → 𝑌1

10 10 10

6.02% 100.00% 100.00%

5 5 5

6.02% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T9: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

3 3 3

1.81% 100.00% 100.00%

2 2 2

2.41% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION Transformations combined

2 0

1.20% 0.00%

2 0

2.41% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.60% 0.00%

1 0

1.20% 0.00%

BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined

1 0

0.60% 0.00%

1 0

1.20% 0.00%

Unexplained by ALL hand errors Transformations combined T12: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T8: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T13: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T15: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 )

42 13 5 3 3 2

25.30% 30.95% 11.90% 7.14% 7.14% 4.76%

37 9 4 2 2 1

44.58% 24.32% 10.81% 5.41% 5.41% 2.70%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

166 124 131 6 13

100.00% 74.70% 78.92% 3.61% 7.83%

83 46 49 6 9

100.00% 55.42% 59.04% 7.23% 10.84%

Table 34: Hand-picked errors and transformations for linguistic operator(s) ONLY_IF Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T1: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1

164 164 164

35.12% 100.00% 100.00%

1 1 1

3.70% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T2: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

156 152 152

33.40% 97.44% 97.44%

2 1 1

7.41% 50.00% 50.00%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T3: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

97 93 93

20.77% 95.88% 95.88%

3 1 1

11.11% 33.33% 33.33%

BUGGY_REPLACE_XOR_BY_REVERSED_IMPLICATION Transformations combined T5: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 T8: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1

14 13 8 5

3.00% 92.86% 57.14% 35.71%

3 2 1 1

11.11% 66.67% 33.33% 33.33%

BUGGY_REPLACE_XOR_BY_IMPLICATION Transformations combined T6: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌1 → 𝑌2

7 6 6

1.50% 85.71% 85.71%

2 1 1

7.41% 50.00% 50.00%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined ⌈ = T7: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 ⌊ = T9: 𝑌2 ∧ 𝑌1 ⇝ 𝑌1 → 𝑌2

4 4 4 4

0.86% 100.00% 100.00% 100.00%

1 1 1 1

3.70% 100.00% 100.00% 100.00%

BUGGY_ADD_NEGATION Transformations combined

1 0

0.21% 0.00%

1 0

3.70% 0.00%

BUGGY_REMOVE_NEGATION Transformations combined

1 0

0.21% 0.00%

1 0

3.70% 0.00%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined ⌈ = T7: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 ⌊ = T9: 𝑌2 ∧ 𝑌1 ⇝ 𝑌1 → 𝑌2

1 1 1 1

0.21% 100.00% 100.00% 100.00%

1 1 1 1

3.70% 100.00% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T4: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → 𝑌1

22 10 10

4.71% 45.45% 45.45%

12 1 1

44.44% 8.33% 8.33%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

467 445 443 12 10

100.00% 95.29% 94.86% 2.57% 2.14%

27 15 9 7 1

100.00% 55.56% 33.33% 25.93% 3.70%

Table 35: Hand-picked errors and transformations for linguistic operator(s) OR_AND_OR Hand-picked mistake / Transformations

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BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T1: (𝑌1 ∨ 𝑌2 ) ↔ (𝑌3 ∨ 𝑌4 ) ⇝ (𝑌3 ∨ 𝑌4 ) ∧ (𝑌1 ∨ 𝑌2 )

5 5 5

13.89% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T7: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌1 ∧ (𝑌2 ∨ 𝑌3 )

4 4 4

11.11% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T4: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

3 3 3

8.33% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T5: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

1 1 1

2.78% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_DISJUNCTION Transformations combined T9: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 ∨ 𝑌2

1 1 1

2.78% 100.00% 100.00%

1 1 1

4.55% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T2: ¬ (𝑌1 ↔ 𝑌2 ) ∧ ¬ (𝑌3 ↔ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T8: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ 𝑌4 ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T3: (𝑌1 ∨ ¬𝑌2 ) ↔ (𝑌3 ∨ ¬𝑌4 ) ⇝ (𝑌4 ∨ 𝑌3 ) ∧ (𝑌2 ∨ 𝑌1 ) T6: (𝑌1 ∧ 𝑌2 ) ↔ (𝑌3 ∧ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T10: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) ∧ ((𝑌3 ∧ ¬𝑌4 ) ∨ (𝑌4 ∧ ¬𝑌3 )) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T11: (𝑌1 ∧ 𝑌2 ) → (𝑌3 ∧ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T12: (𝑌1 ∨ ¬𝑌2 ) ↔ (𝑌3 ∨ 𝑌4 ) ⇝ (𝑌3 ∨ 𝑌4 ) ∧ (𝑌1 ∨ 𝑌2 ) T13: ¬ (𝑌1 ↔ 𝑌2 ) ∧ ¬ (𝑌3 ∧ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T14: (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) ∧ ¬ (𝑌1 ∧ 𝑌2 ) ∧ ¬ (𝑌3 ∧ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T15: ¬ (𝑌1 ↔ 𝑌2 ) ∨ ¬ (𝑌3 ↔ 𝑌4 ) ⇝ (𝑌2 ∨ 𝑌1 ) ∧ (𝑌3 ∨ 𝑌4 ) T16: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌3 ∨ ¬𝑌4 ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌4 ∨ 𝑌3 ) T17: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌3 ∨ ¬𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌4 ∨ 𝑌3 ) T18: (𝑌1 ↔ ¬𝑌2 ) ∨ (𝑌3 ↔ ¬𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌4 ∨ 𝑌3 ) T19: (𝑌1 ↔ ¬𝑌2 ) ∧ (𝑌3 ↔ ¬𝑌4 ) ⇝ (𝑌4 ∨ 𝑌3 ) ∧ (𝑌2 ∨ 𝑌1 ) T20: (𝑌1 ↔ ¬𝑌2 ) ↔ (𝑌3 ↔ ¬𝑌4 ) ⇝ (𝑌4 ∨ 𝑌3 ) ∧ (𝑌1 ∨ 𝑌2 )

22 20 5 2 1 1 1 1 1 1 1 1 1 1 1 1 1

61.11% 90.91% 22.73% 9.09% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55% 4.55%

17 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

77.27% 88.24% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88% 5.88%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

36 14 34 0 20

100.00% 38.89% 94.44% 0.00% 55.56%

22 5 20 0 15

100.00% 22.73% 90.91% 0.00% 68.18%

Table 36: Hand-picked errors and transformations for linguistic operator(s) OR Hand-picked mistake / Transformations

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BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T3: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 ∨ 𝑌3 ∨ 𝑌2 T4: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌3 ∨ 𝑌1 ∨ 𝑌2 T1: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛 T2: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛

30 30 15 9 5 1

83.33% 100.00% 50.00% 30.00% 16.67% 3.33%

5 5 1 2 1 1

62.50% 100.00% 20.00% 40.00% 20.00% 20.00%

BUGGY_REPLACE_IMPLICATION_BY_DISJUNCTION Transformations combined T5: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3

4 4 4

11.11% 100.00% 100.00%

1 1 1

12.50% 100.00% 100.00%

Unexplained by ALL hand errors Transformations combined T6: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌2 ∨ 𝑌1 ∨ 𝑌3 T7: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 ∨ 𝑌1

2 2 1 1

5.56% 100.00% 50.00% 50.00%

2 2 1 1

25.00% 100.00% 50.00% 50.00%

Whole file ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

36 34 36 0 2

100.00% 94.44% 100.00% 0.00% 5.56%

8 6 8 0 2

100.00% 75.00% 100.00% 0.00% 25.00%

B.2

Full Propositional Logic Data Set

The hand-picked mistakes are from [13]. Table 37: Hand-picked errors and transformations for the ungrouped propositional data set. Hand-picked mistake / Transformations

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BUGGY_REPLACE_IMPLICATION_BY_REVERSED_IMPLICATION Transformations combined T169: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T184: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → 𝑌1 T210: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T293: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T280: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T382: 𝑌1 ∨ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∧ 𝑌3 )

1418 1414 1343 172 55 8 6 2

23.22% 99.72% 94.71% 12.13% 3.88% 0.56% 0.42% 0.14%

105 102 80 37 17 1 3 1

6.68% 97.14% 76.19% 35.24% 16.19% 0.95% 2.86% 0.95%

BUGGY_REPLACE_IMPLICATION_BY_EQUIVALENCE Transformations combined T168: 𝑌1 → 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 T167: 𝑌1 → 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T255: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

663 660 360 298 2

10.86% 99.55% 54.30% 44.95% 0.30%

45 43 22 20 1

2.86% 95.56% 48.89% 44.44% 2.22%

BUGGY_REPLACE_EQUIVALENCE_BY_REVERSED_IMPLICATION Transformations combined T94: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T120: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T140: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1 T164: (¬𝑌1 ∨ ¬𝑌2 ) ↔ ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T133: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T113: 𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T106: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 T143: ¬𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 → 𝑌2 T151: ¬ (¬𝑌1 ) ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T157: ¬ (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

446 444 426 201 56 13 1 7 6 3 1 1

7.30% 99.55% 95.52% 45.07% 12.56% 2.91% 0.22% 1.57% 1.35% 0.67% 0.22% 0.22%

45 43 31 3 3 3 1 7 2 1 1 1

2.86% 95.56% 68.89% 6.67% 6.67% 6.67% 2.22% 15.56% 4.44% 2.22% 2.22% 2.22%

BUGGY_REPLACE_DISJUNCTION_BY_CONJUNCTION Transformations combined T318: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T323: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 T274: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T362: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 T367: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 ⌈ = T313: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T428: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2 T303: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T424: ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⌈ = T313: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T428: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2 T87: (¬𝑌1 ∨ ¬𝑌2 ) ∧ ¬𝑌3 ⇝ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3

334 330 201 153 150 128 118 35 35 17 44 35 35 1

5.47% 98.80% 60.18% 45.81% 44.91% 38.32% 35.33% 10.48% 10.48% 5.09% 13.17% 10.48% 10.48% 0.30%

61 58 23 16 15 34 29 9 9 6 13 9 9 1

3.88% 95.08% 37.70% 26.23% 24.59% 55.74% 47.54% 14.75% 14.75% 9.84% 21.31% 14.75% 14.75% 1.64%

BUGGY_REPLACE_EQUIVALENCE_BY_IMPLICATION Transformations combined T92: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 T121: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3 T138: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 T153: ¬ (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T155: ¬ (𝑌1 ∨ 𝑌2 ) ↔ ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

293 288 278 14 5 4 2

4.80% 98.29% 94.88% 4.78% 1.71% 1.37% 0.68%

37 35 30 6 1 3 1

2.35% 94.59% 81.08% 16.22% 2.70% 8.11% 2.70%

Continued on next page 48

Hand-picked mistake / Transformations

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0.34%

2.70%

BUGGY_ADD_NEGATION Transformations combined T1: 𝑌1 ⇝ ¬𝑌1 T225: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T171: 𝑌1 → 𝑌2 ⇝ ¬𝑌1 → 𝑌2 T291: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T233: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T272: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T378: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T204: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) T212: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 )

183 160 126 67 17 15 7 4 4 2 2

3.00% 87.43% 68.85% 36.61% 9.29% 8.20% 3.83% 2.19% 2.19% 1.09% 1.09%

41 34 24 7 7 1 4 1 1 2 1

2.61% 82.93% 58.54% 17.07% 17.07% 2.44% 9.76% 2.44% 2.44% 4.88% 2.44%

BUGGY_REMOVE_NEGATION Transformations combined T317: ¬𝑌1 ⇝ 𝑌1 T198: 𝑌1 → ¬𝑌2 ⇝ 𝑌1 → 𝑌2 T104: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T61: ¬𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 ∧ ¬𝑌2 T251: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → 𝑌2 T135: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T348: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → 𝑌2 T170: 𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1 T292: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T146: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T136: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 T147: ¬𝑌1 ↔ ¬ (¬𝑌2 ) ⇝ 𝑌1 ↔ 𝑌2 T205: 𝑌1 → ¬ (𝑌2 ∧ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌3 ∨ 𝑌2 ) T261: ¬𝑌1 → ¬ (𝑌2 ∧ ¬𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌3 ∨ ¬𝑌2 ) T281: ¬ (𝑌1 ∧ 𝑌2 ) → ¬ (¬𝑌3 ) ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T289: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T327: ¬ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 )

112 107 84 12 5 5 2 1 1 10 3 3 1 1 1 1 1 1 1

1.83% 95.54% 75.00% 10.71% 4.46% 4.46% 1.79% 0.89% 0.89% 8.93% 2.68% 2.68% 0.89% 0.89% 0.89% 0.89% 0.89% 0.89% 0.89%

35 30 19 5 4 1 2 1 1 1 2 1 1 1 1 1 1 1 1

2.23% 85.71% 54.29% 14.29% 11.43% 2.86% 5.71% 2.86% 2.86% 2.86% 5.71% 2.86% 2.86% 2.86% 2.86% 2.86% 2.86% 2.86% 2.86%

BUGGY_REPLACE_CONJUNCTION_BY_IMPLICATION Transformations combined T16: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 T15: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 T19: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2 T40: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T402: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 )

94 89 69 18 8 1 2

1.54% 94.68% 73.40% 19.15% 8.51% 1.06% 2.13%

31 26 19 6 1 1 1

1.97% 83.87% 61.29% 19.35% 3.23% 3.23% 3.23%

BUGGY_REPLACE_DISJUNCTION_BY_XOR Transformations combined T374: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

91 91 91

1.49% 100.00% 100.00%

2 2 2

0.13% 100.00% 100.00%

BUGGY_REPLACE_CONJUNCTION_BY_DISJUNCTION Transformations combined T4: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛 T181: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T58: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 T73: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 ∨ 𝑌3 ∨ 𝑌2 T392: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌3 ∨ 𝑌1 ∨ 𝑌2 T263: ¬𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) T20: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → 𝑌1

70 70 37 24 4 15 9 8 1

1.15% 100.00% 52.86% 34.29% 5.71% 21.43% 12.86% 11.43% 1.43%

22 22 14 8 2 1 2 4 1

1.40% 100.00% 63.64% 36.36% 9.09% 4.55% 9.09% 18.18% 4.55%

BUGGY_PULL_NEGATION_CONJUNCTION

1.10%

0.83%

T218: 𝑌1 → ((𝑌2 ∧ 𝑌3 ) ∨ (¬𝑌2 ∧ ¬𝑌3 )) ⇝ 𝑌1 → (𝑌2 → 𝑌3 )

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67 67 61

100.00% 100.00% 91.04%

13 13 12

100.00% 100.00% 92.31%

BUGGY_REPLACE_CONJUNCTION_BY_REVERSED_IMPLICATION Transformations combined T16: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 T30: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T15: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 T19: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2 T40: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T26: 𝑌1 ∧ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1

48 46 28 10 16 14 2 2

0.79% 95.83% 58.33% 20.83% 33.33% 29.17% 4.17% 4.17%

18 16 11 2 3 1 2 2

1.15% 88.89% 61.11% 11.11% 16.67% 5.56% 11.11% 11.11%

BUGGY_REPLACE_NOR_BY_CONJUNCTION Transformations combined T2: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T226: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T56: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 T331: ¬ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T215: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1

47 47 40 31 6 5 1

0.77% 100.00% 85.11% 65.96% 12.77% 10.64% 2.13%

19 19 13 9 5 4 1

1.21% 100.00% 68.42% 47.37% 26.32% 21.05% 5.26%

BUGGY_REPLACE_XOR_BY_NAND Transformations combined ⌈ = T88: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 ⌊ = T337: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T141: ¬𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T89: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ∧ 𝑌1

44 44 24 24 19 1

0.72% 100.00% 54.55% 54.55% 43.18% 2.27%

5 5 2 2 2 1

0.32% 100.00% 40.00% 40.00% 40.00% 20.00%

BUGGY_REPLACE_DISJUNCTION_BY_NAND Transformations combined T364: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T373: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

39 39 39 35

0.64% 100.00% 100.00% 89.74%

8 8 8 6

0.51% 100.00% 100.00% 75.00%

BUGGY_REPLACE_NAND_BY_CONJUNCTION Transformations combined T363: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⌈ = T297: (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → 𝑌3 ⌊ = T301: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T305: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⌈ = T170: 𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1 ⌊ = T185: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 ⌈ = T317: ¬𝑌1 ⇝ 𝑌1 ⌊ = T330: ¬ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛

36 35 25 22 22 4 9 9 1 1

0.59% 97.22% 69.44% 61.11% 61.11% 11.11% 25.00% 25.00% 2.78% 2.78%

19 18 13 10 10 1 4 4 1 1

1.21% 94.74% 68.42% 52.63% 52.63% 5.26% 21.05% 21.05% 5.26% 5.26%

BUGGY_REPLACE_CONJUNCTION_BY_EQUIVALENCE Transformations combined T12: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T13: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1

35 35 18 17

0.57% 100.00% 51.43% 48.57%

10 10 4 6

0.64% 100.00% 40.00% 60.00%

BUGGY_REPLACE_NOR_BY_NAND Transformations combined ⌈ = T355: ¬ (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⌊ = T362: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⌈ = T357: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌊ = T367: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2

26 26 26 26 22 22

0.43% 100.00% 100.00% 100.00% 84.62% 84.62%

7 7 7 7 6 6

0.45% 100.00% 100.00% 100.00% 85.71% 85.71%

BUGGY_ADD_OPERAND_TO_CONJUNCTION Transformations combined

23 0

0.38% 0.00%

7 0

0.45% 0.00%

Transformations combined T57: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T63: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

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BUGGY_REPLACE_IMPLICATION_BY_CONJUNCTION Transformations combined T166: 𝑌1 → 𝑌2 ⇝ 𝑌2 ∧ 𝑌1 T165: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T246: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3 T351: ¬ (𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T262: ¬𝑌1 → ¬ (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1

21 20 14 5 1 1 1

0.34% 95.24% 66.67% 23.81% 4.76% 4.76% 4.76%

10 9 3 5 1 1 1

0.64% 90.00% 30.00% 50.00% 10.00% 10.00% 10.00%

BUGGY_REPLACE_CONJUNCTION_BY_NAND Transformations combined ⌈ = T1: 𝑌1 ⇝ ¬𝑌1 ⌊ = T3: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T229: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

21 19 14 14 5

0.34% 90.48% 66.67% 66.67% 23.81%

7 5 4 4 1

0.45% 71.43% 57.14% 57.14% 14.29%

BUGGY_REPLACE_EQUIVALENCE_BY_XOR Transformations combined T1: 𝑌1 ⇝ ¬𝑌1

21 21 21

0.34% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_CONJUNCTION Transformations combined T89: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ∧ 𝑌1 T88: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T191: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∧ 𝑌3 ) T339: ¬ (𝑌1 ↔ ¬𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2

18 16 9 6 2 1

0.29% 88.89% 50.00% 33.33% 11.11% 5.56%

11 10 4 5 1 1

0.70% 90.91% 36.36% 45.45% 9.09% 9.09%

BUGGY_REPLACE_IMPLICATION_BY_NAND Transformations combined T174: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T379: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T175: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T260: ¬𝑌1 → ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

18 18 8 7 2 1

0.29% 100.00% 44.44% 38.89% 11.11% 5.56%

5 5 2 1 1 1

0.32% 100.00% 40.00% 20.00% 20.00% 20.00%

BUGGY_REPLACE_XOR_BY_REVERSED_IMPLICATION Transformations combined T139: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T336: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 T343: ¬ (¬𝑌1 ↔ ¬𝑌2 ) ⇝ 𝑌1 → 𝑌2

17 17 8 8 1

0.28% 100.00% 47.06% 47.06% 5.88%

4 4 2 1 1

0.25% 100.00% 50.00% 25.00% 25.00%

BUGGY_REPLACE_NOR_BY_DISJUNCTION Transformations combined T172: 𝑌1 → 𝑌2 ⇝ ¬𝑌2 → 𝑌1 T317: ¬𝑌1 ⇝ 𝑌1 ⌈ = T198: 𝑌1 → ¬𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T209: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T81: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌3 ∨ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 )

15 14 10 4 3 3 1

0.25% 93.33% 66.67% 26.67% 20.00% 20.00% 6.67%

9 8 5 3 2 2 1

0.57% 88.89% 55.56% 33.33% 22.22% 22.22% 11.11%

BUGGY_ADD_PREMISE_FOR_IMPLICATION Transformations combined

13 0

0.21% 0.00%

3 0

0.19% 0.00%

BUGGY_REPLACE_NAND_BY_DISJUNCTION Transformations combined T219: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⌈ = T201: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) | = T319: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⌊ = T325: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∨ 𝑌2 T265: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) T430: ¬ (𝑌1 ∧ 𝑌2 ) ∨ ¬ (𝑌1 ∧ 𝑌3 ) ∨ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

12 9 4 3 3 3 1 1

0.20% 75.00% 33.33% 25.00% 25.00% 25.00% 8.33% 8.33%

8 6 2 2 2 2 1 1

0.51% 75.00% 25.00% 25.00% 25.00% 25.00% 12.50% 12.50%

BUGGY_REPLACE_DISJUNCTION_BY_REVERSED_IMPLICATION

0.20%

0.38%

Continued on next page 51

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Transformations combined T371: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → 𝑌1 T432: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T369: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → 𝑌2

11 7 2 4

91.67% 58.33% 16.67% 33.33%

5 4 2 1

83.33% 66.67% 33.33% 16.67%

BUGGY_REPLACE_NAND_BY_XOR Transformations combined ⌈ = T12: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 ⌊ = T324: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T429: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T13: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1

12 12 9 9 2 1

0.20% 100.00% 75.00% 75.00% 16.67% 8.33%

3 3 1 1 1 1

0.19% 100.00% 33.33% 33.33% 33.33% 33.33%

BUGGY_REPLACE_EQUIVALENCE_BY_NAND Transformations combined T97: 𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

11 10 10

0.18% 90.91% 90.91%

5 4 4

0.32% 80.00% 80.00%

BUGGY_REPLACE_NOR_BY_XOR Transformations combined ⌈ = T358: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) ⌊ = T368: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T64: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 )

10 10 6 6 4

0.16% 100.00% 60.00% 60.00% 40.00%

2 2 1 1 1

0.13% 100.00% 50.00% 50.00% 50.00%

BUGGY_ADD_EQUIVALENCE Transformations combined

9 0

0.15% 0.00%

3 0

0.19% 0.00%

BUGGY_REPLACE_XOR_BY_NOR Transformations combined T334: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2

9 9 9

0.15% 100.00% 100.00%

3 3 3

0.19% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_DISJUNCTION Transformations combined T192: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T145: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 )

8 7 4 3

0.13% 87.50% 50.00% 37.50%

4 3 1 2

0.25% 75.00% 25.00% 50.00%

BUGGY_REPLACE_XOR_BY_IMPLICATION Transformations combined T335: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌1 → 𝑌2 T137: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2

8 8 6 2

0.13% 100.00% 75.00% 25.00%

3 3 1 2

0.19% 100.00% 33.33% 66.67%

BUGGY_REPLACE_DISJUNCTION_BY_IMPLICATION Transformations combined T371: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → 𝑌1 T71: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T418: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T369: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → 𝑌2

7 6 3 1 1 3

0.11% 85.71% 42.86% 14.29% 14.29% 42.86%

6 5 3 1 1 2

0.38% 83.33% 50.00% 16.67% 16.67% 33.33%

BUGGY_ADD_OPERAND_TO_DISJUNCTION Transformations combined

7 0

0.11% 0.00%

5 0

0.32% 0.00%

BUGGY_REPLACE_XOR_BY_DISJUNCTION Transformations combined T206: 𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌3 ∨ 𝑌2 ) T344: ¬ (¬𝑌1 ↔ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 )

7 6 5 1

0.11% 85.71% 71.43% 14.29%

4 3 2 1

0.25% 75.00% 50.00% 25.00%

BUGGY_REPLACE_IMPLICATION_BY_XOR Transformations combined T176: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T381: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) T347: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌2 ↔ 𝑌1

7 7 3 3 1

0.11% 100.00% 42.86% 42.86% 14.29%

3 3 1 1 1

0.19% 100.00% 33.33% 33.33% 33.33%

BUGGY_MOVE_PARENTHESIS2

0.08%

0.25%

Continued on next page 52

Hand-picked mistake / Transformations

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Transformations combined T306: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T214: 𝑌1 → ¬ (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

4 3 1

80.00% 60.00% 20.00%

3 2 1

75.00% 50.00% 25.00%

BUGGY_REPLACE_CONJUNCTION_BY_XOR Transformations combined T17: 𝑌1 ∧ 𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 )

5 5 5

0.08% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REPLACE_EQUIVALENCE_BY_NOR Transformations combined T243: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3

4 2 2

0.07% 50.00% 50.00%

3 2 2

0.19% 66.67% 66.67%

BUGGY_MOVE_PARENTHESIS1 Transformations combined T23: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T359: ¬ (𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 )

4 4 3 1

0.07% 100.00% 75.00% 25.00%

2 2 1 1

0.13% 100.00% 50.00% 50.00%

BUGGY_REPLACE_IMPLICATION_BY_DISJUNCTION Transformations combined T217: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3

4 4 4

0.07% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REMOVE_FIRST_OPERAND_FROM_CONJUNCTION Transformations combined T190: 𝑌1 → ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 )

2 1 1

0.03% 50.00% 50.00%

2 1 1

0.13% 50.00% 50.00%

BUGGY_REPLACE_DISJUNCTION_BY_EQUIVALENCE Transformations combined T368: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2

2 1 1

0.03% 50.00% 50.00%

2 1 1

0.13% 50.00% 50.00%

BUGGY_REPLACE_DISJUNCTION_BY_NOR Transformations combined

2 0

0.03% 0.00%

2 0

0.13% 0.00%

BUGGY_ADD_CONCLUSION_FOR_IMPLICATION Transformations combined

1 0

0.02% 0.00%

1 0

0.06% 0.00%

BUGGY_EXPLICIFY_IMPLICATION_1 Transformations combined T372: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → ¬𝑌1

1 1 1

0.02% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_PULL_NEGATION_DISJUNCTION Transformations combined T314: (¬𝑌1 ∨ ¬𝑌2 ) → ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 )

1 1 1

0.02% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REMOVE_FIRST_FROM_EQUIVALENCE Transformations combined

1 0

0.02% 0.00%

1 0

0.06% 0.00%

BUGGY_REMOVE_FIRST_OPERAND_FROM_DISJUNCTION Transformations combined

1 0

0.02% 0.00%

1 0

0.06% 0.00%

BUGGY_REMOVE_LATER_OPERAND_FROM_DISJUNCTION Transformations combined

1 0

0.02% 0.00%

1 0

0.06% 0.00%

BUGGY_REPLACE_NAND_BY_EQUIVALENCE Transformations combined T361: ¬ (¬𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌2 ↔ 𝑌1 )

1 1 1

0.02% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_EQUIVALENCE Transformations combined T333: ¬ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌2 ↔ 𝑌1 )

1 1 1

0.02% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

BUGGY_REPLACE_NOR_BY_REVERSED_IMPLICATION Transformations combined T350: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌2 → ¬𝑌1

1 1 1

0.02% 100.00% 100.00%

1 1 1

0.06% 100.00% 100.00%

Continued on next page 53

Hand-picked mistake / Transformations

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BUGGY_REPLACE_XOR_BY_CONJUNCTION Transformations combined

1 0

0.02% 0.00%

1 0

0.06% 0.00%

1736 918 114 70 66 31 28 22 22 21 9 1 20 20 18 18 16 14 14 3 12 10 9 9 8 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4

28.43% 52.88% 6.57% 4.03% 3.80% 1.79% 1.61% 1.27% 1.27% 1.21% 0.52% 0.06% 1.15% 1.15% 1.04% 1.04% 0.92% 0.81% 0.81% 0.17% 0.69% 0.58% 0.52% 0.52% 0.46% 0.40% 0.40% 0.40% 0.40% 0.40% 0.40% 0.35% 0.35% 0.35% 0.35% 0.35% 0.35% 0.35% 0.35% 0.35% 0.29% 0.29% 0.29% 0.29% 0.29% 0.29% 0.29% 0.23% 0.23% 0.23% 0.23%

910 366 20 16 6 9 6 7 2 4 1 1 9 8 1 1 1 7 4 2 7 1 6 1 1 4 3 3 3 1 1 4 4 4 4 3 3 2 1 1 4 4 2 1 1 1 1 4 3 3 2

57.89% 40.22% 2.20% 1.76% 0.66% 0.99% 0.66% 0.77% 0.22% 0.44% 0.11% 0.11% 0.99% 0.88% 0.11% 0.11% 0.11% 0.77% 0.44% 0.22% 0.77% 0.11% 0.66% 0.11% 0.11% 0.44% 0.33% 0.33% 0.33% 0.11% 0.11% 0.44% 0.44% 0.44% 0.44% 0.33% 0.33% 0.22% 0.11% 0.11% 0.44% 0.44% 0.22% 0.11% 0.11% 0.11% 0.11% 0.44% 0.33% 0.33% 0.22%

Unexplained by ALL hand errors Transformations combined T407: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T202: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T107: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T390: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ∧ ¬𝑌1 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T365: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T86: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T222: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T105: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → 𝑌1 T124: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T144: ¬𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 T220: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T186: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 ⌈ = T271: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → ¬𝑌3 ⌊ = T275: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T249: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T216: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T224: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T239: (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T111: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T393: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T398: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T227: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T228: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T70: (𝑌1 ∨ 𝑌2 ∨ ¬𝑌3 ) ∧ (𝑌1 ∨ 𝑌3 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ 𝑌3 ∨ ¬𝑌1 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T22: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌈ = T149: ¬ (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T150: ¬ (𝑌2 ∧ 𝑌3 ) ↔ 𝑌1 ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T188: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) T252: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → ¬𝑌2 T38: 𝑌1 ∧ ¬𝑌2 ∧ 𝑌2 ∧ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) ⌈ = T68: (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ∧ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⌊ = T79: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T437: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T300: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T308: ((𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T8: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T5: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 ∧ 𝑌3 ) T380: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T84: (𝑌1 ∨ 𝑌2 ) ∧ (¬𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T277: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T332: ¬ (¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T42: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T99: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T199: 𝑌1 → ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T211: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T264: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 T93: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 T309: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T95: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1

Continued on next page 54

Hand-picked mistake / Transformations T24: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T122: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T182: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) T203: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T223: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T240: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T241: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) T279: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) T115: 𝑌1 ↔ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T258: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T142: ¬𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T299: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T307: ((𝑌1 ∧ 𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T397: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T408: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 ∨ 𝑌2 T31: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T46: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T59: ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌3 ) → ¬𝑌2 T98: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T179: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T180: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 → 𝑌2 ) ∧ 𝑌1 T242: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T247: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 T248: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T417: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T18: 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T25: 𝑌1 ∧ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T60: ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T77: (𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T100: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T108: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T114: 𝑌1 ↔ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T163: (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T230: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T234: (𝑌1 ∧ ¬𝑌2 ) → (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T237: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T282: ¬ (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T302: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T304: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T409: (𝑌1 ∧ ¬𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T21: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 T34: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ 𝑌4 ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T52: (𝑌1 → ¬𝑌2 ) ∧ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T55: (¬𝑌1 → 𝑌2 ) ∧ (¬𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T69: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T159: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ↔ 𝑌4 ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ∨ 𝑌4 ) T161: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T183: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T189: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌2 ) → 𝑌1 T193: 𝑌1 → (𝑌2 ↔ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T194: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ 𝑌2 → (𝑌3 → 𝑌1 ) T207: 𝑌1 → ¬ (¬𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T254: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T259: ¬𝑌1 → (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1

#dst

%dst

4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.23% 0.23% 0.23% 0.23% 0.23% 0.23% 0.23% 0.23% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.17% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12%

1 1 1 1 1 1 1 1 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.33% 0.33% 0.22% 0.22% 0.22% 0.22% 0.22% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11%

Continued on next page 55

Hand-picked mistake / Transformations T273: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T345: ¬ ((𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ) ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T349: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌2 → 𝑌1 T360: ¬ ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T381: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) T413: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌1 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T422: (¬𝑌1 → 𝑌2 ) ∨ (¬𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T6: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 ) T7: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T10: 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ⇝ (𝑌3 ∧ 𝑌1 ) ∨ (𝑌3 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ) T11: 𝑌1 ∧ 𝑌2 ∧ ¬ (𝑌3 ∧ 𝑌4 ) ∧ ¬ (𝑌5 ∧ 𝑌6 ) ⇝ 𝑌1 ∧ 𝑌2 ⌈ = T20: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → 𝑌1 ⌊ = T62: ¬𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 T27: 𝑌1 ∧ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T35: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ ¬𝑌2 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T37: 𝑌1 ∧ 𝑌2 ∧ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T39: 𝑌1 ∧ ¬𝑌2 ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T41: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌1 ⇝ 𝑌2 → 𝑌1 T43: (𝑌1 ↔ ¬𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) T44: (𝑌1 ↔ ¬𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T47: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 → 𝑌3 ) T48: (𝑌1 → 𝑌2 ) ∧ (𝑌1 → 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌3 ) → 𝑌2 T49: (𝑌1 → 𝑌2 ) ∧ (𝑌2 → 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T50: (𝑌1 → 𝑌3 ) ∧ (𝑌2 → 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3 T51: (𝑌1 → ¬𝑌2 ) ∧ (𝑌1 → ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 T53: (𝑌1 → ¬𝑌2 ) ∧ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 ) T54: (𝑌1 → ¬𝑌2 ) ∧ ¬𝑌3 ⇝ (¬𝑌3 ∧ ¬𝑌2 ) → 𝑌1 T65: ¬ (𝑌1 ∧ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 T78: (𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 ⇝ (𝑌3 ∧ 𝑌1 ) ∨ (𝑌3 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ) T80: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T82: (𝑌1 ∨ 𝑌2 ) ∧ (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T83: (𝑌1 ∨ 𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T85: (𝑌1 ∨ ¬𝑌1 ) ∧ (𝑌2 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T96: 𝑌1 ↔ 𝑌2 ⇝ ¬𝑌1 → 𝑌2 T101: 𝑌1 ↔ (𝑌2 ∧ ¬𝑌3 ) ⇝ (¬𝑌3 ∧ ¬𝑌2 ) → 𝑌1 T109: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 ∨ 𝑌1 T110: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 → 𝑌2 ) ∧ 𝑌1 T112: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) T119: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 T123: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) T125: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) T126: (𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T127: (𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T128: (𝑌1 ∧ ¬𝑌2 ) ↔ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T129: (𝑌1 ∧ ¬𝑌2 ) ↔ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T130: (𝑌1 ↔ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T131: (𝑌1 ↔ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) T132: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) T134: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) T156: ¬ (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T160: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T162: (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 T178: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 T195: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ 𝑌3 → (𝑌2 → 𝑌1 )

#dst

%dst

2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.12% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06%

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11% 0.11%

Continued on next page 56

Hand-picked mistake / Transformations T196: 𝑌1 → (¬𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T197: 𝑌1 → ¬𝑌2 ⇝ 𝑌2 ↔ 𝑌1 T200: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T214: 𝑌1 → ¬ (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T221: (𝑌1 ∧ 𝑌2 ) → 𝑌1 ⇝ 𝑌2 → 𝑌1 T231: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T232: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∨ 𝑌1 ) T235: (𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T238: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T244: (¬𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T245: (¬𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 T250: (𝑌1 → ¬𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 T253: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T257: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌1 ) → 𝑌2 T266: ¬𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T278: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 T283: ¬ (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∨ 𝑌1 ) T288: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T298: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) → 𝑌4 ⇝ 𝑌2 → (𝑌1 ∨ 𝑌4 ∨ 𝑌3 ) T306: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T310: ((𝑌1 ↔ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T311: ((𝑌1 ↔ 𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T316: (𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T321: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T322: ¬ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ⇝ ¬𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 T328: ¬ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T329: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T340: ¬ (𝑌1 ↔ ¬𝑌2 ) ⇝ 𝑌1 → 𝑌2 T341: ¬ (𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 )) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T342: ¬ (¬𝑌1 ↔ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 T346: ¬ ((𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ) ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T352: ¬ (𝑌1 → ¬𝑌2 ) ⇝ 𝑌2 ↔ 𝑌1 T366: 𝑌1 ∨ 𝑌2 ∨ ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T370: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 T377: 𝑌1 ∨ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 T385: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T386: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T387: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ) ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T388: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 T389: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) T395: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 ) T396: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) T400: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T401: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T403: (𝑌1 ∧ 𝑌2 ) ∨ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T404: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ↔ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T405: (𝑌1 ∧ 𝑌2 ) ∨ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T410: (𝑌2 ∧ 𝑌1 ) ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) T412: (𝑌1 ↔ 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3 T414: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌3 ↔ 𝑌1 ) ∨ (𝑌3 ↔ 𝑌2 ) ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) T415: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌3 ↔ 𝑌1 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 T419: (𝑌1 → 𝑌2 ) ∨ (𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T420: (𝑌1 → ¬𝑌2 ) ∨ (𝑌2 → ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T421: (𝑌1 → ¬𝑌2 ) ∨ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 )

#dst

%dst

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06% 0.06%

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Continued on next page 57

Hand-picked mistake / Transformations T423: ((𝑌1 ∨ 𝑌2 ) → 𝑌3 ) ∨ 𝑌1 ∨ 𝑌2 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T427: ¬𝑌1 ∨ ¬𝑌2 ∨ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T434: 𝑌1 ∨ 𝑌2 ∨ 𝑌1 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) T436: 𝑌1 ∨ ¬𝑌2 ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T438: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T439: ¬𝑌1 ∨ ¬𝑌2 ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Whole data set ANY hand error explained ANY transformation explained Only explained by hand error Only explained by transformation

#dst

%dst

1 1 1 1 1 1

0.06% 0.06% 0.06% 0.06% 0.06% 0.06%

1 1 1 1 1 1

0.11% 0.11% 0.11% 0.11% 0.11% 0.11%

6106 4370 5156 132 918

100.00% 71.57% 84.44% 2.16% 15.03%

1572 662 955 73 366

100.00% 42.11% 60.75% 4.64% 23.28%

B.3

Propositional Logic: Hierarchical Clustering of Transformations Table 38: Transformation hierarchy clusters Transformations Whole data set ANY transformation explained

#dst

%dst

6106 5156

100.00% 84.44%

1572 955

100.00% 60.75%

1343 172

21.99% 2.82%

80 37

5.09% 2.35%

426 201 56 13 1

6.98% 3.29% 0.92% 0.21% 0.02%

31 3 3 3 1

1.97% 0.19% 0.19% 0.19% 0.06%

360

5.90%

1.40%

298

4.88%

1.27%

278 14

4.55% 0.23%

30 6

1.91% 0.38%

201 153 150

3.29% 2.51% 2.46%

23 16 15

1.46% 1.02% 0.95%

161 67 17 14

2.64% 1.10% 0.28% 0.23%

29 7 7 4

1.84% 0.45% 0.45% 0.25%

154 140 35 35 22 17 44 35 35 26 22

2.52% 2.29% 0.57% 0.57% 0.36% 0.28% 0.72% 0.57% 0.57% 0.43% 0.36%

41 35 9 9 6 6 13 9 9 7 6

2.61% 2.23% 0.57% 0.57% 0.38% 0.38% 0.83% 0.57% 0.57% 0.45% 0.38%

114

1.87%

1.27%

Cluster 1 T169: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T184: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → 𝑌1 Cluster 2 T94: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T120: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T140: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1 T164: (¬𝑌1 ∨ ¬𝑌2 ) ↔ ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) T133: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) Cluster 3 T168: 𝑌1 → 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 Cluster 4 T167: 𝑌1 → 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 Cluster 5 T92: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 T121: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3 Cluster 6 T318: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T323: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 T274: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 7 T1: 𝑌1 ⇝ ¬𝑌1 T225: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T171: 𝑌1 → 𝑌2 ⇝ ¬𝑌1 → 𝑌2 T3: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) Cluster 8 T362: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 T367: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 ⌈ = T313: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T428: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2 T357: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T303: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 T424: ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⌈ = T313: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T428: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬𝑌1 ∧ ¬𝑌2 T355: ¬ (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T357: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 9 T407: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 )

Continued on next page 59

Transformations

#dst

%dst

97 10

1.59% 0.16%

30 2

1.91% 0.13%

1.49%

0.13%

89 15 3 5 5 2 1 1 1 1

1.46% 0.25% 0.05% 0.08% 0.08% 0.03% 0.02% 0.02% 0.02% 0.02%

23 7 2 4 1 2 1 1 1 1

1.46% 0.45% 0.13% 0.25% 0.06% 0.13% 0.06% 0.06% 0.06% 0.06%

1.15%

1.02%

67 61

1.10% 1.00%

13 12

0.83% 0.76%

1.08%

0.38%

0.90%

1.08%

40 31

0.66% 0.51%

13 9

0.83% 0.57%

39 35

0.64% 0.57%

8 6

0.51% 0.38%

37 24 4

0.61% 0.39% 0.07%

14 8 2

0.89% 0.51% 0.13%

34 22 3

0.56% 0.36% 0.05%

9 2 3

0.57% 0.13% 0.19%

0.51%

0.57%

Cluster 10 T16: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 T30: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 11 T374: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 12 T317: ¬𝑌1 ⇝ 𝑌1 T198: 𝑌1 → ¬𝑌2 ⇝ 𝑌1 → 𝑌2 T209: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T104: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T61: ¬𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 ∧ ¬𝑌2 T251: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → 𝑌2 T81: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌3 ∨ 𝑌4 ) ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) T135: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T330: ¬ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T348: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → 𝑌2 Cluster 13 T202: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 14 T57: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T63: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 15 T107: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 16 T210: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 17 T2: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 T226: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 18 T364: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) T373: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 19 T4: 𝑌1 ∧ . . . ∧ 𝑌𝑛 ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛 T181: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T58: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ ¬𝑌1 ∨ . . . ∨ ¬𝑌𝑛 Cluster 20 T15: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 T19: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ¬𝑌2 T40: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 21 T390: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ∧ ¬𝑌1 ) (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

⇝

(𝑌1 ∧ 𝑌2 ) ∨

Cluster 22 Continued on next page 60

Transformations

#dst

%dst

T88: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T337: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) T191: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∧ 𝑌3 )

30 24 2

0.49% 0.39% 0.03%

7 2 1

0.45% 0.13% 0.06%

0.46%

0.38%

27 9

0.44% 0.15%

5 1

0.32% 0.06%

25 22 22 4

0.41% 0.36% 0.36% 0.07%

13 10 10 1

0.83% 0.64% 0.64% 0.06%

0.36%

0.45%

0.36%

0.13%

21 9 1

0.34% 0.15% 0.02%

4 1 1

0.25% 0.06% 0.06%

0.33%

0.57%

0.33%

0.51%

19 9

0.31% 0.15%

5 4

0.32% 0.25%

0.31%

0.13%

0.29%

0.45%

18 18

0.29% 0.29%

1 1

0.06% 0.06%

0.26%

0.06%

0.25%

0.06%

0.25%

0.06%

Cluster 23 T365: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 24 T12: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T324: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 25 T363: 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⇝ ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⌈ = T297: (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → 𝑌3 ⌊ = T301: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T305: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 26 T86: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 27 T222: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 28 T105: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → 𝑌1 T124: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) T144: ¬𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 29 T220: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 30 T186: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 31 T170: 𝑌1 → 𝑌2 ⇝ 𝑌2 → ¬𝑌1 T185: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 Cluster 32 T141: ¬𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 33 T13: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 Cluster 34 ⌈ = T271: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) → 𝑌3 ⇝ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) → ¬𝑌3 ⌊ = T275: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 35 T249: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 36 T73: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 ∨ 𝑌3 ∨ 𝑌2 Cluster 37 T291: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3

Continued on next page 61

Transformations

#dst

%dst

0.23%

0.45%

14 3

0.23% 0.05%

4 2

0.25% 0.13%

0.23%

0.19%

0.20%

0.45%

10 2 1 1

0.16% 0.03% 0.02% 0.02%

7 2 1 1

0.45% 0.13% 0.06% 0.06%

0.16%

0.32%

0.16%

0.32%

0.16%

0.25%

0.16%

0.06%

0.15%

0.38%

0.15%

0.19%

0.15%

0.13%

0.15%

0.06%

0.13%

0.25%

0.13%

0.06%

Cluster 38 T216: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 39 T224: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T239: (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) Cluster 40 T166: 𝑌1 → 𝑌2 ⇝ 𝑌2 ∧ 𝑌1 Cluster 41 T111: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 42 T371: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → 𝑌1 T432: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) T71: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 T418: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) Cluster 43 T89: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ∧ 𝑌1 Cluster 44 T172: 𝑌1 → 𝑌2 ⇝ ¬𝑌2 → 𝑌1 Cluster 45 T97: 𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 46 T393: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 47 T398: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 48 T334: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 Cluster 49 T392: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌3 ∨ 𝑌1 ∨ 𝑌2 Cluster 50 T227: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 51 T263: ¬𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) Cluster 52 T139: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 Cluster 53 T174: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 54 T228: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) Cluster 55 Continued on next page 62

Transformations

#dst

%dst

T293: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 )

0.13%

0.06%

0.13%

0.06%

0.11%

0.45%

0.11%

0.25%

0.11%

0.25%

0.11%

0.19%

7 7

0.11% 0.11%

3 3

0.19% 0.19%

0.11%

0.19%

7 6

0.11% 0.10%

2 1

0.13% 0.06%

0.11%

0.06%

0.11%

0.06%

0.11%

0.06%

6 5

0.10% 0.08%

5 4

0.32% 0.25%

0.10%

0.25%

6 6

0.10% 0.10%

4 4

0.25% 0.25%

0.10%

0.25%

0.10%

0.19%

0.10%

0.19%

Cluster 56 T336: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 Cluster 57 T113: 𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 58 T70: (𝑌1 ∨ 𝑌2 ∨ ¬𝑌3 ) ∧ (𝑌1 ∨ 𝑌3 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ 𝑌3 ∨ ¬𝑌1 ) (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

⇝

(𝑌1 ∧ 𝑌2 ) ∨

Cluster 59 T233: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 Cluster 60 T22: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 61 ⌈ = T149: ¬ (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⌊ = T150: ¬ (𝑌2 ∧ 𝑌3 ) ↔ 𝑌1 ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 62 T369: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 63 T368: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ↔ 𝑌2 T358: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 64 T188: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) Cluster 65 T252: ¬𝑌1 → 𝑌2 ⇝ 𝑌1 → ¬𝑌2 Cluster 66 T379: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 67 T56: ¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ⇝ 𝑌1 ∧ . . . ∧ 𝑌𝑛 T331: ¬ (¬𝑌1 ∧ . . . ∧ ¬𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) Cluster 68 T38: 𝑌1 ∧ ¬𝑌2 ∧ 𝑌2 ∧ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 69 ⌈ = T68: (𝑌1 ∨ . . . ∨ 𝑌𝑛 ) ∧ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⌊ = T79: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 70 T437: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 71 T280: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) Cluster 72 T300: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3

Continued on next page 63

Transformations

#dst

%dst

0.10%

0.19%

0.10%

0.13%

0.10%

0.13%

0.10%

0.06%

0.10%

0.06%

0.10%

0.06%

5 1 1

0.08% 0.02% 0.02%

5 1 1

0.32% 0.06% 0.06%

0.08%

0.25%

0.08%

0.25%

0.08%

0.13%

0.08%

0.13%

0.08%

0.13%

0.08%

0.06%

0.08%

0.06%

0.08%

0.06%

0.08%

0.06%

0.08%

0.06%

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Cluster 73 T308: ((𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 74 T8: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 75 T106: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 76 T5: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 ∧ 𝑌3 ) Cluster 77 T335: ¬ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌1 → 𝑌2 Cluster 78 T380: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 79 T165: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T246: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3 T351: ¬ (𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 80 T84: (𝑌1 ∨ 𝑌2 ) ∧ (¬𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 81 T277: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) Cluster 82 T206: 𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌3 ∨ 𝑌2 ) Cluster 83 T332: ¬ (¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 84 T381: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) Cluster 85 T17: 𝑌1 ∧ 𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) Cluster 86 T42: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) Cluster 87 T99: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 88 T138: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 Cluster 89 T199: 𝑌1 → ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 90 T211: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1

Continued on next page 64

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Cluster 91 T229: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 92 T264: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 Cluster 93 T93: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 Cluster 94 T153: ¬ (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 T155: ¬ (𝑌1 ∨ 𝑌2 ) ↔ ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 95 T306: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 96 T309: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 97 T95: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 98 T219: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 99 T24: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 100 T64: ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 101 T122: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 102 T182: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) Cluster 103 T192: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 104 T203: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 Cluster 105 T217: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 ∨ 𝑌2 ∨ 𝑌3 Cluster 106 T223: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) Cluster 107 T240: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 108 T241: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) Cluster 109 Continued on next page 65

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Cluster 110 T279: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∧ ¬𝑌2 ) Cluster 111 T378: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 112 T115: 𝑌1 ↔ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 113 T258: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 114 T142: ¬𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 115 T145: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 116 ⌈ = T201: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) | = T319: ¬ (𝑌1 ∧ . . . ∧ 𝑌𝑛 ) ⇝ 𝑌1 ∨ . . . ∨ 𝑌𝑛 ⌊ = T325: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∨ 𝑌2 Cluster 117 T292: ¬ (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 118 T299: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 119 T307: ((𝑌1 ∧ 𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 120 T397: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 121 T408: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 ∨ 𝑌2 Cluster 122 T23: 𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 123 T31: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) Cluster 124 T46: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 125 T59: ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌3 ) → ¬𝑌2 Cluster 126 T98: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 127 T143: ¬𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 → 𝑌2

Continued on next page 66

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Cluster 128 T146: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 129 T176: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 130 T179: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 Cluster 131 T180: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 → 𝑌2 ) ∧ 𝑌1 Cluster 132 T242: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 133 T247: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 Cluster 134 T248: (𝑌1 → 𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 135 T417: (𝑌1 → 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌2 → 𝑌1 ) ∧ 𝑌3 Cluster 136 T18: 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 137 T20: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → 𝑌1 T62: ¬𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 138 T25: 𝑌1 ∧ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 139 T26: 𝑌1 ∧ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 140 T60: ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 141 T77: (𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌1 ∧ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 142 T100: 𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 143 T108: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 144 T114: 𝑌1 ↔ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 145 T137: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 146 Continued on next page 67

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Cluster 147 T204: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) Cluster 148 T214: 𝑌1 → ¬ (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 149 T230: (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 150 T234: (𝑌1 ∧ ¬𝑌2 ) → (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 151 T237: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 152 T243: (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 Cluster 153 T282: ¬ (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 154 T302: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) Cluster 155 T304: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 156 T409: (𝑌1 ∧ ¬𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 157 T21: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 158 T34: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ 𝑌4 ⇝ (𝑌1 ∨ 𝑌2 ) ∧ (𝑌3 ∨ 𝑌4 ) Cluster 159 T52: (𝑌1 → ¬𝑌2 ) ∧ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 160 T55: (¬𝑌1 → 𝑌2 ) ∧ (¬𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 161 T69: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 162 T159: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ↔ 𝑌4 ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ∨ 𝑌4 ) Cluster 163 T161: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 164 T175: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 165 Continued on next page 68

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Cluster 166 T189: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌2 ) → 𝑌1 Cluster 167 T193: 𝑌1 → (𝑌2 ↔ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 168 T194: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ 𝑌2 → (𝑌3 → 𝑌1 ) Cluster 169 T207: 𝑌1 → ¬ (¬𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 170 T212: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) Cluster 171 T254: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 172 T255: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 173 T259: ¬𝑌1 → (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 174 T273: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 175 T345: ¬ ((𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ) ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 176 T349: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌2 → 𝑌1 Cluster 177 T360: ¬ ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 178 T382: 𝑌1 ∨ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∧ 𝑌3 ) Cluster 179 T402: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 ) Cluster 180 T413: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌1 ↔ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 181 T422: (¬𝑌1 → 𝑌2 ) ∨ (¬𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 182 T429: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 183 T6: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 ) Cluster 184 Continued on next page 69

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Cluster 185 T10: 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ⇝ (𝑌3 ∧ 𝑌1 ) ∨ (𝑌3 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ) Cluster 186 T11: 𝑌1 ∧ 𝑌2 ∧ ¬ (𝑌3 ∧ 𝑌4 ) ∧ ¬ (𝑌5 ∧ 𝑌6 ) ⇝ 𝑌1 ∧ 𝑌2 Cluster 187 T27: 𝑌1 ∧ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 Cluster 188 T35: 𝑌1 ∧ 𝑌2 ∧ 𝑌3 ∧ ¬𝑌2 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 189 T37: 𝑌1 ∧ 𝑌2 ∧ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 190 T39: 𝑌1 ∧ ¬𝑌2 ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 191 T41: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌1 ⇝ 𝑌2 → 𝑌1 Cluster 192 T43: (𝑌1 ↔ ¬𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) Cluster 193 T44: (𝑌1 ↔ ¬𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 194 T47: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌2 → (𝑌1 → 𝑌3 ) Cluster 195 T48: (𝑌1 → 𝑌2 ) ∧ (𝑌1 → 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌3 ) → 𝑌2 Cluster 196 T49: (𝑌1 → 𝑌2 ) ∧ (𝑌2 → 𝑌3 ) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 197 T50: (𝑌1 → 𝑌3 ) ∧ (𝑌2 → 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) → 𝑌3 Cluster 198 T51: (𝑌1 → ¬𝑌2 ) ∧ (𝑌1 → ¬𝑌3 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → 𝑌1 Cluster 199 T53: (𝑌1 → ¬𝑌2 ) ∧ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 200 T54: (𝑌1 → ¬𝑌2 ) ∧ ¬𝑌3 ⇝ (¬𝑌3 ∧ ¬𝑌2 ) → 𝑌1 Cluster 201 T65: ¬ (𝑌1 ∧ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 Cluster 202 T78: (𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 ⇝ (𝑌3 ∧ 𝑌1 ) ∨ (𝑌3 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ) Cluster 203 Continued on next page 70

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Cluster 204 T82: (𝑌1 ∨ 𝑌2 ) ∧ (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 205 T83: (𝑌1 ∨ 𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 206 T85: (𝑌1 ∨ ¬𝑌1 ) ∧ (𝑌2 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 207 T87: (¬𝑌1 ∨ ¬𝑌2 ) ∧ ¬𝑌3 ⇝ ¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 Cluster 208 T96: 𝑌1 ↔ 𝑌2 ⇝ ¬𝑌1 → 𝑌2 Cluster 209 T101: 𝑌1 ↔ (𝑌2 ∧ ¬𝑌3 ) ⇝ (¬𝑌3 ∧ ¬𝑌2 ) → 𝑌1 Cluster 210 T109: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 ∨ 𝑌1 Cluster 211 T110: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 → 𝑌2 ) ∧ 𝑌1 Cluster 212 T112: 𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) Cluster 213 T119: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 Cluster 214 T123: (𝑌1 ∧ 𝑌2 ) ↔ 𝑌3 ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) Cluster 215 T125: (𝑌1 ∧ 𝑌2 ) ↔ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) Cluster 216 T126: (𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 Cluster 217 T127: (𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 218 T128: (𝑌1 ∧ ¬𝑌2 ) ↔ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 219 T129: (𝑌1 ∧ ¬𝑌2 ) ↔ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 220 T130: (𝑌1 ↔ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 221 T131: (𝑌1 ↔ 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌2 ∧ 𝑌1 ) Cluster 222 Continued on next page 71

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T132: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌1 → (𝑌2 → 𝑌3 )

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Cluster 223 T134: (𝑌1 → 𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) Cluster 224 T136: ¬𝑌1 ↔ 𝑌2 ⇝ 𝑌2 ↔ 𝑌1 Cluster 225 T147: ¬𝑌1 ↔ ¬ (¬𝑌2 ) ⇝ 𝑌1 ↔ 𝑌2 Cluster 226 T151: ¬ (¬𝑌1 ) ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 Cluster 227 T156: ¬ (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 228 T157: ¬ (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 229 T160: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 230 T162: (𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ (¬𝑌2 ∧ ¬𝑌1 ) → 𝑌3 Cluster 231 T178: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 Cluster 232 T190: 𝑌1 → ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 233 T195: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ 𝑌3 → (𝑌2 → 𝑌1 ) Cluster 234 T196: 𝑌1 → (¬𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 235 T197: 𝑌1 → ¬𝑌2 ⇝ 𝑌2 ↔ 𝑌1 Cluster 236 T200: 𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 237 T205: 𝑌1 → ¬ (𝑌2 ∧ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌3 ∨ 𝑌2 ) Cluster 238 T215: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ (𝑌2 ∧ 𝑌3 ) → ¬𝑌1 Cluster 239 T218: 𝑌1 → ((𝑌2 ∧ 𝑌3 ) ∨ (¬𝑌2 ∧ ¬𝑌3 )) ⇝ 𝑌1 → (𝑌2 → 𝑌3 ) Cluster 240 T221: (𝑌1 ∧ 𝑌2 ) → 𝑌1 ⇝ 𝑌2 → 𝑌1 Cluster 241 Continued on next page 72

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Cluster 242 T232: (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∨ 𝑌1 ) Cluster 243 T235: (𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 244 T238: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 Cluster 245 T244: (¬𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 246 T245: (¬𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 Cluster 247 T250: (𝑌1 → ¬𝑌2 ) → 𝑌3 ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 248 T253: ¬𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 249 T257: ¬𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌3 ∧ ¬𝑌1 ) → 𝑌2 Cluster 250 T260: ¬𝑌1 → ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 251 T261: ¬𝑌1 → ¬ (𝑌2 ∧ ¬𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌3 ∨ ¬𝑌2 ) Cluster 252 T262: ¬𝑌1 → ¬ (𝑌2 → 𝑌3 ) ⇝ (𝑌2 → 𝑌3 ) ∧ 𝑌1 Cluster 253 T265: ¬𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ ¬𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) Cluster 254 T266: ¬𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 255 T278: ¬ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ (𝑌1 → 𝑌3 ) ∧ 𝑌2 Cluster 256 T281: ¬ (𝑌1 ∧ 𝑌2 ) → ¬ (¬𝑌3 ) ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 257 T283: ¬ (𝑌1 ↔ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌2 ∨ 𝑌1 ) Cluster 258 T288: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 259 T289: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 Cluster 260 Continued on next page 73

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Cluster 261 T310: ((𝑌1 ↔ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 )) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 262 T311: ((𝑌1 ↔ 𝑌2 ) ∨ 𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 263 T314: (¬𝑌1 ∨ ¬𝑌2 ) → ¬𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 264 T316: (𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ) → 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 265 T321: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 266 T322: ¬ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ⇝ ¬𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 Cluster 267 T327: ¬ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 268 T328: ¬ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 269 T329: ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 270 T333: ¬ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌2 ↔ 𝑌1 ) Cluster 271 T339: ¬ (𝑌1 ↔ ¬𝑌2 ) ⇝ ¬𝑌1 ∧ ¬𝑌2 Cluster 272 T340: ¬ (𝑌1 ↔ ¬𝑌2 ) ⇝ 𝑌1 → 𝑌2 Cluster 273 T341: ¬ (𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 )) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 Cluster 274 T342: ¬ (¬𝑌1 ↔ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 Cluster 275 T343: ¬ (¬𝑌1 ↔ ¬𝑌2 ) ⇝ 𝑌1 → 𝑌2 Cluster 276 T344: ¬ (¬𝑌1 ↔ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 277 T346: ¬ ((𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ) ⇝ ¬𝑌3 → (¬𝑌1 ∨ ¬𝑌2 ) Cluster 278 T347: ¬ (𝑌1 → 𝑌2 ) ⇝ 𝑌2 ↔ 𝑌1 Cluster 279 Continued on next page 74

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Cluster 280 T352: ¬ (𝑌1 → ¬𝑌2 ) ⇝ 𝑌2 ↔ 𝑌1 Cluster 281 T359: ¬ (𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌2 ∧ 𝑌1 ) Cluster 282 T361: ¬ (¬𝑌1 ∨ ¬𝑌2 ) ⇝ ¬ (𝑌2 ↔ 𝑌1 ) Cluster 283 T366: 𝑌1 ∨ 𝑌2 ∨ ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 284 T370: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 Cluster 285 T372: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 286 T377: 𝑌1 ∨ ¬𝑌2 ⇝ 𝑌2 → ¬𝑌1 Cluster 287 T385: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 288 T386: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 )

⇝

(𝑌1 ∧ 𝑌2 ) ∨

Cluster 289 T387: (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ) ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 ) Cluster 290 T388: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌4 ) ⇝ 𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ∧ ¬𝑌4 Cluster 291 T389: (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) Cluster 292 T395: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌2 → (𝑌1 ∨ 𝑌3 ) Cluster 293 T396: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) Cluster 294 T400: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 295 T401: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 296 T403: (𝑌1 ∧ 𝑌2 ) ∨ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 297 T404: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ↔ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 )

Continued on next page 75

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0.02%

0.06%

Cluster 298 T405: (𝑌1 ∧ 𝑌2 ) ∨ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 299 T410: (𝑌2 ∧ 𝑌1 ) ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 → 𝑌2 ) Cluster 300 T412: (𝑌1 ↔ 𝑌2 ) ∨ 𝑌3 ⇝ (𝑌1 → 𝑌2 ) ∧ 𝑌3 Cluster 301 T414: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌3 ↔ 𝑌1 ) ∨ (𝑌3 ↔ 𝑌2 ) ⇝ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌3 ∧ 𝑌2 ) Cluster 302 T415: (𝑌1 ↔ 𝑌2 ) ∨ (𝑌3 ↔ 𝑌1 ) ⇝ (¬𝑌2 ∧ ¬𝑌3 ) → ¬𝑌1 Cluster 303 T419: (𝑌1 → 𝑌2 ) ∨ (𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 304 T420: (𝑌1 → ¬𝑌2 ) ∨ (𝑌2 → ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 305 T421: (𝑌1 → ¬𝑌2 ) ∨ (¬𝑌1 → 𝑌2 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 306 T423: ((𝑌1 ∨ 𝑌2 ) → 𝑌3 ) ∨ 𝑌1 ∨ 𝑌2 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 307 T427: ¬𝑌1 ∨ ¬𝑌2 ∨ ¬𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 308 T430: ¬ (𝑌1 ∧ 𝑌2 ) ∨¬ (𝑌1 ∧ 𝑌3 ) ∨¬ (𝑌2 ∧ 𝑌3 ) ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 309 T434: 𝑌1 ∨ 𝑌2 ∨ 𝑌1 ∨ 𝑌3 ⇝ (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ∨ (𝑌2 ∧ 𝑌3 ) Cluster 310 T436: 𝑌1 ∨ ¬𝑌2 ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∧ 𝑌2 ) Cluster 311 T438: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 312 T439: ¬𝑌1 ∨ ¬𝑌2 ∨ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∧ 𝑌2 )

B.4

Modal Logic: Hierarchical Clustering of Transformations Table 39: Transformation hierarchy clusters Transformations Whole data set ANY transformation explained

#dst

%dst

12482 9910

100.00% 79.39%

3951 2333

100.00% 59.05%

707 32 16 4 3 4 4 3 2 1 1

5.66% 0.26% 0.13% 0.03% 0.02% 0.03% 0.03% 0.02% 0.02% 0.01% 0.01%

162 1 1 2 1 1 1 1 1 1 1

4.10% 0.03% 0.03% 0.05% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

421 421

3.37% 3.37%

1 1

0.03% 0.03%

290 168 115 6

2.32% 1.35% 0.92% 0.05%

4 1 1 1

0.10% 0.03% 0.03% 0.03%

287 131 130 6 4 5 4 3 1

2.30% 1.05% 1.04% 0.05% 0.03% 0.04% 0.03% 0.02% 0.01%

12 1 1 3 1 1 1 1 1

0.30% 0.03% 0.03% 0.08% 0.03% 0.03% 0.03% 0.03% 0.03%

264 219 216 2 11 7 3 2 1 1 1

2.12% 1.75% 1.73% 0.02% 0.09% 0.06% 0.02% 0.02% 0.01% 0.01% 0.01%

14 4 2 1 1 2 1 1 1 1 1

0.35% 0.10% 0.05% 0.03% 0.03% 0.05% 0.03% 0.03% 0.03% 0.03% 0.03%

Cluster 1 T960: 𝑌1 → 𝑌2 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1035: 𝑌1 → ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1758: ^ (¬𝑌1 ) ∨ □ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) T981: 𝑌1 → (¬𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T331: ¬𝑌1 ∧ ^ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) ⌈ = T530: □ (□𝑌1 ) ⇝ ^ (^𝑌1 ) ⌊ = T545: □ (□ (¬𝑌1 )) ⇝ ^ (^ (¬𝑌1 )) T1810: ¬𝑌1 ∨ □ (□ (¬𝑌1 )) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) T1423: ¬𝑌1 → ^ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) T987: 𝑌1 → □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1104: 𝑌1 → (¬ (^𝑌1 ) ∨ ^ (^𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 2 ⌈ = T1226: □𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) ⌊ = T1587: 𝑌1 ∨ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 3 T1708: □𝑌1 ∨ 𝑌2 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1715: □𝑌1 ∨ □ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1717: □𝑌1 ∨ □ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1709: □𝑌1 ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 4 T637: □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T643: □ (𝑌1 ∨ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T636: □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □𝑌2 T649: □ (𝑌1 ∨ ^ (^𝑌2 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T646: □ (𝑌1 ∨ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T641: □ (𝑌1 ∨ □ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T642: □ (𝑌1 ∨ □ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T633: □ (𝑌1 ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T651: □ (𝑌1 ∨ ¬𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 5 T68: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T92: 𝑌1 ∧ ^ (¬𝑌2 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T86: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T94: 𝑌1 ∧ ^ (¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) T61: 𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T74: 𝑌1 ∧ ^ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T71: 𝑌1 ∧ ^ (𝑌1 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T82: 𝑌1 ∧ ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T72: 𝑌1 ∧ ^ (𝑌1 ∧ ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) T78: 𝑌1 ∧ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) T96: 𝑌1 ∧ ^ (𝑌1 ∨ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 6 Continued on next page 77

Transformations T446: □𝑌1 ⇝ ^𝑌1 T1533: ¬ (□𝑌1 ) ⇝ ¬ (^𝑌1 ) T30: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 T621: □ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) T529: □ (□𝑌1 ) ⇝ □ (^𝑌1 ) T639: □ (𝑌1 ∨ 𝑌2 ) ⇝ ^ (𝑌1 ∨ 𝑌2 )

#dst

%dst

262 95 82 45 15 2

2.10% 0.76% 0.66% 0.36% 0.12% 0.02%

15 3 2 3 1 1

0.38% 0.08% 0.05% 0.08% 0.03% 0.03%

238 25

1.91% 0.20%

85 2

2.15% 0.05%

13 11 10 7 3 1 1 1 1

0.10% 0.09% 0.08% 0.06% 0.02% 0.01% 0.01% 0.01% 0.01%

7 1 1 1 1 1 1 1 1

0.18% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

217 217 216 1

1.74% 1.74% 1.73% 0.01%

3 3 2 1

0.08% 0.08% 0.05% 0.03%

199 168

1.59% 1.35%

6 1

0.15% 0.03%

193 22 14 1 10 2

1.55% 0.18% 0.11% 0.01% 0.08% 0.02%

16 1 4 1 1 1

0.40% 0.03% 0.10% 0.03% 0.03% 0.03%

178 20 20 11 11 7 6 2 2

1.43% 0.16% 0.16% 0.09% 0.09% 0.06% 0.05% 0.02% 0.02%

9 1 1 1 1 1 2 1 1

0.23% 0.03% 0.03% 0.03% 0.03% 0.03% 0.05% 0.03% 0.03%

163

1.31%

0.03%

144

1.15%

0.38%

Cluster 7 T1152: □𝑌1 → 𝑌2 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) (^ (¬𝑌2 ) ∨ ^ (^ (¬𝑌2 ))) T1235: □𝑌1 → ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1176: □𝑌1 → □𝑌2 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1201: □𝑌1 → ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1224: □𝑌1 → (𝑌1 ∨ ^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1232: □𝑌1 → (^𝑌1 ∨ ^ (^𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1210: □𝑌1 → ¬𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1157: □𝑌1 → (𝑌1 ∧ □𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1193: □𝑌1 → ^𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1205: □𝑌1 → ^ ((𝑌2 → ¬𝑌1 ) ∨ ¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1237: □𝑌1 → (¬𝑌1 ∨ ^ (□ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 8 ⌈ = T18: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T67: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 → ^𝑌2 T86: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T142: 𝑌1 ∧ 𝑌2 ∧ ^ (𝑌1 ∨ 𝑌2 ) ⇝ (𝑌1 ∧ 𝑌2 ) → ^ (𝑌1 ∨ 𝑌2 ) Cluster 9 T1580: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T1715: □𝑌1 ∨ □ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 10 T676: ^𝑌1 ⇝ □𝑌1 T753: ^ (^𝑌1 ) ⇝ □ (^𝑌1 ) T1317: ^𝑌1 → 𝑌2 ⇝ □𝑌1 → 𝑌2 T1364: ^ (𝑌1 ∧ 𝑌2 ) → (𝑌2 ∨ ^𝑌3 ) ⇝ □ (𝑌1 ∧ 𝑌2 ) → (𝑌2 ∨ ^𝑌3 ) T1552: ¬ (^𝑌1 ) ⇝ ¬ (□𝑌1 ) T772: ^ (𝑌1 → 𝑌2 ) ⇝ □ (𝑌1 → 𝑌2 ) Cluster 11 T2: 𝑌1 ⇝ □𝑌1 ⌈ = T954: 𝑌1 → 𝑌2 ⇝ □ (𝑌1 → 𝑌2 ) ⌊ = T1453: ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) T677: ^𝑌1 ⇝ □ (^𝑌1 ) T1519: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) T1518: ¬𝑌1 ⇝ ¬ (□𝑌1 ) T1079: 𝑌1 → (𝑌2 ∨ 𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) T41: 𝑌1 ∧ □ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T963: 𝑌1 → 𝑌2 ⇝ □𝑌1 → 𝑌2 Cluster 12 T1747: ^ (□𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 13 T5: 𝑌1 ⇝ ¬𝑌1

Continued on next page 78

Transformations T822: 𝑌1 ↔ 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) T680: ^𝑌1 ⇝ ^ (¬𝑌1 ) ⌈ = T65: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) ⌊ = T684: ^𝑌1 ⇝ ¬ (^𝑌1 ) T459: □ (𝑌1 ∧ 𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) T1590: 𝑌1 ∨ □𝑌1 ⇝ 𝑌1 ∨ □ (¬𝑌1 )

#dst

%dst

56 28 18 18 10 8

0.45% 0.22% 0.14% 0.14% 0.08% 0.06%

1 4 3 3 1 1

0.03% 0.10% 0.08% 0.08% 0.03% 0.03%

139 130 3 1

1.11% 1.04% 0.02% 0.01%

7 1 1 1

0.18% 0.03% 0.03% 0.03%

138 53 30 14 13 8 8 1

1.11% 0.42% 0.24% 0.11% 0.10% 0.06% 0.06% 0.01%

14 3 2 1 3 2 2 1

0.35% 0.08% 0.05% 0.03% 0.08% 0.05% 0.05% 0.03%

131 96 16 14 3 1

1.05% 0.77% 0.13% 0.11% 0.02% 0.01%

6 1 1 1 1 1

0.15% 0.03% 0.03% 0.03% 0.03% 0.03%

130

1.04%

0.08%

120 120

0.96% 0.96%

2 2

0.05% 0.05%

114 93 89 3 6 1 1 1

0.91% 0.75% 0.71% 0.02% 0.05% 0.01% 0.01% 0.01%

11 3 1 1 1 1 1 1

0.28% 0.08% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

113 105

0.91% 0.84%

7 4

0.18% 0.10%

108

0.87%

0.03%

Cluster 14 T640: □ (𝑌1 ∨ □𝑌2 ) ⇝ □𝑌2 ∧ □ (□𝑌2 ) T636: □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □𝑌2 T664: □ (^𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T662: □ (□𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 15 T3: 𝑌1 ⇝ ^𝑌1 T109: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) T1083: 𝑌1 → (𝑌2 ∨ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T15: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 T957: 𝑌1 → 𝑌2 ⇝ 𝑌1 → ^𝑌2 ⌈ = T679: ^𝑌1 ⇝ ^ (^𝑌1 ) ⌊ = T1732: ^𝑌1 ∨ ^𝑌1 ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) T932: ¬𝑌1 ↔ (𝑌1 ∧ □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 16 T113: 𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T123: 𝑌1 ∧ ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T107: 𝑌1 ∧ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T114: 𝑌1 ∧ ¬ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T132: 𝑌1 ∧ ¬ (^ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) T131: 𝑌1 ∧ ¬ (^ (□𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 17 T793: ^ (¬𝑌1 ) ⇝ ¬ (^𝑌1 ) Cluster 18 ⌈ = T1588: 𝑌1 ∨ 𝑌2 ⇝ ^𝑌1 ∨ ^𝑌2 ⌊ = T1604: 𝑌1 ∨ ^𝑌1 ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) Cluster 19 T204: ^𝑌1 ∧ □𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) T262: ^ (¬𝑌1 ) ∧ □𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) T259: ^ (¬𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T271: ^ (¬ (^𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T201: ^𝑌1 ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T244: ^ (□ (𝑌1 → ¬𝑌2 )) ∧ □𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) T245: ^ (□ (¬𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T249: ^ (^𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 20 T21: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → 𝑌1 T376: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 21 T1466: (𝑌1 ∨ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 )

Continued on next page 79

Transformations

#dst

%dst

106 3 2 1 1 1

0.85% 0.02% 0.02% 0.01% 0.01% 0.01%

24 1 1 1 1 1

0.61% 0.03% 0.03% 0.03% 0.03% 0.03%

97 89 8 5

0.78% 0.71% 0.06% 0.04%

3 1 2 1

0.08% 0.03% 0.05% 0.03%

94 2 2

0.75% 0.02% 0.02%

9 1 1

0.23% 0.03% 0.03%

86 63 11 2 2 2 2 1

0.69% 0.50% 0.09% 0.02% 0.02% 0.02% 0.02% 0.01%

16 3 4 2 2 1 1 1

0.40% 0.08% 0.10% 0.05% 0.05% 0.03% 0.03% 0.03%

83 35 11 11 9 1 1

0.66% 0.28% 0.09% 0.09% 0.07% 0.01% 0.01%

20 1 1 1 1 1 1

0.51% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

79 16 4 1

0.63% 0.13% 0.03% 0.01%

31 3 1 1

0.78% 0.08% 0.03% 0.03%

79 23 6 2 1

0.63% 0.18% 0.05% 0.02% 0.01%

9 1 1 1 1

0.23% 0.03% 0.03% 0.03% 0.03%

0.62%

0.53%

Cluster 22 T377: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (^ (¬𝑌3 ) ∨ ^ (^ (¬𝑌3 ))) T439: (^ (¬𝑌1 ) ∨ □ (^ (¬𝑌1 ))) ∧ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T421: (□ (¬𝑌1 ) ∨ □ (^ (¬𝑌1 ))) ∧ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T400: (𝑌1 ∨ ^ (^𝑌1 )) ∧ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T412: (□𝑌1 ∨ □ (□𝑌1 )) ∧ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T424: (^𝑌1 ∨ □ (□𝑌1 )) ∧ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 23 T37: 𝑌1 ∧ □𝑌2 ⇝ 𝑌2 → 𝑌1 T259: ^ (¬𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T380: (𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) ∧ T440: (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

□𝑌1

⇝

𝑌1

→

Cluster 24 T961: 𝑌1 → 𝑌2 ⇝ 𝑌2 → 𝑌1 T1378: ^ (¬𝑌1 ) → 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1451: ¬ (𝑌1 ↔ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 25 T444: □𝑌1 ⇝ 𝑌1 T1150: □𝑌1 → 𝑌2 ⇝ 𝑌1 → 𝑌2 T589: □ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → 𝑌2 T528: □ (□𝑌1 ) ⇝ □𝑌1 T1088: 𝑌1 → (□𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) T729: ^ (□𝑌1 ) ⇝ ^𝑌1 T1170: □𝑌1 → □𝑌1 ⇝ □𝑌1 → 𝑌1 T627: □ (¬ (□𝑌1 )) ⇝ ¬ (□𝑌1 ) Cluster 26 T32: 𝑌1 ∧ □𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) T240: ^ (□𝑌1 ) ∧ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) ⌈ = T17: 𝑌1 ∧ 𝑌2 ⇝ □𝑌1 ∧ □𝑌2 ⌊ = T25: 𝑌1 ∧ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T248: ^ (^𝑌1 ) ∧ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T351: ¬ (□ (^𝑌1 )) ∧ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T393: (𝑌1 ∨ ^𝑌1 ) ∧ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 27 T953: 𝑌1 → 𝑌2 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1014: 𝑌1 → □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1076: 𝑌1 → (𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1091: 𝑌1 → (^𝑌1 ∨ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 28 T689: ^ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T704: ^ (𝑌1 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T694: ^ (𝑌1 ∧ ^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T692: ^ (𝑌1 ∧ □ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) T703: ^ (𝑌1 ∧ ^ (𝑌1 ∨ ¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 29 T20: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

Continued on next page 80

Transformations T87: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T43: 𝑌1 ∧ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T79: 𝑌1 ∧ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T42: 𝑌1 ∧ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T75: 𝑌1 ∧ ^ (¬𝑌1 ∧ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T76: 𝑌1 ∧ ^ (□𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T27: 𝑌1 ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T45: 𝑌1 ∧ □ (^ (¬𝑌1 ∨ ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T58: 𝑌1 ∧ □ (^ (¬𝑌1 ) ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T104: 𝑌1 ∧ ^ (¬𝑌1 ∨ ^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T124: 𝑌1 ∧ ¬ (^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

#dst

%dst

9 4 3 2 2 2 1 1 1 1 1

0.07% 0.03% 0.02% 0.02% 0.02% 0.02% 0.01% 0.01% 0.01% 0.01% 0.01%

1 1 1 1 1 1 1 1 1 1 1

0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

75 50

0.60% 0.40%

9 1

0.23% 0.03%

0.60%

0.05%

0.60%

0.03%

0.58%

0.03%

0.58%

0.03%

67 40 27

0.54% 0.32% 0.22%

2 1 1

0.05% 0.03% 0.03%

65 39 4 17 4 1 1 1

0.52% 0.31% 0.03% 0.14% 0.03% 0.01% 0.01% 0.01%

12 5 1 2 1 1 1 1

0.30% 0.13% 0.03% 0.05% 0.03% 0.03% 0.03% 0.03%

65 24 18 18 11 4 1

0.52% 0.19% 0.14% 0.14% 0.09% 0.03% 0.01%

10 1 1 1 1 1 1

0.25% 0.03% 0.03% 0.03% 0.03% 0.03% 0.03%

65 2

0.52% 0.02%

8 2

0.20% 0.05%

Cluster 30 T312: ¬𝑌1 ∧ □𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) T353: ¬ (^𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 31 T1583: 𝑌1 ∨ 𝑌2 ⇝ 𝑌2 ↔ ¬𝑌1 Cluster 32 T1248: □ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 33 T1136: (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 34 T445: □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 35 T1531: ¬ (□𝑌1 ) ⇝ □ (¬𝑌1 ) T1535: ¬ (□ (𝑌1 ∧ 𝑌2 )) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) T1630: 𝑌1 ∨ ¬ (□𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 36 T1151: □𝑌1 → 𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1195: □𝑌1 → ^𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1192: □𝑌1 → ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1223: □𝑌1 → ¬ (^𝑌2 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T1184: □𝑌1 → □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T1169: □𝑌1 → □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌈ = T1175: □𝑌1 → □𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T1182: □𝑌1 → □ (^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 37 T134: 𝑌2 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T358: ¬ (^𝑌1 ) ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌈ = T319: ¬𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌊ = T326: ¬𝑌1 ∧ ^𝑌2 ⇝ 𝑌2 → ^ (¬𝑌1 ) T61: 𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T211: ^𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T148: 𝑌1 ∧ ¬𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 38 T951: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T1069: 𝑌1 → ¬ (^𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 )

Continued on next page 81

Transformations

#dst

%dst

59 32 7 1 1 1

0.47% 0.26% 0.06% 0.01% 0.01% 0.01%

9 1 1 1 1 1

0.23% 0.03% 0.03% 0.03% 0.03% 0.03%

58 47

0.46% 0.38%

5 1

0.13% 0.03%

0.44%

0.03%

0.44%

0.03%

0.43%

0.05%

0.42%

0.05%

0.41%

0.08%

0.41%

0.05%

0.41%

0.03%

0.41%

0.03%

50 1 1

0.40% 0.01% 0.01%

5 1 1

0.13% 0.03% 0.03%

49 49 49 42

0.39% 0.39% 0.39% 0.34%

6 6 6 2

0.15% 0.15% 0.15% 0.05%

0.34%

0.05%

0.34%

0.03%

0.34%

0.51%

Cluster 39 T23: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ∨ 𝑌2 T48: 𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) T1522: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ∨ 𝑌2 ) ⌈ = T308: ¬𝑌1 ∧ 𝑌2 ⇝ ¬𝑌1 ∨ 𝑌2 ⌊ = T504: □ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) T551: □ (^ (𝑌1 ∧ 𝑌2 ∧ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 40 T35: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T47: 𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 41 T895: □𝑌1 ↔ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 42 T1728: ^𝑌1 ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 43 T479: □ (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 44 T1648: (𝑌1 ∧ 𝑌2 ) ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 45 T1550: ¬ (^𝑌1 ) ⇝ ^ (¬𝑌1 ) Cluster 46 T1227: □𝑌1 → (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 47 T791: ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 48 T1057: 𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 49 T389: (𝑌1 ∨ □𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ □𝑌3 ) T388: (𝑌1 ∨ □𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) T390: (𝑌1 ∨ □ (^𝑌2 )) ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 50 ⌈ = T688: ^ (𝑌1 ∧ 𝑌2 ) ⇝ ^𝑌2 | = T711: ^ (𝑌1 ∧ ¬𝑌2 ) ⇝ ^ (¬𝑌2 ) ⌊ = T714: ^ (𝑌2 ∧ ¬𝑌1 ) ⇝ ^ (¬𝑌1 ) T722: ^ ((𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 51 T995: 𝑌1 → □𝑌2 ⇝ □𝑌1 → 𝑌2 Cluster 52 T470: □ (𝑌1 ∧ ^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 53 T674: ^𝑌1 ⇝ 𝑌1

Continued on next page 82

Transformations T547: □ (^𝑌1 ) ⇝ □𝑌1 T1337: ^𝑌1 → ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T1358: ^𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) ⌈ = T727: ^ (□𝑌1 ) ⇝ □𝑌1 ⌊ = T1616: 𝑌1 ∨ ^ (□𝑌2 ) ⇝ 𝑌1 ∨ □𝑌2 T1024: 𝑌1 → ^𝑌2 ⇝ 𝑌1 → 𝑌2 T1194: □𝑌1 → ^𝑌1 ⇝ □𝑌1 → 𝑌1 T754: ^ (^𝑌1 ) ⇝ ^𝑌1 T696: ^ (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 T775: ^ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → 𝑌2 T1092: 𝑌1 → (^𝑌2 ∨ □𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 )

#dst

%dst

10 9 5 3 3 3 1 2 2 1 1

0.08% 0.07% 0.04% 0.02% 0.02% 0.02% 0.01% 0.02% 0.02% 0.01% 0.01%

6 1 2 2 2 2 1 2 1 1 1

0.15% 0.03% 0.05% 0.05% 0.05% 0.05% 0.03% 0.05% 0.03% 0.03% 0.03%

40 8 8 5 2

0.32% 0.06% 0.06% 0.04% 0.02%

16 1 1 1 1

0.40% 0.03% 0.03% 0.03% 0.03%

0.31%

0.03%

0.30%

0.03%

0.30%

0.03%

33 32

0.26% 0.26%

2 1

0.05% 0.03%

0.26%

0.05%

0.26%

0.03%

0.26%

0.03%

0.26%

0.03%

31 3 2

0.25% 0.02% 0.02%

4 2 1

0.10% 0.05% 0.03%

0.24%

0.05%

0.24%

0.03%

Cluster 54 T1144: □𝑌1 → 𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) ⌈ = T1296: □ (𝑌1 ∨ 𝑌2 ) → 𝑌1 ⇝ □𝑌1 ∧ □𝑌2 ⌊ = T1302: □ (𝑌1 ∨ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1142: □𝑌1 → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1243: □ (𝑌1 ∧ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 55 T1353: ^𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 56 T356: ¬ (^𝑌1 ) ∧ □ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 57 T1749: ^ (^𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 58 T133: 𝑌2 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) T48: 𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 59 T1018: 𝑌1 → □ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 60 T527: □ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 61 T623: □ (¬𝑌1 ) ⇝ ¬ (□𝑌1 ) Cluster 62 T1010: 𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 63 T159: □𝑌1 ∧ 𝑌2 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T177: □𝑌1 ∧ □ (¬𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T174: □𝑌1 ∧ □ (¬𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 64 T952: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 65 T223: ^𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 66 Continued on next page 83

Transformations

#dst

%dst

T465: □ (𝑌1 ∧ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 )

0.24%

0.03%

0.24%

0.03%

29 19 1 1 1

0.23% 0.15% 0.01% 0.01% 0.01%

10 4 1 1 1

0.25% 0.10% 0.03% 0.03% 0.03%

0.22%

0.03%

0.22%

0.03%

0.22%

0.03%

27 27

0.22% 0.22%

2 2

0.05% 0.05%

0.21%

0.03%

0.21%

0.03%

0.21%

0.03%

0.20%

0.03%

24 13 8 8 2 1 8

0.19% 0.10% 0.06% 0.06% 0.02% 0.01% 0.06%

8 5 2 2 1 1 1

0.20% 0.13% 0.05% 0.05% 0.03% 0.03% 0.03%

0.19%

0.10%

0.19%

0.03%

24 24

0.19% 0.19%

1 1

0.03% 0.03%

Cluster 67 T1138: (¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 68 T950: 𝑌1 → 𝑌2 ⇝ 𝑌2 T773: ^ (𝑌1 → 𝑌2 ) ⇝ ^𝑌2 T1063: 𝑌1 → ¬ (□𝑌1 ) ⇝ ¬ (□𝑌1 ) T1406: (𝑌1 → □ (𝑌2 ∧ 𝑌3 )) → 𝑌4 ⇝ □ (𝑌2 ∧ 𝑌3 ) → 𝑌4 T1566: ¬ (𝑌1 → □𝑌1 ) ⇝ ¬ (□𝑌1 ) Cluster 69 T16: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 70 T110: 𝑌1 ∧ ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 71 T339: ¬ (□𝑌1 ) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 72 ⌈ = T100: 𝑌1 ∧ ^ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^𝑌3 ) ⌊ = T102: 𝑌1 ∧ ^ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 73 T381: (𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 74 T1249: □ (¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 75 T1551: ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 76 T546: □ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 77 T1586: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1609: 𝑌1 ∨ ^𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌈ = T1585: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T1620: 𝑌1 ∨ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T1603: 𝑌1 ∨ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) T1615: 𝑌1 ∨ ^ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T1632: 𝑌1 ∨ ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 78 T702: ^ (𝑌1 ∧ ^ (¬𝑌2 )) ⇝ 𝑌2 → ^ (¬𝑌2 ) Cluster 79 T50: 𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 80 ⌈ = T1584: 𝑌1 ∨ 𝑌2 ⇝ ¬𝑌1 ↔ 𝑌2 ⌊ = T1612: 𝑌1 ∨ ^ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ 𝑌2 )

Continued on next page 84

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23 23 16

0.18% 0.18% 0.13%

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22 7 1 1

0.18% 0.06% 0.01% 0.01%

8 1 1 1

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20 20

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20 20

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Cluster 81 T1818: ¬𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌1 ∨ 𝑌2 ) Cluster 82 ⌈ = T447: □𝑌1 ⇝ ^ (^𝑌1 ) ⌊ = T1093: 𝑌1 → (^𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T1758: ^ (¬𝑌1 ) ∨ □ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) Cluster 83 T667: □ (^𝑌1 ∨ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 84 T818: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 T840: 𝑌1 ↔ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌈ = T584: □ (¬ (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) ⌊ = T930: ¬𝑌1 ↔ 𝑌2 ⇝ ¬𝑌1 → 𝑌2 Cluster 85 T1589: 𝑌1 ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 86 T1475: (𝑌1 ∨ □𝑌1 ) → 𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) T1460: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌1 ∧ 𝑌2 T1474: (𝑌1 ∨ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 87 ⌈ = T108: 𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ⇝ ¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ⌊ = T476: □ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 88 T231: ^ (𝑌1 ∧ 𝑌2 ) ∧ ^ (□𝑌3 ) ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 89 T510: □ (¬ (^𝑌1 ) ∧ ^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 90 T759: ^ (^ ((𝑌1 ∨ 𝑌2 ) ∧ □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 91 ⌈ = T1145: □𝑌1 → 𝑌2 ⇝ □ (𝑌1 → 𝑌2 ) ⌊ = T1293: □ (¬ (𝑌1 ∨ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 92 T432: (^ (□𝑌1 ) ∨ □𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 93 T726: ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 94 T1016: 𝑌1 → □ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 95 T1159: □𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 96 T483: □ (𝑌1 ∧ ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 )

Continued on next page 85

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16 8 8 1

0.13% 0.06% 0.06% 0.01%

8 2 2 1

0.20% 0.05% 0.05% 0.03%

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13 9

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10 6

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Cluster 97 T708: ^ (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 98 T1733: ^𝑌1 ∨ ^𝑌2 ∨ ^𝑌3 ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 99 T480: □ (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 100 T1002: 𝑌1 → □ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 101 T1215: □𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 102 T1308: □ (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 103 T1735: ^𝑌1 ∨ ^𝑌2 ⇝ ^𝑌2 ∨ ^ (^𝑌2 ) ⌈ = T679: ^𝑌1 ⇝ ^ (^𝑌1 ) ⌊ = T1732: ^𝑌1 ∨ ^𝑌1 ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) T1097: 𝑌1 → (^ (^𝑌2 → 𝑌1 ) ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 104 T119: 𝑌1 ∧ ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 105 T507: □ (¬ (□𝑌1 ) ∧ ^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 106 T989: 𝑌1 → □𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 107 T1025: 𝑌1 → ^𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 108 T84: 𝑌1 ∧ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 109 T1650: (𝑌1 ∧ 𝑌2 ) ∨ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 110 T681: ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 111 T1281: □ (¬𝑌1 ) → 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 112 T1536: ¬ (□ (𝑌1 ∧ 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 113 T962: 𝑌1 → 𝑌2 ⇝ 𝑌2 → (^ (¬𝑌2 ) ∨ ^ (^ (¬𝑌2 ))) T1463: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → (^ (¬𝑌3 ) ∨ ^ (^ (¬𝑌3 ))) Cluster 114 Continued on next page 86

Transformations

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T1593: 𝑌1 ∨ □𝑌2 ⇝ 𝑌1 ∨ □ (¬𝑌1 ) T1590: 𝑌1 ∨ □𝑌1 ⇝ 𝑌1 ∨ □ (¬𝑌1 )

13 8

0.10% 0.06%

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13 13

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3 3

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12 1

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9 1

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12 8 1 1

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12 11

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Cluster 115 ⌈ = T31: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) ⌊ = T451: □𝑌1 ⇝ ¬ (^𝑌1 ) Cluster 116 T1008: 𝑌1 → □ (¬𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 117 T394: (𝑌1 ∨ ^𝑌1 ) ∧ □𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 118 T959: 𝑌1 → 𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 119 T206: ^𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 120 T228: ^ (𝑌1 ∧ 𝑌2 ) ∧ □𝑌3 ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 121 T567: □ (^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 122 T758: ^ (^ (𝑌1 ∧ ¬ (□𝑌2 ))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 123 T1080: 𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 124 T1225: □𝑌1 → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1229: □𝑌1 → ((𝑌1 ∧ 𝑌1 ) ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 125 T1459: (𝑌1 ∨ 𝑌2 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1474: (𝑌1 ∨ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1485: (𝑌1 ∨ ^ (□𝑌1 )) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1486: (𝑌1 ∨ ¬𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 126 T1043: 𝑌1 → ^ (𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 127 T1096: 𝑌1 → (^ (^𝑌2 ) ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 128 T1196: □𝑌1 → ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T1201: □𝑌1 → ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 129 T175: □𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 130 T609: □ (𝑌1 → (¬𝑌1 ∨ ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

Continued on next page 87

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0.10%

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11 1 1

0.09% 0.01% 0.01%

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Cluster 131 T869: (¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 132 T1186: □𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 133 T1622: 𝑌1 ∨ ^ (¬𝑌2 ) ⇝ 𝑌2 → (^ (¬𝑌2 ) ∨ ^ (^ (¬𝑌2 ))) T1621: 𝑌1 ∨ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1707: (¬ (^𝑌1 ) ∧ □𝑌1 ) ∨ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 134 T1204: □𝑌1 → ^ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 135 T1639: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 136 T263: ^ (¬𝑌1 ) ∧ □ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 137 T765: ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 138 T1171: □𝑌1 → □𝑌2 ⇝ □ (𝑌1 → 𝑌2 ) Cluster 139 T1253: □ (¬𝑌1 ∧ ¬𝑌2 ) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 140 T1365: ^ (¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ) Cluster 141 T1464: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 142 T1153: □𝑌1 → 𝑌2 ⇝ □𝑌1 → 𝑌1 T1170: □𝑌1 → □𝑌1 ⇝ □𝑌1 → 𝑌1 T1194: □𝑌1 → ^𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 143 T57: 𝑌1 ∧ □ (𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 144 T69: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T87: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 145 T93: 𝑌1 ∧ ^ (¬𝑌2 ) ⇝ 𝑌2 → (^ (¬𝑌2 ) ∨ ^ (^ (¬𝑌2 ))) T87: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 146 T392: (𝑌1 ∨ ^𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 147 T396: (𝑌1 ∨ ^𝑌2 ) ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 )

Continued on next page 88

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9 2 1 2 1 1

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Cluster 148 ⌈ = T690: ^ (𝑌1 ∧ 𝑌2 ) ⇝ ^𝑌1 ∨ ^𝑌2 | = T695: ^ (𝑌1 ∧ ^𝑌1 ) ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) ⌊ = T1031: 𝑌1 → ^ (𝑌2 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 149 T1643: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ □𝑌2 ) ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 150 T40: 𝑌1 ∧ □ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 151 T115: 𝑌1 ∧ ¬ (□𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 152 T172: □𝑌1 ∧ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 153 T514: □ ((𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 154 T1331: ^𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 155 T1366: ^ (¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 156 T1432: ¬ (□𝑌1 ) → 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 157 T1602: 𝑌1 ∨ ^𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 158 T1516: ¬𝑌1 ⇝ 𝑌1 T790: ^ (¬𝑌1 ) ⇝ ^𝑌1 T1562: ¬ (^ (¬𝑌1 )) ⇝ ¬ (^𝑌1 ) T1549: ¬ (^𝑌1 ) ⇝ ^𝑌1 T1081: 𝑌1 → (𝑌2 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) T1530: ¬ (□𝑌1 ) ⇝ □𝑌1 Cluster 159 T1465: (𝑌1 ∨ 𝑌2 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1476: (𝑌1 ∨ □𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1481: (𝑌1 ∨ ^𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1487: (𝑌1 ∨ ¬𝑌1 ) → □𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) Cluster 160 T1064: 𝑌1 → ¬ (□𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 161 T1105: 𝑌2 → (□𝑌1 ∨ □ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 162 T111: 𝑌1 ∧ ¬𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 )

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Cluster 163 T128: 𝑌1 ∧ ¬ (^𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 164 T161: □𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 165 T343: ¬ (□𝑌1 ) ∧ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 166 T373: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ¬𝑌4 ⇝ ^ (¬𝑌4 ) Cluster 167 T404: (𝑌1 ∨ ¬𝑌2 ) ∧ (𝑌2 ∨ ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 168 T683: ^𝑌1 ⇝ ¬ (□𝑌1 ) Cluster 169 T804: ^ (𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 170 T806: ^ (𝑌1 ∨ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 171 ⌈ = T991: 𝑌1 → □𝑌2 ⇝ □ (𝑌1 → 𝑌2 ) ⌊ = T1454: ¬ (𝑌1 ∨ 𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 172 T1149: □𝑌1 → 𝑌2 ⇝ □ (¬𝑌2 ∨ 𝑌1 ) Cluster 173 T1513: (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 174 T1736: ^𝑌1 ∨ ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 175 T837: 𝑌1 ↔ ^𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T840: 𝑌1 ↔ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 176 T1771: (𝑌1 → 𝑌2 ) ∨ ^𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 177 T1: 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 178 T8: 𝑌1 ∧ 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 179 T51: 𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 180 T98: 𝑌1 ∧ ^ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 )

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Cluster 181 T706: ^ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 182 T779: ^ (𝑌1 → ¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 183 T858: 𝑌1 ↔ ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 184 T1203: □𝑌1 → ^ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 185 T1444: ¬ (^𝑌1 ) → 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 186 T1468: (𝑌1 ∨ 𝑌2 ) → ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 187 T1155: □𝑌1 → 𝑌2 ⇝ □𝑌2 → 𝑌2 T1285: □ (¬𝑌1 ) → 𝑌2 ⇝ □𝑌2 → 𝑌2 Cluster 188 T912: ^𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T919: ^𝑌1 ↔ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 189 T803: ^ (𝑌1 ∨ 𝑌2 ) ⇝ ^𝑌2 T814: ^ ((𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ∨ (𝑌2 ∧ ¬𝑌1 ∧ ¬𝑌3 ) ∨ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ) ∨ ¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 190 T11: 𝑌1 ∧ 𝑌2 ∧ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 191 T203: ^𝑌1 ∧ □𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 192 T810: ^ (𝑌1 ∨ ¬𝑌2 ) ⇝ 𝑌2 → ^ (¬𝑌2 ) T809: ^ (𝑌1 ∨ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 193 T1082: 𝑌1 → (𝑌2 ∨ ^ (□𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 194 T1598: 𝑌1 ∨ □ (¬𝑌2 ) ⇝ 𝑌2 ∨ □ (¬𝑌2 ) Cluster 195 T1814: ¬𝑌1 ∨ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 196 T4: 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 197 T126: 𝑌1 ∧ ¬ (^𝑌2 ) ⇝ 𝑌2 ∧ ¬ (^𝑌1 )

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Cluster 198 T363: ¬ (^𝑌1 ) ∧ ^ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 199 T491: □ (^𝑌1 ∧ ^ (¬𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 200 T515: □ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 201 T1228: □𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 202 T1344: ^𝑌1 → (𝑌2 ↔ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 203 T1409: ¬𝑌1 → 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 204 T1413: ¬𝑌1 → 𝑌2 ⇝ ¬𝑌2 ∨ ¬𝑌1 Cluster 205 T1431: ¬ (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 206 T1490: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 207 T1521: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 208 T1573: ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 209 T1775: (𝑌1 → 𝑌2 ) ∨ (𝑌1 → □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) T1791: (𝑌1 → ¬𝑌1 ) ∨ (𝑌1 → □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 210 T815: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 Cluster 211 T325: ¬𝑌1 ∧ ^𝑌2 ⇝ ^ (¬𝑌1 ) Cluster 212 T816: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ 𝑌2 T829: 𝑌1 ↔ □ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 213 T66: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 214 T179: □𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 215 T645: □ (𝑌1 ∨ ^ (□𝑌2 )) ⇝ □𝑌2 ∧ □ (□𝑌2 )

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Cluster 216 T958: 𝑌1 → 𝑌2 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 217 T1313: ^𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 218 T1764: (𝑌1 ↔ 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 219 T214: ^𝑌1 ∧ ^𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 220 T473: □ (𝑌1 ∧ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 221 T516: □ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (□𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 222 T548: □ (^ (𝑌1 ∧ 𝑌2 ∧ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 223 T573: □ (^ (𝑌1 ∨ ^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 224 T599: □ (𝑌1 → ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 225 T628: □ (¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 226 T675: ^𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 227 T894: □𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 228 T969: 𝑌1 → (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 229 T976: 𝑌1 → (□𝑌1 ∧ □𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 230 T1028: 𝑌1 → ^ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 231 T1058: 𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 232 T1189: □𝑌1 → □ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 233 T1346: ^𝑌1 → ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 234 T1350: ^𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 ))

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Cluster 235 T1382: ^ (¬ (𝑌1 ∨ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 236 T1449: ¬ (^ (𝑌1 ∧ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 237 T1532: ¬ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 238 T1760: ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 239 T821: 𝑌1 ↔ 𝑌2 ⇝ 𝑌2 → 𝑌1 T828: 𝑌1 ↔ □𝑌1 ⇝ □𝑌1 → 𝑌1 T945: (𝑌1 ∨ ^𝑌1 ) ↔ □𝑌2 ⇝ □𝑌2 → (𝑌1 ∨ ^𝑌1 ) T926: ^ (¬𝑌1 ) ↔ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 240 T22: 𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 241 T62: 𝑌1 ∧ ^𝑌1 ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) T331: ¬𝑌1 ∧ ^ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) Cluster 242 T316: ¬𝑌1 ∧ □ (¬𝑌2 ) ⇝ 𝑌2 → ^ (¬𝑌2 ) T314: ¬𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 243 T838: 𝑌1 ↔ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 244 T1315: ^𝑌1 → 𝑌2 ⇝ ^𝑌2 Cluster 245 T1334: ^𝑌1 → ^𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 246 T1610: 𝑌1 ∨ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T1621: 𝑌1 ∨ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 247 T53: 𝑌1 ∧ □ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 248 T90: 𝑌1 ∧ ^ (¬𝑌2 ) ⇝ 𝑌2 ∧ ¬ (^𝑌1 ) Cluster 249 T91: 𝑌1 ∧ ^ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 250 T129: 𝑌1 ∧ ¬ (^𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 251 T158: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3

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Cluster 252 T189: □ (¬𝑌1 ) ∧ □ (□𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 253 T324: ¬𝑌1 ∧ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 254 T449: □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 255 T613: □ (¬𝑌1 → 𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 256 T650: □ (𝑌1 ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 257 T663: □ (□𝑌1 ∨ □ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 258 T794: ^ (¬ (𝑌1 ∧ 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 259 T850: 𝑌1 ↔ ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 260 T856: 𝑌1 ↔ ¬ (^𝑌1 ∧ □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 261 T898: □ (¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 262 T968: 𝑌1 → 𝑌2 ⇝ 𝑌1 ∨ 𝑌2 Cluster 263 T1056: 𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 264 T1168: □𝑌1 → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 265 T1197: □𝑌1 → ^𝑌2 ⇝ □𝑌1 → (𝑌2 ∨ ^𝑌2 ) Cluster 266 T1199: □𝑌1 → ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 267 T1220: □𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 268 T1318: ^𝑌1 → 𝑌2 ⇝ □𝑌1 → ¬𝑌2 Cluster 269 T1325: ^𝑌1 → □𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 270 T1415: ¬𝑌1 → □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 ))

Continued on next page 95

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Cluster 271 T1439: ¬ (□ (𝑌1 ∧ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 272 T1637: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 273 T1640: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 274 T1713: □𝑌1 ∨ □𝑌2 ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 275 T1781: (𝑌1 → □𝑌2 ) ∨ (𝑌3 → □𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 276 T335: ¬𝑌1 ∧ ¬𝑌2 ⇝ 𝑌1 → ^ (¬𝑌1 ) T336: ¬𝑌1 ∧ ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) T334: ¬𝑌1 ∧ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 277 T597: □ (𝑌1 → □𝑌2 ) ⇝ □𝑌2 ∧ □ (□𝑌2 ) T596: □ (𝑌1 → □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 278 T1027: 𝑌1 → ^𝑌2 ⇝ □𝑌2 → 𝑌1 T1022: 𝑌1 → ^𝑌1 ⇝ □𝑌1 → 𝑌1 T1452: ¬ (𝑌1 ↔ 𝑌2 ) → ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 279 T1730: ^𝑌1 ∨ □𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) Cluster 280 T34: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 281 T73: 𝑌1 ∧ ^ (𝑌2 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T75: 𝑌1 ∧ ^ (¬𝑌1 ∧ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 282 T77: 𝑌1 ∧ ^ (□𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T79: 𝑌1 ∧ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 283 T604: □ (𝑌1 → ¬𝑌2 ) ⇝ ^ (¬𝑌2 ) Cluster 284 T655: □ ((𝑌1 ∧ □𝑌2 ) ∨ 𝑌2 ) ⇝ □𝑌2 ∧ □ (□𝑌2 ) T654: □ ((𝑌1 ∧ □𝑌1 ) ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 285 T1039: 𝑌1 → ^ (¬ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 286 T1078: 𝑌1 → (𝑌1 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 ))

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Cluster 287 T1592: 𝑌1 ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 288 T1635: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 289 T1769: (𝑌1 → 𝑌2 ) ∨ (𝑌1 ∧ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ ^𝑌2 ) Cluster 290 ⌈ = T24: 𝑌1 ∧ 𝑌2 ⇝ ¬𝑌1 ∨ ¬𝑌2 ⌊ = T460: □ (𝑌1 ∧ 𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 291 T120: 𝑌1 ∧ ¬ (□𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 292 T187: □ (^ (𝑌1 ∨ 𝑌2 )) ∧ □ (^ (^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 293 T205: ^𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 294 T230: ^ (𝑌1 ∧ 𝑌2 ) ∧ □ (□𝑌3 ) ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 295 T239: ^ (¬𝑌1 ∧ ¬𝑌2 ) ∧ ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 296 T254: ^ (^ (¬𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 297 T260: ^ (¬𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 298 T273: ^ (¬ (𝑌1 ∨ 𝑌2 )) ∧ 𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ) Cluster 299 T309: ¬𝑌1 ∧ □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 300 T333: ¬𝑌1 ∧ ^ (𝑌2 ∨ 𝑌3 ) ⇝ ^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 )) Cluster 301 T500: □ (¬𝑌1 ∧ ^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 302 T509: □ (¬ (^𝑌1 ) ∧ □ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 303 T555: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 304 T580: □ (𝑌1 ↔ ¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 305 Continued on next page 97

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Cluster 306 T607: □ (𝑌1 → (𝑌2 ∨ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 307 T610: □ ((¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 308 T615: □ (¬𝑌1 → 𝑌2 ) ⇝ ^ (¬𝑌1 ) Cluster 309 T631: □ (¬ (𝑌1 → 𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 310 T724: ^ ((𝑌1 ∨ 𝑌2 ) ∧ □ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 311 T725: ^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (□𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 312 T728: ^ (□𝑌1 ) ⇝ □ (^𝑌1 ) Cluster 313 T738: ^ (□ ((𝑌1 ∨ 𝑌2 ) ∧ □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 314 T741: ^ (□ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (□𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 315 T760: ^ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ^𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 316 T785: ^ ((¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ) Cluster 317 T801: ^ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4 ∨ 𝑌5 ) ⇝ □ (𝑌2 ∨ 𝑌3 ∨ 𝑌4 ) Cluster 318 T900: □ (¬𝑌1 ) ↔ 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 319 T1009: 𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 320 T1011: 𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 321 T1017: 𝑌1 → □ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 322 T1065: 𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 323 T1111: (𝑌1 ∧ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 324 Continued on next page 98

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T1125: (□ (¬𝑌1 ) ∧ □ (¬𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 )

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Cluster 325 T1164: □𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 326 T1179: □𝑌1 → □ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 327 T1202: □𝑌1 → ^ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 328 T1239: □ (𝑌1 ∧ 𝑌2 ) → 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 329 T1274: □ (^𝑌1 ) → ¬ (□𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 330 T1294: □ (¬ (𝑌1 ∨ 𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 331 T1297: □ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 332 T1351: ^𝑌1 → ¬ (^𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 333 T1362: ^ (𝑌1 ∧ 𝑌2 ) → ^ (□𝑌3 ) ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 334 T1396: (𝑌1 ↔ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 335 T1400: (𝑌1 ↔ ¬𝑌2 ) → 𝑌3 ⇝ 𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 336 T1467: (𝑌1 ∨ 𝑌2 ) → □ (^𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 337 T1477: (𝑌1 ∨ □𝑌1 ) → □𝑌2 ⇝ □𝑌2 → (𝑌1 ∨ ^𝑌1 ) Cluster 338 T1627: 𝑌1 ∨ ¬𝑌2 ⇝ ¬ (𝑌2 ↔ 𝑌1 ) Cluster 339 T1711: □𝑌1 ∨ □𝑌2 ∨ □𝑌3 ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 340 T1721: □ (^𝑌1 ) ∨ □ (^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 341 T1298: □ (𝑌1 ∨ 𝑌2 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 342 T1691: (𝑌2 ∧ □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 343 Continued on next page 99

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T64: 𝑌1 ∧ ^𝑌2 ∧ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T85: 𝑌1 ∧ ^ (¬𝑌1 ) ∧ ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

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Cluster 344 T590: □ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → ^𝑌2 T603: □ (𝑌1 → ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 345 T591: □ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 346 T707: ^ (𝑌1 ∧ ¬𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 347 T1054: 𝑌1 → (𝑌2 → ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T1423: ¬𝑌1 → ^ (¬𝑌1 ) ⇝ ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) Cluster 348 T1055: 𝑌1 → ¬𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) T565: □ (^ ((𝑌1 ∨ 𝑌2 ) → ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 349 T1245: □ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) → 𝑌4 ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌4 ) Cluster 350 T1480: (𝑌1 ∨ ^𝑌1 ) → 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 351 T1482: (𝑌1 ∨ ^𝑌2 ) → 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 352 T1483: (𝑌1 ∨ ^𝑌2 ) → □𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) T1481: (𝑌1 ∨ ^𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 353 T1501: (^𝑌1 ∨ ^𝑌2 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) T1504: (^𝑌1 ∨ ^ (^𝑌1 )) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 354 T1681: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 ∨ 𝑌2 Cluster 355 T1789: (𝑌1 → ^ (^𝑌2 )) ∨ (𝑌1 → □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 356 T26: 𝑌1 ∧ □𝑌1 ⇝ □𝑌1 Cluster 357 T97: 𝑌1 ∧ ^ (𝑌2 ∨ 𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 358 T101: 𝑌1 ∧ ^ (𝑌2 ∨ ^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 359 T117: 𝑌1 ∧ ¬ (□𝑌2 ) ∧ □ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌3 ) ∧ ¬ (^𝑌2 )))

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Cluster 360 T157: ¬𝑌1 ∧ ¬𝑌2 ∧ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 361 T165: □𝑌1 ∧ □𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 362 T182: □ (𝑌1 ∧ 𝑌2 ) ∧ □𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 363 T194: □ (¬ (^𝑌1 )) ∧ □ (□ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 364 T227: ^ (𝑌1 ∧ 𝑌2 ) ∧ □𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 365 T246: ^ (□ (¬𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 366 T264: ^ (¬𝑌1 ) ∧ □ (^𝑌2 ∨ ^𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 367 T347: ¬ (□ (□𝑌1 )) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 368 T355: ¬ (^𝑌1 ) ∧ □ (□𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 369 T357: ¬ (^𝑌1 ) ∧ □ (^𝑌2 ∨ ^𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 370 T370: ¬ (𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 371 T436: (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 372 T453: □ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 373 T456: □ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 374 T462: □ (𝑌1 ∧ □ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 375 T478: □ (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 376 T522: □ ((^𝑌1 ∨ ^𝑌2 ) ∧ ^ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 377 T549: □ (^ (𝑌1 ∧ ¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 378 T560: □ (^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 )

Continued on next page 101

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Cluster 379 T619: □ ((¬𝑌1 ∨ ¬𝑌2 ) → ^𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 380 T622: □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 381 T672: □ (¬𝑌1 ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 382 T686: ^ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌2 ∧ ^𝑌1 Cluster 383 T752: ^ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 384 T766: ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 385 T778: ^ (𝑌1 → ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 386 T782: ^ (𝑌1 → (𝑌2 ∨ ^𝑌1 )) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 387 ⌈ = T819: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → ¬𝑌2 ⌊ = T889: □𝑌1 ↔ (𝑌2 ↔ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 388 T826: 𝑌1 ↔ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 389 T827: 𝑌1 ↔ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 390 T831: 𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 391 T874: □𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 392 T882: □𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 393 T886: □𝑌1 ↔ ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 394 T909: □ (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 395 T913: ^𝑌1 ↔ (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 396 T940: ¬ (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 397 Continued on next page 102

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Cluster 398 T977: 𝑌1 → (^𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 399 T994: 𝑌1 → □𝑌2 ⇝ 𝑌2 → 𝑌1 Cluster 400 T1005: 𝑌1 → □ (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 401 T1007: 𝑌1 → □ (¬𝑌1 ) ⇝ ¬ (□𝑌1 ) Cluster 402 T1048: 𝑌1 → (𝑌2 ↔ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 403 T1075: 𝑌1 → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 404 T1099: 𝑌1 → ((𝑌1 → ^𝑌2 ) ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 405 T1108: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 406 T1115: (𝑌1 ∧ 𝑌2 ) → ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 407 T1124: (□𝑌1 ∧ □𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 408 T1126: (□ (¬𝑌1 ) ∧ □ (¬𝑌2 )) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 409 T1140: (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 410 T1187: □𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 411 T1222: □𝑌1 → ¬ (^𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 412 T1255: □ (¬𝑌1 ∧ ¬𝑌2 ) → ¬𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 413 T1259: □ (□𝑌1 ) → □ (¬𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 414 T1268: □ (^𝑌1 ) → □𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 415 T1286: □ (¬𝑌1 ) → □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 416 Continued on next page 103

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Cluster 417 T1383: ^ (¬ (𝑌1 ∨ 𝑌2 )) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 418 T1384: ^ (¬ (𝑌1 ∨ 𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 419 T1386: ^ (𝑌1 ∨ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 420 T1394: ^ (¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 421 T1411: ¬𝑌1 → 𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 422 T1436: ¬ (□𝑌1 ) → ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 423 T1441: ¬ (□ (𝑌1 ∨ 𝑌2 )) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 424 T1456: ¬ (𝑌1 ∨ 𝑌2 ) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 425 T1508: (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 426 T1512: (¬𝑌1 ∨ ^ (¬𝑌1 )) → □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 427 T1542: ¬ (□ (𝑌1 → 𝑌2 )) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 428 T1543: ¬ (□ (𝑌1 → 𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 429 T1545: ¬ (□ (𝑌1 → 𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 430 T1594: 𝑌1 ∨ □ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 431 T1611: 𝑌1 ∨ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 432 T1614: 𝑌1 ∨ ^ (𝑌2 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 433 T1712: □𝑌1 ∨ □𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 434 T1748: ^ (□𝑌1 ) ∨ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 435 Continued on next page 104

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Cluster 436 T1795: (□𝑌1 → ((𝑌1 ∨ 𝑌2 ) → 𝑌2 )) ∨ (□𝑌1 → 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 437 T1812: ¬𝑌1 ∨ □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 438 T1822: ¬ (^𝑌1 ) ∨ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 439 T1824: ¬ (^𝑌1 ) ∨ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 440 T1829: 𝑌1 ∨ 𝑌2 ∨ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 441 T12: 𝑌1 ∧ 𝑌2 ∧ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) T106: 𝑌1 ∧ ¬𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 442 T54: 𝑌1 ∧ □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) T55: 𝑌1 ∧ □ (𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 443 T63: 𝑌1 ∧ ^𝑌2 ∧ □𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 444 T70: 𝑌1 ∧ ^𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) T320: ¬𝑌1 ∧ ^𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 445 T136: 𝑌1 ∧ 𝑌2 ∧ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 446 T291: (𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 ∧ ^𝑌2 T303: (^𝑌1 → 𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 ∧ ^𝑌2 Cluster 447 T305: (¬𝑌1 → 𝑌2 ) ∧ □ (^𝑌3 ) ⇝ □ (^ (𝑌3 ∧ ¬ (^𝑌2 ))) Cluster 448 T422: (^𝑌1 ∨ □𝑌2 ) ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 449 T431: (^𝑌1 ∨ ^ (¬𝑌2 )) ∧ 𝑌2 ⇝ 𝑌2 → ^ (¬𝑌2 ) Cluster 450 T461: □ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌2 → 𝑌1 Cluster 451 T653: □ (𝑌1 ∨ ¬𝑌2 ) ⇝ 𝑌2 → ^ (¬𝑌2 ) Cluster 452 T691: ^ (𝑌1 ∧ □𝑌2 ) ⇝ 𝑌2 → ^ (¬𝑌2 )

Continued on next page 105

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Cluster 453 T834: 𝑌1 ↔ □ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 454 T835: 𝑌1 ↔ □ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 455 T1015: 𝑌1 → □ (𝑌2 ∨ 𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 456 T1098: 𝑌1 → ((𝑌1 → 𝑌2 ) ∨ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 457 T1190: □𝑌1 → □ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 458 T1233: □𝑌1 → (^𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 459 T1280: □ (¬𝑌1 ) → 𝑌2 ⇝ 𝑌2 ∧ ¬ (^𝑌1 ) Cluster 460 T1307: □ (^𝑌1 ∨ ^ (^𝑌1 )) → □𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) Cluster 461 T1357: ^𝑌1 → (^𝑌2 ∨ ^ (^𝑌1 )) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 462 T1402: ((¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ) → 𝑌4 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) T1403: ((¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ) → ^𝑌4 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 463 T1412: ¬𝑌1 → 𝑌2 ⇝ □𝑌2 → 𝑌2 T1530: ¬ (□𝑌1 ) ⇝ □𝑌1 Cluster 464 T1686: (𝑌1 ∧ ¬ (□𝑌2 )) ∨ 𝑌2 ⇝ 𝑌2 ∨ □ (¬𝑌2 ) Cluster 465 T1695: (^𝑌1 ∧ □𝑌2 ) ∨ 𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) T1694: (^𝑌1 ∧ □𝑌1 ) ∨ 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 466 T1714: □𝑌1 ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 467 T1770: (𝑌1 → 𝑌2 ) ∨ □ (^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 468 T1780: (𝑌1 → □𝑌2 ) ∨ (𝑌1 → □ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 469 T1785: (𝑌1 → ^𝑌2 ) ∨ (𝑌1 → □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 470 T1806: ((𝑌1 ∨ 𝑌2 ) → □𝑌3 ) ∨ □𝑌3 ⇝ □𝑌3 ∧ □ (□𝑌3 )

Continued on next page 106

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Cluster 471 T28: 𝑌1 ∧ □𝑌1 ⇝ 𝑌1 ∨ ^𝑌1 Cluster 472 T29: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 473 T36: 𝑌1 ∧ □𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 474 T56: 𝑌1 ∧ □ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 475 T60: 𝑌1 ∧ ^𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 476 T99: 𝑌1 ∧ ^ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ 𝑌3 ) Cluster 477 T112: 𝑌1 ∧ ¬𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 478 T135: 𝑌1 ∧ 𝑌2 ∧ □𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 479 T154: ^𝑌1 ∧ ^𝑌2 ∧ 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 480 T156: ¬𝑌1 ∧ ^𝑌1 ∧ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 481 T170: □𝑌1 ∧ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 482 T171: □𝑌1 ∧ □ (^ ((𝑌1 ∨ 𝑌2 ) → ¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 483 T181: □𝑌1 ∧ □ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 484 T191: □ (¬𝑌1 ) ∧ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 485 T199: □ (¬𝑌1 ∨ ¬𝑌2 ) ∧ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 486 T200: ^𝑌1 ∧ 𝑌2 ⇝ ^ (𝑌1 ∧ 𝑌2 ) Cluster 487 T207: ^𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 488 T209: ^𝑌1 ∧ □ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 489 T213: ^𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 )

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Cluster 490 T219: ^𝑌1 ∧ ^ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 491 T220: ^𝑌1 ∧ ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 492 T222: ^𝑌1 ∧ ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 493 T232: ^ (𝑌1 ∧ ^𝑌1 ) ∧ □𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 494 T250: ^ (^𝑌1 ) ∧ □ (¬ (^𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 495 T258: ^ (¬𝑌1 ) ∧ 𝑌2 ⇝ ^ (¬𝑌1 ) Cluster 496 T261: ^ (¬𝑌1 ) ∧ □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 497 T270: ^ (¬ (□𝑌1 )) ∧ ^ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 498 T274: ^ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ^ (¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 499 T275: ^ (𝑌1 ∨ 𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 500 T277: ^ (𝑌1 ∨ ^𝑌1 ) ∧ ^𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 501 T310: ¬𝑌1 ∧ □𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 502 T311: ¬𝑌1 ∧ □𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 503 T332: ¬𝑌1 ∧ ^ (𝑌2 ∨ 𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 504 T368: ¬ (^ (^ (^𝑌1 ))) ∧ ^ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 505 T383: (𝑌1 ∨ 𝑌2 ) ∧ ^𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 506 T397: (𝑌1 ∨ ^𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 507 T409: (□𝑌1 ∨ □𝑌2 ) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 508 T413: (□ (𝑌1 ∧ ¬𝑌2 ) ∨ □ (𝑌2 ∧ ¬𝑌1 )) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 509 T429: (^𝑌1 ∨ ^𝑌2 ) ∧ ¬ (^ (^𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 510 T434: (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) ∧ □𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 511 T435: (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 512 T452: □ (𝑌1 ∧ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 513 T455: □ (𝑌1 ∧ 𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 514 T466: □ (𝑌1 ∧ ^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 515 T468: □ (𝑌1 ∧ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 516 T469: □ (𝑌1 ∧ ^ (^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 517 T482: □ (𝑌1 ∧ ¬ (□𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 518 T487: □ (□ (𝑌1 ∨ 𝑌2 ) ∧ □ (□ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 519 T489: □ (^𝑌1 ∧ ^𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 520 T493: □ (^𝑌1 ∧ ^ (¬ (□𝑌2 ))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 521 T501: □ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 522 T508: □ (¬ (□𝑌1 ) ∧ ^ (𝑌2 ∧ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 523 T517: □ ((𝑌1 ∨ □𝑌1 ) ∧ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 524 T523: □ ((^𝑌1 ∨ ^𝑌2 ) ∧ ¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 525 T525: □ ((¬𝑌1 ∨ ¬𝑌2 ) ∧ 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 526 T539: □ (□ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 527 T550: □ (^ (𝑌1 ∧ ¬𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 )

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Cluster 528 T568: □ (^ (¬𝑌1 )) ⇝ ¬ (□𝑌1 ) Cluster 529 T587: □ (𝑌1 → 𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 530 T608: □ (𝑌1 → (𝑌2 ∨ ^𝑌1 )) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 531 T616: □ (¬𝑌1 → (¬𝑌2 ∧ ¬𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 532 T618: □ ((¬𝑌1 ∨ ¬𝑌2 ) → □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 533 T634: □ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 534 T671: □ (¬𝑌1 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 535 T678: ^𝑌1 ⇝ □ (¬𝑌1 ) Cluster 536 T685: ^ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 537 T699: ^ (𝑌1 ∧ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 538 T719: ^ (¬𝑌1 ∧ ^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 539 T720: ^ (¬ (□𝑌1 ) ∧ ^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 540 T732: ^ (□ (𝑌1 ∧ 𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 541 T733: ^ (□ (𝑌1 ∧ ¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 542 T734: ^ (□ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 )) ⇝ ^ (¬𝑌2 ∧ ¬𝑌3 ∧ □𝑌1 ) Cluster 543 T739: ^ (□ ((𝑌1 ∨ 𝑌2 ) ∧ ^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 544 T755: ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 545 T757: ^ (^ (𝑌1 ∧ ^ (¬𝑌1 ))) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 546 T761: ^ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 547 T784: ^ ((¬𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 548 T796: ^ (¬ ((𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ 𝑌4 )) ⇝ ^ (¬𝑌4 ) Cluster 549 T797: ^ (¬ ((𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 )) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 550 T798: ^ (¬ (□𝑌1 )) ⇝ □ (¬𝑌1 ) Cluster 551 T808: ^ (𝑌1 ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 552 T811: ^ (𝑌1 ∨ ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 553 T832: 𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 554 T836: 𝑌1 ↔ □ (¬𝑌1 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 555 T843: 𝑌1 ↔ ^ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 556 T851: 𝑌1 ↔ ¬𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 557 T863: 𝑌1 ↔ (¬𝑌1 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 558 T864: 𝑌1 ↔ (¬𝑌1 ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 559 T871: □𝑌1 ↔ 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 560 T887: □𝑌1 ↔ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 561 T888: □𝑌1 ↔ ^ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 562 T893: □𝑌1 ↔ ¬ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 563 T905: □ (¬ (𝑌1 ∨ 𝑌2 )) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 564 T910: □ (¬𝑌1 ∨ ¬𝑌2 ) ↔ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 565 T923: ^ (¬𝑌1 ∧ ¬𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 )

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Cluster 566 T942: (𝑌1 ∨ 𝑌2 ) ↔ 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 567 T948: (¬𝑌1 ∨ ¬𝑌2 ) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 568 ⌈ = T955: 𝑌1 → 𝑌2 ⇝ □ (¬𝑌1 → 𝑌2 ) ⌊ = T1461: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 569 T964: 𝑌1 → 𝑌2 ⇝ □𝑌1 → ¬𝑌2 Cluster 570 ⌈ = T967: 𝑌1 → 𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) ⌊ = T1207: □𝑌1 → (𝑌2 → 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 571 T970: 𝑌1 → (𝑌1 ∧ □𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 572 T983: 𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ ¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 Cluster 573 T992: 𝑌1 → □𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 574 T1006: 𝑌1 → □ (𝑌2 ↔ ¬𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 575 T1021: 𝑌1 → □ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 576 T1044: 𝑌1 → ^ (𝑌2 ∨ ^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 577 T1045: 𝑌1 → ^ (𝑌2 ∨ ¬𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 578 T1052: 𝑌1 → (^𝑌2 ↔ ¬ (^𝑌3 )) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 579 T1059: 𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 580 T1061: 𝑌1 → ¬ (^𝑌2 ∧ ^𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 581 T1066: 𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 582 T1068: 𝑌1 → ¬ (□ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 583 T1087: 𝑌1 → ((𝑌2 ∧ ^𝑌1 ) ∨ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 ))

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Cluster 584 T1102: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 585 T1109: (𝑌1 ∧ 𝑌2 ) → □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 586 T1110: (𝑌1 ∧ 𝑌2 ) → □𝑌3 ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 587 T1116: (𝑌1 ∧ 𝑌2 ) → ^ (^𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 588 T1134: (^ (¬𝑌1 ) ∧ ^ (¬𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 589 T1181: □𝑌1 → □ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 590 T1183: □𝑌1 → □ (𝑌2 ↔ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 591 T1214: □𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 592 T1234: □𝑌1 → (^𝑌2 ∨ ^𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 593 T1236: □𝑌1 → (((𝑌1 ∨ 𝑌2 ) → ^𝑌2 ) ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 594 T1238: □𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 595 T1250: □ (¬𝑌1 ∧ ¬𝑌2 ) → □ (□𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 596 T1254: □ (¬𝑌1 ∧ ¬𝑌2 ) → ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 597 T1257: □ (□𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 598 T1266: □ (^𝑌1 ) → 𝑌2 ⇝ □𝑌1 → ¬𝑌2 Cluster 599 T1267: □ (^𝑌1 ) → (¬ (^𝑌2 ) ∧ ^ (𝑌3 ∨ 𝑌4 )) ⇝ □ (^ ((𝑌3 ∨ 𝑌4 ) ∧ ¬ (^𝑌2 ))) Cluster 600 T1271: □ (^𝑌1 ) → ^ (¬𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 601 T1272: □ (^𝑌1 ) → (𝑌2 ↔ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 602 T1300: □ (𝑌1 ∨ 𝑌2 ) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 603 T1309: □ (¬𝑌1 ∨ ¬𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 604 T1311: □ (¬𝑌1 ∨ ¬𝑌2 ) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 605 T1312: ^𝑌1 → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 606 T1314: ^𝑌1 → 𝑌2 ⇝ ^𝑌1 Cluster 607 T1320: ^𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 608 T1324: ^𝑌1 → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 609 T1328: ^𝑌1 → □ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 610 T1339: ^𝑌1 → ^ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 611 T1348: ^𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 612 T1361: ^ (𝑌1 ∧ 𝑌2 ) → ^𝑌3 ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 613 T1372: ^ (□𝑌1 ) → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 614 T1387: ^ (𝑌1 ∨ 𝑌2 ) → □ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 615 T1392: ^ (𝑌1 ∨ ^𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 616 T1410: ¬𝑌1 → 𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 617 T1418: ¬𝑌1 → □𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 618 T1433: ¬ (□𝑌1 ) → 𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 619 T1434: ¬ (□𝑌1 ) → □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 620 T1440: ¬ (□ (𝑌1 ∨ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 621 T1446: ¬ (^𝑌1 ) → ^𝑌2 ⇝ □ (¬𝑌1 → 𝑌2 )

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Cluster 622 T1455: ¬ (𝑌1 ∨ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 623 T1469: (𝑌1 ∨ 𝑌2 ) → ^ (□𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 624 T1489: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 Cluster 625 T1495: (□𝑌1 ∨ □ (^𝑌1 )) → □𝑌2 ⇝ □𝑌2 ∧ □ (□𝑌2 ) Cluster 626 T1502: (^𝑌1 ∨ ^𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 627 T1503: (^𝑌1 ∨ ^ (^𝑌1 )) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 628 (^ (□𝑌1 ) ∨ ^ (□𝑌2 )) T1506: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

→

^ (□ (□ (¬𝑌3 )))

⇝

Cluster 629 T1523: ¬ (𝑌1 ∧ □𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 630 T1527: ¬ (𝑌1 ∧ ^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 631 T1528: ¬ (𝑌1 ∧ ^𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 632 T1544: ¬ (□ (𝑌1 → 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 633 T1546: ¬ (□ (𝑌1 → ¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 634 T1547: ¬ (□ (𝑌1 → ¬𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 635 T1558: ¬ (^ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 636 T1559: ¬ (^ (𝑌1 → 𝑌2 )) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 637 T1560: ¬ (^ (𝑌1 → 𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 638 T1578: 𝑌1 ∨ 𝑌2 ∨ □𝑌3 ⇝ □ (𝑌1 ∨ 𝑌3 ∨ 𝑌2 ) Cluster 639 T1599: 𝑌1 ∨ □ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 640 Continued on next page 115

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Cluster 641 T1618: 𝑌1 ∨ ^ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 642 T1641: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 643 T1657: (𝑌1 ∧ □𝑌2 ) ∨ (𝑌3 ∧ □𝑌2 ) ∨ (¬𝑌1 ∧ ¬𝑌3 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 644 T1658: (𝑌1 ∧ □𝑌2 ) ∨ (𝑌3 ∧ □𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 645 T1678: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 646 T1684: (𝑌1 ∧ ¬𝑌2 ) ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 647 (□ (^𝑌1 ) ∧ □ (^𝑌2 )) T1693: □ (^ ((𝑌1 ∨ 𝑌3 ) ∧ ¬ (^𝑌2 )))

∨

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⇝

Cluster 648 T1696: (^𝑌1 ∧ □𝑌2 ) ∨ (^ (^𝑌1 ) ∧ □𝑌2 ) ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 649 T1724: □ (¬𝑌1 ) ∨ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 650 T1725: □ (𝑌1 ∨ 𝑌2 ) ∨ □ (𝑌3 ∧ 𝑌4 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌4 ) Cluster 651 T1738: ^𝑌1 ∨ ^ (^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 652 T1752: ^ (^ (□𝑌1 )) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 653 T1759: ^ (¬𝑌1 ) ∨ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 654 T1762: ^ (𝑌1 ∨ 𝑌2 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 655 T1765: (𝑌1 ↔ 𝑌2 ) ∨ ^𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 656 T1768: (𝑌1 → 𝑌2 ) ∨ 𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 657 T1773: (𝑌1 → 𝑌2 ) ∨ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 → (𝑌2 ∨ ^𝑌2 ) Cluster 658 T1790: (𝑌1 → ^ ((𝑌1 ∨ 𝑌2 ) → ^𝑌2 )) ∨ (𝑌1 → ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 ))

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Cluster 659 T1792: (𝑌1 → ¬𝑌1 ) ∨ (𝑌1 → ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 660 T1793: (□𝑌1 → 𝑌1 ) ∨ (□ (□𝑌1 ) → 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 661 T1794: (□𝑌1 → 𝑌2 ) ∨ (□𝑌1 → 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 662 T1813: ¬𝑌1 ∨ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 663 T1826: ¬ (𝑌1 ∨ 𝑌2 ) ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 664 T1827: ¬ (𝑌1 ∨ 𝑌2 ) ∨ ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 665 T6: 𝑌1 ⇝ ¬ (□𝑌1 ) Cluster 666 T7: 𝑌1 ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 667 T9: 𝑌1 ∧ 𝑌1 ∧ ¬ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 668 T10: 𝑌1 ∧ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 669 T13: 𝑌1 ∧ 𝑌2 ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 670 T14: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 671 T19: 𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 672 T33: 𝑌1 ∧ □𝑌2 ⇝ □ (𝑌1 ∧ 𝑌2 ) Cluster 673 T38: 𝑌1 ∧ □ (𝑌2 ∧ 𝑌1 ) ⇝ 𝑌2 ∧ □ (𝑌2 ∧ 𝑌1 ) Cluster 674 T39: 𝑌1 ∧ □ (𝑌2 ∧ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 675 T44: 𝑌1 ∧ □ (^ (𝑌1 ∨ 𝑌2 )) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 676 T46: 𝑌1 ∧ □ (¬𝑌1 ) ∧ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 677 T49: 𝑌1 ∧ □ (¬𝑌2 ) ⇝ 𝑌2 ∧ ¬ (^𝑌1 )

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Cluster 678 T52: 𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 679 T59: 𝑌1 ∧ ^𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 680 T80: 𝑌1 ∧ ^ (□ (𝑌1 ∨ 𝑌2 )) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 681 T81: 𝑌1 ∧ ^ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 682 T83: 𝑌1 ∧ ^ (^𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 683 T88: 𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 684 T89: 𝑌1 ∧ ^ (¬𝑌2 ) ∧ □ (¬𝑌2 ) ⇝ 𝑌2 ∧ ¬ (^𝑌1 ) Cluster 685 T95: 𝑌1 ∧ ^ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 686 T103: 𝑌1 ∧ ^ ((𝑌2 → ^𝑌2 ) ∨ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 687 T105: 𝑌1 ∧ ^ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 688 T116: 𝑌1 ∧ ¬ (□𝑌2 ) ∧ □𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌3 ) ∧ ¬ (^𝑌2 ))) Cluster 689 T118: 𝑌1 ∧ ¬ (□𝑌2 ) ⇝ 𝑌2 ∧ ¬ (^𝑌1 ) Cluster 690 T121: 𝑌1 ∧ ¬ (^𝑌1 ) ∧ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 691 T122: 𝑌1 ∧ ¬ (^𝑌1 ) ∧ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 692 T125: 𝑌1 ∧ ¬ (^𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 693 T127: 𝑌1 ∧ ¬ (^𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 694 T130: 𝑌1 ∧ ¬ (^ (𝑌2 ∧ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 695 T137: 𝑌1 ∧ 𝑌2 ∧ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 696 T138: 𝑌1 ∧ 𝑌2 ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 697 T139: 𝑌1 ∧ 𝑌2 ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 698 T140: 𝑌1 ∧ 𝑌2 ∧ ^ (𝑌2 ∧ ¬𝑌1 ) ⇝ (𝑌1 ∧ 𝑌2 ) → ^ (𝑌1 ∨ 𝑌2 ) Cluster 699 T141: 𝑌1 ∧ 𝑌2 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 700 T143: 𝑌1 ∧ 𝑌2 ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 701 T144: 𝑌1 ∧ □𝑌2 ∧ 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 702 T145: 𝑌1 ∧ □ (□𝑌2 ) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 703 T146: 𝑌1 ∧ ^𝑌1 ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 704 T147: 𝑌1 ∧ ^𝑌2 ∧ ¬ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 705 T149: 𝑌1 ∧ ¬𝑌2 ∧ ^ (𝑌2 ∨ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 706 T150: 𝑌1 ∧ ¬ (^𝑌1 ) ∧ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 707 T151: □𝑌1 ∧ □𝑌2 ∧ □𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 708 T152: □ (¬𝑌1 ) ∧ □ (¬𝑌2 ) ∧ ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 709 T153: ^𝑌1 ∧ □𝑌1 ∧ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 710 T155: ^𝑌1 ∧ ^ (□𝑌2 ) ∧ 𝑌1 ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 711 T160: □𝑌1 ∧ □𝑌2 ∧ □𝑌3 ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 712 T162: □𝑌1 ∧ □𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 713 T163: □𝑌1 ∧ □𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 714 T164: □𝑌1 ∧ □𝑌2 ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 715 T166: □𝑌1 ∧ □ (□𝑌1 ) ⇝ □𝑌1

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Cluster 716 T167: □𝑌1 ∧ □ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 717 T168: □𝑌1 ∧ □ (□ (□ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 718 T169: □𝑌1 ∧ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 719 T173: □𝑌1 ∧ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 720 T176: □𝑌1 ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 721 T178: □𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬𝑌1 → 𝑌2 ) Cluster 722 T180: □𝑌1 ∧ □ (¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 723 T183: □ (□ (𝑌1 ∨ 𝑌2 )) ∧ □ (□ (^𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 724 ⌈ = T184: □ (^𝑌1 ) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌1 ∧ 𝑌2 )) ⌊ = T188: □ (^ (𝑌1 ∨ 𝑌2 )) ∧ □ (^ (¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 725 T185: □ (^ (𝑌1 ∨ 𝑌2 )) ∧ □ (□ (□𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 726 T186: □ (^ (𝑌1 ∨ 𝑌2 )) ∧ □ (□ (□ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 727 T190: □ (¬𝑌1 ) ∧ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 728 T192: □ (¬𝑌1 ) ∧ □ (¬𝑌2 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 729 T193: □ (¬ (□𝑌1 )) ∧ □ (□ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 730 T195: □ (¬ (^𝑌1 )) ∧ □ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 731 T196: □ (¬ (^ (□ (¬𝑌1 )))) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 732 T197: □ (𝑌1 ∨ 𝑌2 ) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 733 T198: □ (^𝑌1 ∨ ^𝑌2 ) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 734 Continued on next page 120

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Cluster 735 T208: ^𝑌1 ∧ □ (¬𝑌2 ) ⇝ ^ (¬𝑌2 ) Cluster 736 T210: ^𝑌1 ∧ □ (𝑌2 ∨ 𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 737 T212: ^𝑌1 ∧ ^𝑌2 ∧ ^𝑌3 ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 738 ⌈ = T215: ^𝑌1 ∧ ^𝑌2 ⇝ ^ (𝑌1 ∧ 𝑌2 ) ⌊ = T218: ^𝑌1 ∧ ^ (□𝑌2 ) ⇝ ^ (𝑌1 ∧ □𝑌2 ) Cluster 739 T216: ^𝑌1 ∧ ^𝑌2 ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 740 T217: ^𝑌1 ∧ ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 741 T221: ^𝑌1 ∧ ^ (^ (¬𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 742 T224: ^𝑌1 ∧ ^ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 743 T225: ^𝑌1 ∧ ^ (¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 744 T226: ^𝑌1 ∧ ^ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 745 T229: ^ (𝑌1 ∧ 𝑌2 ) ∧ □ (□𝑌3 ) ∧ □ (□ (¬𝑌3 )) ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 746 T233: ^ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ∧ ^ (𝑌2 ∧ ¬𝑌1 ∧ ¬𝑌3 ) ∧ ^ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ) □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 )

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Cluster 747 T234: ^ (𝑌1 ∧ ¬𝑌2 ) ∧ ^ (^𝑌3 ) ∧ ^ (□𝑌3 ) ⇝ ^ (¬𝑌2 ∧ ¬𝑌1 ∧ □𝑌3 ) Cluster 748 T235: ^ (𝑌1 ∧ ¬𝑌2 ) ∧ ^ (^𝑌3 ) ⇝ ^ (¬𝑌2 ∧ ¬𝑌1 ∧ □𝑌3 ) Cluster 749 T236: ^ (¬𝑌1 ∧ ¬𝑌2 ) ∧ □ (□ (¬𝑌3 )) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 750 T237: ^ (¬𝑌1 ∧ ¬𝑌2 ) ∧ □ (□ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 )) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 751 T238: ^ (¬𝑌1 ∧ ¬𝑌2 ) ∧ ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 752 T241: ^ (□𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

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Cluster 753 T242: ^ (□𝑌1 ) ∧ □𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 754 T243: ^ (□𝑌1 ) ∧ ^ (□ (¬𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 755 T247: ^ (□ (¬𝑌1 )) ∧ □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 756 T251: ^ (^𝑌1 ) ∧ ^ (□ (¬𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 757 T252: ^ (^ (□𝑌1 )) ∧ ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 758 T253: ^ (^ (□ (¬𝑌1 ))) ∧ □ (□ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 759 T255: ^ (^ (¬𝑌1 )) ∧ □ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 760 T256: ^ (^ (𝑌1 ∨ 𝑌2 )) ∧ ^ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 761 T257: ^ (^ (𝑌1 ∨ 𝑌2 )) ∧ ^ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 762 T265: ^ (¬𝑌1 ) ∧ ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 763 T266: ^ (¬ (𝑌1 ∧ 𝑌2 )) ∧ □ (□𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 764 T267: ^ (¬ (𝑌1 ∧ 𝑌2 )) ∧ ^ (□𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 765 T268: ^ (¬ (□𝑌1 )) ∧ □ (^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 766 T269: ^ (¬ (□𝑌1 )) ∧ ^ (□𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 767 T272: ^ (¬ (^𝑌1 )) ∧ ^ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 768 T276: ^ (𝑌1 ∨ □𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 769 T278: ^ (𝑌1 ∨ ^𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 770 T279: (𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 771 T280: (𝑌1 ↔ 𝑌2 ) ∧ □𝑌3 ⇝ □ (¬ (𝑌2 ∨ 𝑌1 ) → 𝑌3 )

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Cluster 772 T281: (𝑌1 ↔ 𝑌2 ) ∧ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 773 T282: (𝑌1 ↔ 𝑌2 ) ∧ (𝑌3 ↔ 𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 774 T283: (𝑌1 ↔ 𝑌2 ) ∧ ¬ (^ (𝑌3 ∨ 𝑌4 )) ⇝ □ (¬ (𝑌3 ∨ 𝑌4 ) → 𝑌2 ) Cluster 775 T284: (𝑌1 ↔ (𝑌2 ∧ ¬𝑌3 )) ∧ □ (𝑌1 ↔ (𝑌2 ∧ ¬𝑌3 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌3 ) Cluster 776 T285: (𝑌1 ↔ ^𝑌3 ) ∧ (𝑌2 ↔ ^𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 777 T286: (□𝑌1 ↔ □𝑌2 ) ∧ 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 778 T287: (^𝑌1 ↔ ^𝑌2 ) ∧ (^𝑌3 ↔ ^𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 779 T288: (¬𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 780 T289: (¬𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ⇝ ¬ (𝑌2 ∨ 𝑌1 ) → 𝑌3 Cluster 781 T290: (¬𝑌1 ↔ ^𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 782 T292: (𝑌1 → 𝑌2 ) ∧ (𝑌1 → 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 783 T293: (𝑌1 → 𝑌2 ) ∧ (𝑌1 ∨ ^𝑌1 ) ⇝ □𝑌2 → (𝑌1 ∨ ^𝑌1 ) Cluster 784 T294: (𝑌1 → 𝑌3 ) ∧ (𝑌2 → 𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 785 T295: (𝑌1 → □𝑌3 ) ∧ (𝑌2 → □𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 786 T296: (𝑌1 → ^𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 787 T297: (𝑌1 → ^𝑌2 ) ∧ (𝑌1 → ^𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 788 T298: (𝑌1 → ^𝑌2 ) ∧ (𝑌3 → ^𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 789 T299: (𝑌1 → ¬𝑌2 ) ∧ □ (𝑌1 → ¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 790 T300: (𝑌1 → (𝑌2 ∨ ^ (^𝑌2 ))) ∧ 𝑌1 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 ))

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Cluster 791 T301: (□𝑌1 → 𝑌1 ) ∧ (□ (□𝑌1 ) → 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 792 T302: (□𝑌1 → ¬𝑌2 ) ∧ ¬𝑌1 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 793 T304: (^𝑌1 → ¬𝑌2 ) ∧ ¬𝑌1 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 794 ⌈ = T306: (¬𝑌1 → 𝑌2 ) ∧ (¬𝑌2 → 𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) ⌊ = T1163: □𝑌1 → ((¬𝑌2 → 𝑌3 ) ∧ (¬𝑌3 → 𝑌2 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 795 ⌈ = T307: ¬𝑌1 ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 ⌊ = T330: ¬𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 796 T313: ¬𝑌1 ∧ □ (^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 797 T315: ¬𝑌1 ∧ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 798 T317: ¬𝑌1 ∧ □ (𝑌1 ∨ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 799 T318: ¬𝑌1 ∧ ^𝑌1 ∧ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 800 T321: ¬𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 801 T322: ¬𝑌1 ∧ ^𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 802 T323: ¬𝑌1 ∧ ^𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 803 T327: ¬𝑌1 ∧ ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 804 T328: ¬𝑌1 ∧ ^ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 805 T329: ¬𝑌1 ∧ ^ (¬𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 806 T337: ¬ (□𝑌1 ) ∧ □ (□ (𝑌2 ∧ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 807 T338: ¬ (□𝑌1 ) ∧ □ (□ (^ (𝑌2 ∧ 𝑌3 ))) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 808 T340: ¬ (□𝑌1 ) ∧ □ (𝑌2 ∨ ^𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 )))

Continued on next page 124

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T350: ¬ (□ (□ (□ (^𝑌1 ))))∧□ (□ (□ (𝑌2 ∨ 𝑌3 ))) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 )))

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Cluster 809 T341: ¬ (□𝑌1 ) ∧ □ (□𝑌2 ∨ □𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 810 T342: ¬ (□𝑌1 ) ∧ ^𝑌2 ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 811 T344: ¬ (□𝑌1 ) ∧ ^ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ ^ (¬𝑌2 ∧ ¬𝑌3 ∧ □𝑌1 ) Cluster 812 T345: ¬ (□𝑌1 ) ∧ ^ (□ (𝑌2 ∧ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 813 T346: ¬ (□𝑌1 ) ∧ ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 814 T348: ¬ (□ (□𝑌1 )) ∧ ^𝑌2 ⇝ ^ (𝑌2 ∧ ¬ (^𝑌1 )) Cluster 815 T349: ¬ (□ (□ (□ (□ (□ (□𝑌1 )))))) □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 )))

∧

□ (^ (𝑌2 ∨ 𝑌3 ))

Cluster 816

Cluster 817 T352: ¬ (□ (^𝑌1 )) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 818 T354: ¬ (^𝑌1 ) ∧ □𝑌2 ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 819 T359: ¬ (^𝑌1 ) ∧ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 820 T360: ¬ (^𝑌1 ) ∧ ^𝑌2 ⇝ ¬𝑌2 ∨ ¬𝑌1 Cluster 821 T361: ¬ (^𝑌1 ) ∧ ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 822 T362: ¬ (^𝑌1 ) ∧ ^ (^𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 823 T364: ¬ (^𝑌1 ) ∧ ¬ (^ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 824 T365: ¬ (^ (𝑌1 ∧ 𝑌2 )) ∧ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 825 T366: ¬ (^ (□𝑌1 )) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 826 T367: ¬ (^ (^𝑌1 )) ∧ 𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 827 Continued on next page 125

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Cluster 828 T371: ¬ (𝑌1 ∨ 𝑌2 ) ∧ ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 829 T372: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4 ∨ 𝑌5 ) ∧ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4 ∨ 𝑌5 ) ⇝ □ (𝑌2 ∨ 𝑌3 ∨ 𝑌4 ) Cluster 830 T374: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 831 T375: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 832 T378: (𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 833 T379: (𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 834 T382: (𝑌1 ∨ 𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 835 T384: (𝑌1 ∨ 𝑌2 ) ∧ ^ (¬𝑌1 ∧ ¬𝑌2 ) ∧ ^ (□𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 836 T385: (𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 ⇝ ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 Cluster 837 T386: (𝑌1 ∨ 𝑌2 ) ∧ ¬ (𝑌1 ∧ 𝑌3 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 838 T387: (𝑌1 ∨ □𝑌1 ) ∧ 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 839 T391: (𝑌1 ∨ □ (^𝑌2 )) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 840 T395: (𝑌1 ∨ ^𝑌1 ) ∧ ¬𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 841 T398: (𝑌1 ∨ ^𝑌2 ) ∧ ¬ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 842 T399: (𝑌1 ∨ ^ (□𝑌2 )) ∧ (¬𝑌1 ∨ ^ (□𝑌2 )) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 843 T401: (𝑌1 ∨ ¬𝑌1 ) ∧ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 844 T402: (𝑌1 ∨ ¬𝑌2 ) ∧ 𝑌3 ⇝ ¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 Cluster 845 T403: (𝑌1 ∨ ¬𝑌2 ) ∧ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 846 Continued on next page 126

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Cluster 847 T406: ((𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ¬𝑌2 )) ∧ 𝑌3 ⇝ 𝑌1 → 𝑌3 Cluster 848 T407: (((𝑌1 ∨ 𝑌2 ) ∧ ^𝑌2 ) ∨ ^𝑌2 ) ∧ ^𝑌1 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 849 T408: (□𝑌1 ∨ □𝑌2 ) ∧ □𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 850 T410: (□𝑌1 ∨ □𝑌2 ) ∧ ¬𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 851 T411: (□𝑌1 ∨ □𝑌2 ) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 852 T414: (□ (□𝑌1 ) ∨ □ (□𝑌2 )) ∧ ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 853 T415: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ □ (□ (¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 854 T416: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ □ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 855 T417: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ □ (^ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 856 T418: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 857 T419: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ ^ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 858 T420: (□ (^𝑌1 ) ∨ □ (^𝑌2 )) ∧ ¬ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 859 T423: (^𝑌1 ∨ □𝑌2 ) ∧ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 860 T425: (^𝑌1 ∨ ^𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 861 T426: (^𝑌1 ∨ ^𝑌2 ) ∧ 𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 862 T427: (^𝑌1 ∨ ^𝑌2 ) ∧ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 863 T428: (^𝑌1 ∨ ^𝑌2 ) ∧ ¬𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 864 T430: (^𝑌1 ∨ ^ (^ (^𝑌1 ))) ∧ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 865 Continued on next page 127

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Cluster 866 T437: (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) ∧ ¬ (^ (^ (□𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 867 T438: (^ (^ (¬𝑌1 )) ∨ □𝑌1 ) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 868 T441: (¬𝑌1 ∨ □ (¬𝑌1 )) ∧ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 869 T442: (¬𝑌1 ∨ ¬𝑌2 ) ∧ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 870 T443: (¬ (^𝑌1 ) ∨ ¬ (□𝑌1 )) ∧ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 871 T448: □𝑌1 ⇝ ^ (¬𝑌1 ) Cluster 872 T450: □𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 873 T454: □ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 874 T457: □ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌2 ∧ ^𝑌1 Cluster 875 T458: □ (𝑌1 ∧ 𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 876 T463: □ (𝑌1 ∧ □ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 877 T464: □ (𝑌1 ∧ □ (¬𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 878 T467: □ (𝑌1 ∧ ^ (𝑌1 ∧ ^ (𝑌1 ∧ ^𝑌1 ))) ⇝ □𝑌1 Cluster 879 T471: □ (𝑌1 ∧ ^ (¬𝑌2 )) ⇝ ^ (¬𝑌2 ) Cluster 880 T472: □ (𝑌1 ∧ ¬𝑌1 ∧ ^ (¬𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 881 T474: □ (𝑌1 ∧ ¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 882 T475: □ (𝑌1 ∧ ¬𝑌2 ∧ ^𝑌3 ) ⇝ □ (^ (𝑌3 ∧ ¬ (^𝑌1 ))) Cluster 883 T477: □ (𝑌1 ∧ ¬𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 884 Continued on next page 128

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Cluster 885 T484: □ (𝑌1 ∧ ¬ (^𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 886 T485: □ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 887 T486: □ (□𝑌1 ∧ □ (¬ (^𝑌2 ))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 888 T488: □ (^𝑌1 ∧ ^𝑌2 ) ⇝ □ (𝑌1 ∧ 𝑌2 ) Cluster 889 T490: □ (^𝑌1 ∧ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 890 T492: □ (^𝑌1 ∧ ^ (¬𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 891 T494: □ (^ (𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 892 T495: □ (^ (𝑌1 ∨ 𝑌2 ) ∧ ^ (^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 893 T496: □ ((𝑌1 ↔ 𝑌2 ) ∧ 𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 894 T497: □ ((𝑌1 → 𝑌2 ) ∧ ¬ (□𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌1 ) ∧ ¬ (^𝑌3 ))) Cluster 895 T498: □ (¬𝑌1 ∧ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 896 T499: □ (¬𝑌1 ∧ □𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 897 T502: □ (¬𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 898 T503: □ (¬𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 899 T505: □ (¬𝑌1 ∧ ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 900 T506: □ (¬ (□𝑌1 ) ∧ □𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 901 T511: □ (¬ (^𝑌1 ) ∧ ^𝑌2 ) ⇝ ^ (𝑌2 ∧ ¬ (^𝑌1 )) Cluster 902 T512: □ (¬ (^ (□𝑌1 )) ∧ ^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 903 Continued on next page 129

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Cluster 904 T518: □ ((𝑌1 ∨ ^𝑌2 ) ∧ ¬ (□𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 905 T519: □ ((𝑌1 ∨ ^𝑌2 ) ∧ ¬ (^ (□𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 906 T520: □ ((𝑌1 ∨ ¬𝑌2 ) ∧ ^𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 907 T521: □ ((□𝑌1 ∨ □𝑌2 ) ∧ □ (□ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 908 T524: □ ((^𝑌1 ∨ ^𝑌2 ) ∧ ¬ (□𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 909 T526: □ ((¬𝑌1 ∨ ¬𝑌2 ) ∧ □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 910 T531: □ (□𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 911 T532: □ (□𝑌1 ) ⇝ ¬ (□𝑌1 ) Cluster 912 T533: □ (□𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 913 T534: □ (□ (𝑌1 ∧ □ (¬𝑌1 ))) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 914 T535: □ (□ ((𝑌1 ↔ ¬𝑌2 ) ∧ ^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 915 T536: □ (□ ((𝑌1 ↔ ¬𝑌2 ) ∧ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 916 T537: □ (□ ((𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 917 T538: □ (□ ((𝑌1 ∨ 𝑌2 ) ∧ ^𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 918 T540: □ (□ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 919 T541: □ (□ (□𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 920 T542: □ (□ ((𝑌1 ∨ 𝑌2 ) → □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 921 T543: □ (□ ((𝑌1 ∨ 𝑌2 ) → ¬ (□𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 922 Continued on next page 130

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T544: □ (□ ((𝑌1 ∨ 𝑌2 ) → ¬ (□ (¬𝑌3 )))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 923 T552: □ (^ ((𝑌1 ↔ ¬𝑌2 ) ∧ □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌2 ∨ 𝑌1 ) ∧ ¬ (^𝑌3 ))) Cluster 924 T553: □ (^ (¬𝑌1 ∧ □𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 925 T554: □ (^ (¬𝑌1 ∧ ¬𝑌2 ∧ □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 926 T556: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ^ (¬ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ∨ 𝑌4 )))) □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌4 ))) Cluster 927 T557: □ (^ (((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) ∧ ¬ (□𝑌3 ))) □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 928 T558: □ (^ (□ (𝑌1 ∧ ^ (¬𝑌2 )))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 929 T559: □ (^ (□ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 930 T561: □ (^ (𝑌1 → □ (¬𝑌2 ))) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 931 T562: □ (^ (𝑌1 → ¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 932 T563: □ (^ (□𝑌1 → □ (¬𝑌2 ))) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 933 T564: □ (^ (^ (¬𝑌1 ) → (𝑌2 ∨ 𝑌3 ))) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 934 T566: □ (^ ((𝑌1 ∨ ^𝑌1 ) → □𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 935 T569: □ (^ (𝑌1 ∨ 𝑌2 ∨ ^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 936 T570: □ (^ (𝑌1 ∨ 𝑌2 ∨ ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 937 T571: □ (^ (𝑌1 ∨ 𝑌2 ∨ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 938 T572: □ (^ (𝑌1 ∨ □𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 939 T574: □ (^ (¬𝑌1 ∨ ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 940 T575: □ (𝑌1 ↔ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 )

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Cluster 941 T576: □ (𝑌1 ↔ 𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 942 T577: □ (𝑌1 ↔ □ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 943 T578: □ (𝑌1 ↔ □ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 944 T579: □ (𝑌1 ↔ ¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 945 T581: □ (𝑌1 ↔ ¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 946 T583: □ (¬𝑌1 ↔ ¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 947 T585: □ ((𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ↔ ¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 948 T586: □ (𝑌1 → 𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 949 T588: □ (𝑌1 → 𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 950 T592: □ (𝑌1 → 𝑌2 ) ⇝ □𝑌2 → 𝑌1 Cluster 951 T593: □ (𝑌1 → (𝑌2 ∧ □ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 952 T594: □ (𝑌1 → (□ (¬𝑌1 ) ∧ □ (□ (¬𝑌1 )))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 953 T595: □ (𝑌1 → (^ (¬𝑌1 ) ∧ ^ (^ (¬𝑌1 )))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 954 T598: □ (𝑌1 → ^ (¬𝑌1 ∧ ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 955 T600: □ (𝑌1 → ^ (𝑌1 ∨ ^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 956 T601: □ (𝑌1 → (𝑌2 ↔ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 957 T602: □ (𝑌1 → (¬𝑌2 ↔ 𝑌3 )) ⇝ □ (¬ (𝑌3 ∨ 𝑌2 ) → 𝑌1 ) Cluster 958 T605: □ (𝑌1 → ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 959 T606: □ (𝑌1 → ¬ (^𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

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Cluster 960 T611: □ (^𝑌1 → 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 961 T612: □ (^𝑌1 → (𝑌2 ∨ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 962 T614: □ (¬𝑌1 → 𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 963 T617: □ (¬𝑌1 → ¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 964 T620: □ (¬𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 965 T624: □ (¬ (𝑌1 ∧ 𝑌2 )) ⇝ ^ (¬𝑌2 ) Cluster 966 T625: □ (¬ ((𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 )) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 967 T626: □ (¬ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 )) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 968 T629: □ (¬ (𝑌1 → 𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 969 T630: □ (¬ (𝑌1 → 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 970 T632: □ (¬ (𝑌1 ∨ 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 971 T635: □ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 972 T638: □ (𝑌1 ∨ 𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 973 T644: □ (𝑌1 ∨ ^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 974 T647: □ (𝑌1 ∨ ^ (^𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 975 ⌈ = T648: □ (𝑌1 ∨ ^ (^𝑌1 )) ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) ⌊ = T1019: 𝑌1 → □ (𝑌2 ∨ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 976 T652: □ (𝑌1 ∨ ¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 977 T656: □ ((𝑌1 ∧ ^𝑌1 ) ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 978 Continued on next page 133

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Cluster 979 T658: □ ((𝑌1 ∧ ¬ (^𝑌3 )) ∨ (𝑌2 ∧ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 980 T659: □ ((¬𝑌1 ∧ ^𝑌1 ) ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 981 T660: □ ((¬𝑌1 ∧ ¬𝑌2 ) ∨ 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 982 T661: □ ((¬ (^𝑌1 ) ∧ ^𝑌2 ) ∨ (¬ (^𝑌1 ) ∧ ^𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 983 T665: □ (^𝑌1 ∨ □ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 984 T666: □ (^𝑌1 ∨ ^𝑌2 ∨ ^𝑌3 ) ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 985 T668: □ (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 986 T669: □ ((𝑌1 → ^𝑌1 ) ∨ (𝑌1 → □𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 987 T670: □ ((𝑌1 → ¬𝑌1 ) ∨ (𝑌1 → ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 988 T673: □ (𝑌1 ∨ 𝑌2 ∨ □ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 989 T682: ^𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 990 T687: ^ (𝑌1 ∧ 𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 991 T693: ^ (𝑌1 ∧ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 992 T697: ^ (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌2 ∧ □ (𝑌2 ∧ 𝑌1 ) Cluster 993 T698: ^ (𝑌1 ∧ ^𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 994 T700: ^ (𝑌1 ∧ ^ (¬𝑌1 )) ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) Cluster 995 T701: ^ (𝑌1 ∧ ^ (¬𝑌2 )) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 996 T705: ^ (𝑌1 ∧ ¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

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Cluster 997 T709: ^ (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 998 T710: ^ (𝑌1 ∧ ¬𝑌2 ) ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 999 T712: ^ (𝑌1 ∧ ¬ (^𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1000 T713: ^ (𝑌1 ∧ ¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1001 T715: ^ (𝑌1 ∧ 𝑌2 ∧ ^𝑌3 ) ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 1002 T716: ^ (𝑌1 ∧ ¬𝑌2 ∧ 𝑌2 ∧ ¬𝑌1 ) ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1003 T717: ^ (𝑌1 ∧ ¬𝑌2 ∧ ^𝑌3 ) ⇝ ^ (¬𝑌2 ∧ ¬𝑌1 ∧ □𝑌3 ) Cluster 1004 T718: ^ ((𝑌1 → ^𝑌2 ) ∧ 𝑌2 ) ⇝ ^𝑌2 ∨ ^ (^𝑌2 ) Cluster 1005 T721: ^ (¬ (^ (^𝑌1 )) ∧ ^ (𝑌2 ∨ 𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 1006 T723: ^ ((𝑌1 ∨ 𝑌2 ) ∧ 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1007 T730: ^ (□𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1008 T731: ^ (□𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 1009 T735: ^ (□ (𝑌1 ∧ ¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1010 T736: ^ (□ (¬𝑌1 ∧ ^ (^ (^𝑌2 )))) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 1011 T737: ^ (□ ((𝑌1 ∨ 𝑌2 ) ∧ □𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1012 T740: ^ (□ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1013 T742: ^ (□ ((𝑌1 ∨ ^𝑌2 ) ∧ ^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1014 T743: ^ (□ ((^𝑌1 ∨ ^𝑌2 ) ∧ ^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1015 T744: ^ (□ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 )

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Cluster 1016 T745: ^ (□ (^ ((𝑌1 ∨ 𝑌2 ) ∧ □ (¬𝑌3 )))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1017 T746: ^ (□ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ^ (¬𝑌3 )))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1018 T747: ^ (□ (^ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 )))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1019 T748: ^ (□ (𝑌1 → ¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1020 T749: ^ (□ ((𝑌1 ∨ 𝑌2 ) → □ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1021 T750: ^ (□ (𝑌1 ∨ 𝑌2 ∨ ¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1022 T751: ^ (□ (𝑌1 ∨ 𝑌2 ∨ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1023 T756: ^ (^ (𝑌1 ∧ 𝑌2 )) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1024 T762: ^ (^ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ^ (¬𝑌3 )))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1025 T763: ^ (^ (^ (^ (^ (^ (^ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ))))))) ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 1026 T764: ^ (^ (𝑌1 → □ (¬𝑌2 ))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 1027 T767: ^ (^ (𝑌1 ∨ 𝑌2 ∨ ^𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1028 T768: ^ (𝑌1 ↔ 𝑌2 ) ⇝ □ (𝑌1 ∧ 𝑌2 ) Cluster 1029 T769: ^ (𝑌1 ↔ 𝑌2 ) ⇝ □ (𝑌1 → 𝑌2 ) Cluster 1030 T770: ^ (□𝑌1 ↔ □ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1031 T771: ^ ((¬𝑌1 ↔ 𝑌2 ) ↔ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌1 ) → 𝑌3 ) Cluster 1032 T774: ^ (𝑌1 → 𝑌2 ) ⇝ ^ (¬𝑌2 ) Cluster 1033 T776: ^ (𝑌1 → □𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 1034 T777: ^ (𝑌1 → □ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 ))

Continued on next page 136

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Cluster 1035 T780: ^ (𝑌1 → ¬ (□𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1036 T781: ^ (𝑌1 → ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1037 T783: ^ (𝑌1 → (¬𝑌1 ∨ ^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1038 T786: ^ ((¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1039 T787: ^ ((¬𝑌1 ∧ ¬𝑌2 ) → ^ (□𝑌3 )) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1040 T788: ^ ((¬𝑌1 ↔ 𝑌2 ) → 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌1 ) → 𝑌3 ) Cluster 1041 T789: ^ ((¬𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1042 T792: ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1043 ⌈ = T795: ^ (¬ (𝑌1 ∧ ¬𝑌2 )) ⇝ ^ (𝑌1 ↔ ¬𝑌2 ) ⌊ = T1040: 𝑌1 → ^ (¬ (𝑌2 ∧ ¬𝑌3 )) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1044 T799: ^ (¬ ((𝑌1 ∨ 𝑌2 ) → □𝑌3 )) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1045 T800: ^ (𝑌1 ∨ 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1046 T802: ^ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ⇝ ^ (𝑌1 ∨ 𝑌2 ) Cluster 1047 T805: ^ (𝑌1 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1048 T807: ^ (𝑌1 ∨ ^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1049 T812: ^ ((𝑌1 ∧ 𝑌2 ) ∨ 𝑌1 ) ⇝ □ (𝑌1 ∧ 𝑌2 ) Cluster 1050 T813: ^ ((𝑌1 ∧ ¬ (^𝑌3 )) ∨ (𝑌2 ∧ ¬ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1051 T817: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1052 T820: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1053 Continued on next page 137

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Cluster 1054 T824: 𝑌1 ↔ (^𝑌1 ∧ □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1055 T825: 𝑌1 ↔ (^ (¬𝑌1 ) ∧ ^ (□𝑌2 )) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1056 T830: 𝑌1 ↔ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1057 T833: 𝑌1 ↔ □ (𝑌1 ∨ □𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1058 T839: 𝑌1 ↔ ^ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1059 T841: 𝑌1 ↔ ^ (¬𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 1060 T842: 𝑌1 ↔ ^ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1061 T844: 𝑌1 ↔ ^ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1062 T845: 𝑌1 ↔ ^ (𝑌2 ∨ ¬𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1063 T846: 𝑌1 ↔ (^ (¬𝑌1 ) → □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1064 T847: 𝑌1 ↔ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1065 T848: 𝑌1 ↔ ¬𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1066 T849: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1067 T852: 𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌2 ∨ 𝑌1 ) Cluster 1068 T853: 𝑌1 ↔ ¬𝑌2 ⇝ 𝑌2 ∨ 𝑌1 Cluster 1069 T854: 𝑌1 ↔ ¬ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ 𝑌2 ) Cluster 1070 T855: 𝑌1 ↔ ¬ (𝑌1 ∧ ^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1071 T857: 𝑌1 ↔ ¬ (^𝑌1 ∧ ^ (□𝑌2 )) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1072 Continued on next page 138

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Cluster 1073 T860: 𝑌1 ↔ (𝑌2 ∨ ^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1074 T861: 𝑌1 ↔ (𝑌2 ∨ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1075 T862: 𝑌1 ↔ (^𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1076 T865: (𝑌1 ∧ 𝑌2 ) ↔ □ (^𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1077 T866: (𝑌1 ∧ 𝑌2 ) ↔ ^ (□𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1078 T867: (□𝑌1 ∧ □𝑌2 ) ↔ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1079 T868: (^ (^𝑌1 ) ∧ ^ (□ (¬𝑌2 ))) □ (^ ((𝑌1 ∨ 𝑌3 ) ∧ ¬ (^𝑌2 )))

↔

(^ (^𝑌3 ) ∧ ^ (□ (¬𝑌2 )))

⇝

Cluster 1080 T870: (¬𝑌1 ∧ ¬𝑌2 ) ↔ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1081 T872: □𝑌1 ↔ 𝑌2 ⇝ □ (¬𝑌1 → 𝑌2 ) Cluster 1082 T873: □𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 1083 T875: □𝑌1 ↔ (□𝑌2 ∧ □𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1084 T876: □𝑌1 ↔ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1085 T877: □𝑌1 ↔ □𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1086 T878: □𝑌1 ↔ □ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1087 T879: □𝑌1 ↔ □ (^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1088 T880: □𝑌1 ↔ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1089 T881: □𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1090 T883: □𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 )

Continued on next page 139

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Cluster 1091 T884: □𝑌1 ↔ □ (¬𝑌2 ) ⇝ ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1092 T885: □𝑌1 ↔ □ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1093 T890: □𝑌1 ↔ ¬𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1094 T891: □𝑌1 ↔ ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1095 T892: □𝑌1 ↔ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1096 T896: □𝑌1 ↔ (𝑌2 ∨ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1097 T897: □𝑌1 ↔ (□ (¬𝑌1 ) ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1098 T899: □ (^ (𝑌1 ∧ 𝑌2 )) ↔ ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1099 T901: □ (¬𝑌1 ) ↔ 𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1100 T902: □ (¬𝑌1 ) ↔ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1101 T903: □ (¬𝑌1 ) ↔ ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1102 T904: □ (¬𝑌1 ) ↔ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1103 T906: □ (¬ (𝑌1 ∨ 𝑌2 )) ↔ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1104 T907: □ (𝑌1 ∨ 𝑌2 ) ↔ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1105 T908: □ (^𝑌1 ∨ □𝑌1 ) ↔ 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1106 T911: □ (¬𝑌1 ∨ ¬𝑌2 ) ↔ ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1107 T914: ^𝑌1 ↔ (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1108 T915: ^𝑌1 ↔ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1109 T916: ^𝑌1 ↔ □𝑌1 ⇝ □𝑌1 → 𝑌1

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Cluster 1110 T917: ^𝑌1 ↔ □ (𝑌1 ∨ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1111 T918: ^𝑌1 ↔ ^𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1112 T920: ^𝑌1 ↔ ¬ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1113 T921: ^𝑌1 ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1114 T922: ^ (𝑌1 ∧ 𝑌2 ) ↔ □𝑌3 ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) Cluster 1115 T924: ^ (¬𝑌1 ∧ ¬𝑌2 ) ↔ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1116 T925: ^ (□ (𝑌1 ∧ 𝑌2 )) ↔ ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1117 T927: ^ (¬ (𝑌1 ∧ 𝑌2 )) ↔ □𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1118 T928: ^ (¬ (𝑌1 ∨ 𝑌2 )) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1119 T929: (¬𝑌1 ↔ 𝑌2 ) ↔ □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1120 T931: ¬𝑌1 ↔ 𝑌2 ⇝ ¬𝑌2 ∨ ¬𝑌1 Cluster 1121 T933: ¬𝑌1 ↔ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1122 T934: ¬𝑌1 ↔ ¬𝑌2 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1123 T935: ¬ (□𝑌1 ) ↔ 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1124 T936: ¬ (□𝑌1 ) ↔ ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1125 T937: ¬ (^𝑌1 ) ↔ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1126 T938: ¬ (^𝑌1 ) ↔ 𝑌2 ⇝ 𝑌2 ∧ ¬ (^𝑌1 ) Cluster 1127 T939: ¬ (^ (𝑌1 ∧ 𝑌2 )) ↔ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1128 T941: ¬ (𝑌1 ∨ 𝑌2 ) ↔ ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 )

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Cluster 1129 T943: (𝑌1 ∨ 𝑌2 ) ↔ □𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ 𝑌2 ) Cluster 1130 T944: (𝑌1 ∨ 𝑌2 ) ↔ ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1131 T946: ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 )) ↔ 𝑌3 ⇝ 𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1132 T947: (□𝑌1 ∨ □𝑌2 ) ↔ □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1133 T949: 𝑌1 → 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1134 T956: 𝑌1 → 𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1135 T965: 𝑌1 → 𝑌2 ⇝ □𝑌2 → 𝑌1 Cluster 1136 T971: 𝑌1 → (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1137 T972: 𝑌1 → (𝑌1 ∧ ¬ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1138 T973: 𝑌1 → (𝑌2 ∧ 𝑌3 ) ⇝ ¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 Cluster 1139 T974: 𝑌1 → (𝑌2 ∧ ¬𝑌1 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1140 T975: 𝑌1 → (□𝑌1 ∧ 𝑌2 ) ⇝ □𝑌1 → 𝑌1 Cluster 1141 T978: 𝑌1 → (^ (¬𝑌1 ) ∧ 𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ 𝑌2 ) Cluster 1142 T979: 𝑌1 → ((□𝑌1 → 𝑌1 ) ∧ □𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 1143 T980: 𝑌1 → ((□𝑌1 → 𝑌1 ) ∧ ^𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 1144 T982: 𝑌1 → (¬𝑌2 ∧ □ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1145 T984: 𝑌1 → (¬ (□𝑌2 ) ∧ □𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 1146 T985: 𝑌1 → (¬ (^𝑌2 ) ∧ ^𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1147 T986: 𝑌1 → ((𝑌2 ∨ ¬𝑌3 ) ∧ ^ (𝑌3 ∨ ¬𝑌2 )) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 )

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Cluster 1148 T988: 𝑌1 → □𝑌1 ⇝ ¬ (□𝑌1 ) Cluster 1149 T990: 𝑌1 → □𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 1150 T993: 𝑌1 → □𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1151 T996: 𝑌1 → □ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1152 T997: 𝑌1 → □ (𝑌1 ∧ ¬𝑌2 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1153 T998: 𝑌1 → □ (𝑌1 ∧ ¬𝑌2 ) ⇝ □𝑌1 → 𝑌1 Cluster 1154 T999: 𝑌1 → □ (𝑌2 ∧ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1155 T1000: 𝑌1 → □ (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1156 T1001: 𝑌1 → □ ((□𝑌1 → 𝑌1 ) ∧ 𝑌1 ) ⇝ □𝑌1 → 𝑌1 Cluster 1157 T1003: 𝑌1 → □ (^𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1158 T1004: 𝑌1 → □ (^ (¬𝑌1 )) ⇝ ¬ (□𝑌1 ) Cluster 1159 T1012: 𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1160 T1013: 𝑌1 → □ (¬ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1161 T1020: 𝑌1 → □ (𝑌2 ∨ ¬𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1162 T1023: 𝑌1 → ^𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1163 T1026: 𝑌1 → ^𝑌2 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1164 T1029: 𝑌1 → ^ (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1165 T1030: 𝑌1 → ^ (𝑌2 ∧ 𝑌3 ) ⇝ ^ (𝑌2 ∧ 𝑌3 ∧ □𝑌1 ) Cluster 1166 T1032: 𝑌1 → ^ ((𝑌1 → 𝑌2 ) ∧ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 ))

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Cluster 1167 T1033: 𝑌1 → ^ ((𝑌2 ∨ ¬𝑌3 ) ∧ (𝑌3 ∨ ¬𝑌2 )) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1168 T1034: 𝑌1 → ^ (𝑌2 → ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1169 T1036: 𝑌1 → ^ (¬𝑌1 ) ⇝ ¬ (□𝑌1 ) Cluster 1170 T1037: 𝑌1 → ^ (¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1171 T1038: 𝑌1 → ^ (¬ (𝑌2 ∧ 𝑌3 )) ⇝ ^ (¬𝑌2 ∧ ¬𝑌3 ∧ □𝑌1 ) Cluster 1172 T1041: 𝑌1 → ^ (𝑌1 ∨ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1173 T1042: 𝑌1 → ^ (𝑌2 ∨ 𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1174 T1046: 𝑌1 → ^ ((^𝑌2 → 𝑌1 ) ∨ 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1175 T1047: 𝑌1 → (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1176 T1049: 𝑌1 → (□𝑌2 ↔ □𝑌3 ) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1177 T1050: 𝑌1 → (^𝑌1 ↔ ^𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1178 T1051: 𝑌1 → (^𝑌2 ↔ ^ (¬𝑌3 )) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1179 T1053: 𝑌1 → (𝑌2 → 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌3 ↔ 𝑌2 ) Cluster 1180 T1060: 𝑌1 → ¬ (□𝑌2 ∧ □𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1181 T1062: 𝑌1 → ¬ ((𝑌2 ∨ 𝑌3 ) ∧ (¬𝑌2 ∨ ¬𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1182 T1067: 𝑌1 → ¬ (□𝑌2 ) ⇝ 𝑌1 ∨ □ (¬𝑌2 ) Cluster 1183 T1070: 𝑌1 → ¬ (^𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1184 T1071: 𝑌1 → ¬ (^𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1185 T1072: 𝑌1 → ¬ (^ (𝑌2 ∧ 𝑌3 )) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 )

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Cluster 1186 T1073: 𝑌1 → ¬ (^ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1187 T1074: 𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1188 T1077: 𝑌1 → (𝑌1 ∨ □𝑌2 ) ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1189 T1084: 𝑌1 → (𝑌2 ∨ ^ (^ (^𝑌2 ))) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1190 T1085: 𝑌1 → (𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → ^ (𝑌2 ↔ ¬𝑌3 ) Cluster 1191 T1086: 𝑌1 → ((𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ ^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1192 T1089: 𝑌1 → (□𝑌2 ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1193 T1090: 𝑌1 → (□𝑌2 ∨ □𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1194 T1094: 𝑌1 → (^𝑌2 ∨ ^𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1195 T1095: 𝑌1 → (^ ((𝑌1 ∨ 𝑌2 ) ∧ ^𝑌2 ) ∨ □𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1196 T1100: 𝑌1 → ((𝑌2 → 𝑌3 ) ∨ (𝑌3 → 𝑌2 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1197 T1101: 𝑌1 → (¬𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1198 T1103: 𝑌1 → (¬𝑌2 ∨ ¬𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1199 T1106: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌1 → 𝑌3 Cluster 1200 T1107: (𝑌1 ∧ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1201 T1112: (𝑌1 ∧ 𝑌2 ) → □ (□𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1202 T1113: (𝑌1 ∧ 𝑌2 ) → □ (𝑌1 ∨ ^𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1203 T1114: (𝑌1 ∧ 𝑌2 ) → □ (𝑌1 ∨ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1204 T1117: (𝑌1 ∧ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 )

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Cluster 1205 T1118: (𝑌1 ∧ □𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1206 T1119: (𝑌1 ∧ □𝑌1 ) → (¬𝑌1 ∨ ^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1207 T1120: (𝑌1 ∧ ^𝑌1 ) → ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1208 T1121: (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) → 𝑌4 ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌4 ) Cluster 1209 T1122: (□𝑌1 ∧ □𝑌2 ) → 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1210 T1123: (□𝑌1 ∧ □𝑌2 ) → 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1211 T1127: (□ (¬𝑌1 ) ∧ □ (¬𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1212 T1128: (□ (¬𝑌1 ) ∧ □ (¬𝑌2 )) → ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1213 T1129: (^𝑌1 ∧ ^𝑌2 ) → 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1214 T1130: (^𝑌1 ∧ ^𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1215 T1131: (^𝑌1 ∧ ^𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1216 T1132: (^ (¬𝑌1 ) ∧ ^ (¬𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1217 T1133: (^ (¬𝑌1 ) ∧ ^ (¬𝑌2 )) → □𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1218 T1135: ((□𝑌1 → 𝑌1 ) ∧ □𝑌1 ) → 𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 1219 T1137: (¬𝑌1 ∧ ¬𝑌2 ) → □𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1220 T1139: (¬𝑌1 ∧ ¬𝑌2 ) → ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1221 T1141: (¬ (^𝑌1 ) ∧ ¬ (^𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1222 T1143: □𝑌1 → 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1223 T1146: □𝑌1 → 𝑌2 ⇝ □ (𝑌2 → 𝑌1 )

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Cluster 1224 T1147: □𝑌1 → 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1225 T1148: □𝑌1 → 𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1226 T1154: □𝑌1 → 𝑌2 ⇝ □𝑌2 → 𝑌1 Cluster 1227 T1156: □𝑌1 → (𝑌1 ∧ 𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1228 T1158: □𝑌1 → (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1229 T1160: □𝑌1 → (𝑌2 ∧ □𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌3 ↔ 𝑌2 ) Cluster 1230 T1161: □𝑌1 → (𝑌2 ∧ 𝑌3 ∧ (𝑌2 ∨ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1231 T1162: □𝑌1 → (^𝑌2 ∧ ^𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1232 T1165: □𝑌1 → (¬𝑌2 ∧ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1233 T1166: □𝑌1 → ((𝑌2 ∨ 𝑌3 ) ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1234 T1167: □𝑌1 → ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (𝑌1 ∨ 𝑌4 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1235 T1172: □𝑌1 → □𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1236 T1173: □𝑌1 → □𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1237 T1174: □𝑌1 → □𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1238 T1177: □𝑌1 → □ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1239 T1178: □𝑌1 → □ (𝑌2 ∧ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1240 T1180: □𝑌1 → □ (□ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1241 T1185: □𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1242 T1188: □𝑌1 → □ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 )

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Cluster 1243 T1191: □𝑌1 → ^𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1244 T1198: □𝑌1 → ^𝑌2 ⇝ □𝑌2 → (𝑌1 ∨ ^𝑌1 ) Cluster 1245 T1200: □𝑌1 → ^ (𝑌2 ↔ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1246 T1206: □𝑌1 → (^𝑌2 ↔ ^ (¬𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1247 T1208: □𝑌1 → (𝑌2 → ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1248 T1209: □𝑌1 → (^𝑌1 → 𝑌2 ) ⇝ □ (^𝑌2 ) Cluster 1249 T1211: □𝑌1 → ¬𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1250 T1212: □𝑌1 → ¬𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 1251 T1213: □𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌1 → 𝑌2 ) Cluster 1252 T1216: □𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1253 T1217: □𝑌1 → ¬𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1254 T1218: □𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1255 T1219: □𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1256 T1221: □𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1257 T1230: □𝑌1 → ((𝑌2 ∧ 𝑌3 ) ∨ (¬𝑌2 ∧ ¬𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1258 T1231: □𝑌1 → (□ (^𝑌2 ) ∨ □ (^𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1259 T1240: □ (𝑌1 ∧ 𝑌2 ) → □𝑌3 ⇝ ^ (𝑌1 ∧ 𝑌2 ∧ □𝑌3 ) Cluster 1260 T1241: □ (𝑌1 ∧ 𝑌2 ) → ¬𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1261 T1242: □ (𝑌1 ∧ 𝑌2 ) → (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 )

Continued on next page 148

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Cluster 1262 T1244: □ (𝑌1 ∧ ^𝑌1 ) → ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1263 T1246: □ (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) → □𝑌1 ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1264 T1247: □ (𝑌1 ∧ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1265 T1251: □ (¬𝑌1 ∧ ¬𝑌2 ) → □ (^𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1266 T1252: □ (¬𝑌1 ∧ ¬𝑌2 ) → □ (¬𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1267 T1256: □ (□𝑌1 ) → 𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1268 T1258: □ (□𝑌1 ) → □𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1269 T1260: □ (□𝑌1 ) → ¬ (□𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 1270 T1261: □ (□𝑌1 ) → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1271 T1262: □ (□ (¬𝑌1 )) → 𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1272 T1263: □ (□ (¬𝑌1 )) → □ (¬𝑌2 ∨ ¬𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 1273 T1264: □ (□ (𝑌1 ∨ 𝑌2 )) → □ (□ (□ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1274 T1265: □ (^𝑌1 ) → 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1275 T1269: □ (^𝑌1 ) → □ (𝑌2 ↔ ¬𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1276 T1270: □ (^𝑌1 ) → ^ (^𝑌2 ) ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1277 T1273: □ (^𝑌1 ) → ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1278 T1275: □ (^ (𝑌1 ∧ 𝑌2 )) → ¬ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1279 T1276: □ (^ (𝑌1 ∨ 𝑌2 )) → □ (^ (^ (¬𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1280 T1277: □ (^ (𝑌1 ∨ 𝑌2 )) → ¬ (□ (^ (^𝑌3 ))) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 1281 T1278: □ (^ (𝑌1 ∨ 𝑌2 )) □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

→

¬ (□ (^ (^ (¬𝑌3 ))))

⇝

Cluster 1282 ⌈ = T1279: □ (¬𝑌1 ) → 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) ⌊ = T1284: □ (¬𝑌1 ) → 𝑌2 ⇝ 𝑌2 → ^ (¬𝑌1 ) Cluster 1283 T1282: □ (¬𝑌1 ) → 𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1284 T1283: □ (¬𝑌1 ) → 𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1285 T1287: □ (¬𝑌1 ) → ^𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1286 T1288: □ (¬𝑌1 ) → (𝑌2 ↔ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1287 T1289: □ (¬𝑌1 ) → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

(^ (¬𝑌1 ) ∨ ^ (□ (¬𝑌1 )))

⇝

𝑌1

→

Cluster 1288 T1290: □ (¬ (𝑌1 ∧ 𝑌2 )) → ^𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1289 T1291: □ (¬ (𝑌1 ∧ 𝑌2 )) → ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1290 T1292: □ (¬ (𝑌1 ∧ 𝑌2 )) → ¬𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1291 T1295: □ (𝑌1 ∨ 𝑌2 ) → 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1292 T1299: □ (𝑌1 ∨ 𝑌2 ) → ^𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1293 T1301: □ (𝑌1 ∨ 𝑌2 ) → (^𝑌1 ∨ ^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1294 T1303: □ (𝑌1 ∨ □𝑌2 ) → □ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌2 ∨ 𝑌1 ) ∧ ¬ (^𝑌3 ))) Cluster 1295 T1304: □ (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1296 T1305: □ (^𝑌1 ∨ ^𝑌2 ) → ^ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1297 T1306: □ (^𝑌1 ∨ ^𝑌2 ) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1298 T1310: □ (¬𝑌1 ∨ ¬𝑌2 ) → □ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

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Cluster 1299 T1316: ^𝑌1 → 𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1300 T1319: ^𝑌1 → (𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1301 T1321: ^𝑌1 → ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (𝑌1 ∨ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1302 T1322: ^𝑌1 → ((𝑌2 ∨ ¬𝑌3 ) ∧ (𝑌3 ∨ ¬𝑌2 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1303 T1323: ^𝑌1 → ((□𝑌2 ∨ □𝑌3 ) ∧ ¬𝑌4 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌4 ))) Cluster 1304 T1326: ^𝑌1 → □𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1305 T1327: ^𝑌1 → □𝑌2 ⇝ □𝑌1 → ¬𝑌2 Cluster 1306 T1329: ^𝑌1 → □ (□ (¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1307 T1330: ^𝑌1 → □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1308 T1332: ^𝑌1 → □ (¬𝑌2 ) ⇝ ^ (¬𝑌2 ) Cluster 1309 T1333: ^𝑌1 → □ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1310 T1335: ^𝑌1 → ^𝑌2 ⇝ 𝑌1 → 𝑌2 Cluster 1311 T1336: ^𝑌1 → ^ (□ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1312 T1338: ^𝑌1 → ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1313 T1340: ^𝑌1 → ^ (¬𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1314 T1341: ^𝑌1 → ^ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1315 T1342: ^𝑌1 → ^ (𝑌2 ∨ ^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1316 T1343: ^𝑌1 → ^ (𝑌2 ∨ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1317 T1345: ^𝑌1 → (□𝑌2 ↔ □𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 )

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Cluster 1318 T1349: ^𝑌1 → ¬ (□𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1319 T1352: ^𝑌1 → (𝑌1 ∨ ¬ (^𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1320 T1354: ^𝑌1 → (𝑌2 ∨ ^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1321 T1355: ^𝑌1 → ((𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1322 T1356: ^𝑌1 → (^𝑌1 ∨ □ (□𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1323 (^ (^ (□ (¬𝑌1 ))) ∨ ^ (□ (¬𝑌1 ))) T1359: ^𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

⇝

𝑌1

→

Cluster 1324 T1360: ^𝑌1 → (¬𝑌1 ∨ □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1325 T1363: ^ (𝑌1 ∧ 𝑌2 ) → ^ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1326 T1367: ^ (¬𝑌1 ∧ ¬𝑌2 ) → ^ (𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1327 T1368: ^ (¬𝑌1 ∧ ¬𝑌2 ) → ^ (¬𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1328 T1369: ^ (□𝑌1 ) → □𝑌1 ⇝ □𝑌1 → 𝑌1 Cluster 1329 T1370: ^ (□𝑌1 ) → □ (□ (¬𝑌2 )) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 1330 T1371: ^ (□𝑌1 ) → ¬ (^𝑌2 ) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 1331 T1373: ^ (^𝑌1 ) → □ (□ (□ (¬𝑌2 ))) ⇝ □ (^ (𝑌1 ∧ ¬ (^𝑌2 ))) Cluster 1332 T1374: ^ (^ (𝑌1 ∧ 𝑌2 )) → ^ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1333 T1375: ^ (^ (¬𝑌1 )) → □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1334 T1376: ^ (^ (𝑌1 ∨ 𝑌2 )) → ^ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1335 T1377: ^ (^ (𝑌1 ∨ 𝑌2 )) → ¬ (^𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1336 Continued on next page 152

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Cluster 1337 T1380: ^ (¬ (𝑌1 ∧ 𝑌2 )) → ^𝑌3 ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1338 T1381: ^ (¬ (𝑌1 ↔ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌2 ∨ 𝑌1 ) → 𝑌3 ) Cluster 1339 T1385: ^ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1340 T1388: ^ (𝑌1 ∨ 𝑌2 ) → ^𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1341 T1389: ^ (𝑌1 ∨ 𝑌2 ) → ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1342 T1390: ^ (𝑌1 ∨ 𝑌2 ) → ^ (¬𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1343 T1391: ^ (𝑌1 ∨ 𝑌2 ) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1344 T1393: ^ (𝑌1 ∨ ^𝑌1 ) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1345 T1395: ^ (¬𝑌1 ∨ ¬𝑌2 ) → □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1346 T1397: (𝑌1 ↔ 𝑌2 ) → □ (^𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1347 T1398: (𝑌1 ↔ 𝑌2 ) → ^𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1348 T1399: (𝑌1 ↔ □𝑌1 ) → □𝑌2 ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1349 T1401: (𝑌1 ↔ ¬ (^𝑌1 )) → □𝑌2 ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1350 T1404: (¬𝑌1 ↔ 𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1351 T1405: (¬𝑌1 ↔ □𝑌1 ) → ^𝑌2 ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1352 T1407: ((𝑌1 ∨ 𝑌2 ) → (^𝑌1 ∨ ^ (^𝑌1 ))) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1353 T1408: ((𝑌1 ∨ 𝑌2 ) → (^ (𝑌1 ∨ 𝑌2 ) ∨ ^ (^ (𝑌1 ∨ 𝑌2 )))) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1354 T1414: ¬𝑌1 → (^𝑌2 ∧ ^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1355 Continued on next page 153

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Cluster 1356 T1417: ¬𝑌1 → □𝑌2 ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1357 T1419: ¬𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1358 T1420: ¬𝑌1 → □ (¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1359 T1421: ¬𝑌1 → ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1360 T1422: ¬𝑌1 → ^𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1361 T1424: ¬𝑌1 → ^ (𝑌2 ∨ 𝑌3 ) ⇝ 𝑌1 → ^ (𝑌3 ↔ ¬𝑌2 ) Cluster 1362 T1425: ¬𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1363 T1426: ¬𝑌1 → ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1364 T1427: ¬𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1365 T1428: ¬𝑌1 → ¬ (𝑌2 ∧ 𝑌3 ) ⇝ ¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 Cluster 1366 T1429: ¬𝑌1 → (□ (¬𝑌1 ) ∨ □ (□ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1367 T1430: ¬𝑌1 → (□ (¬𝑌1 ) ∨ □ (^ (¬𝑌1 ))) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1368 T1435: ¬ (□𝑌1 ) → □𝑌2 ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1369 T1437: ¬ (□𝑌1 ) → ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1370 T1438: ¬ (□𝑌1 ) → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1371 T1442: ¬ (^𝑌1 ) → 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1372 T1443: ¬ (^𝑌1 ) → 𝑌2 ⇝ 𝑌2 ∧ ^𝑌1 Cluster 1373 T1445: ¬ (^𝑌1 ) → 𝑌2 ⇝ □𝑌1 → ¬𝑌2 Cluster 1374 Continued on next page 154

Transformations

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T1447: ¬ (^𝑌1 ) → ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 ))

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Cluster 1375 T1448: ¬ (^𝑌1 ) → ¬ (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1376 T1450: ¬ (^ (𝑌1 ∨ 𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1377 T1457: (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) → ¬𝑌4 ⇝ ^ (¬𝑌4 ) Cluster 1378 T1458: (𝑌1 ∨ 𝑌2 ) → 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1379 T1462: (𝑌1 ∨ 𝑌2 ) → 𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1380 T1470: (𝑌1 ∨ 𝑌2 ) → ¬𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1381 T1471: (𝑌1 ∨ 𝑌2 ) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1382 T1472: (𝑌1 ∨ 𝑌2 ) → ¬ (^𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1383 T1473: (𝑌1 ∨ □𝑌1 ∨ 𝑌2 ) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1384 T1478: (𝑌1 ∨ □𝑌1 ) → ^𝑌2 ⇝ □𝑌2 → (𝑌1 ∨ ^𝑌1 ) Cluster 1385 T1479: (𝑌1 ∨ □𝑌1 ) → (𝑌2 ∨ 𝑌3 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1386 T1484: (𝑌1 ∨ ^𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1387 T1488: (𝑌1 ∨ ¬𝑌2 ) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1388 T1491: (□𝑌1 ∨ □𝑌2 ) → 𝑌3 ⇝ 𝑌3 → ^ (𝑌2 ↔ ¬𝑌1 ) Cluster 1389 T1492: (□𝑌1 ∨ □𝑌2 ) → 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1390 T1493: (□𝑌1 ∨ □𝑌2 ) → □ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1391 T1494: (□𝑌1 ∨ □𝑌2 ) → ¬ (□𝑌3 ) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1392 T1496: (□𝑌1 ∨ □ (¬𝑌1 )) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1393 Continued on next page 155

Transformations

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T1497: (□ (𝑌1 ∧ 𝑌2 ) ∨ □ (^ (𝑌1 ∧ 𝑌2 ))) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 )

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Cluster 1394 (□ (^𝑌1 ) ∨ □ (^𝑌2 )) T1498: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

^ (□ (□ (¬𝑌3 )))

→

⇝

Cluster 1395 T1499: (□ (¬𝑌1 ) ∨ □ (¬𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1396 T1500: (^𝑌1 ∨ □𝑌1 ) → 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 1397 T1505: (^ (□𝑌1 ) ∨ ^ (□𝑌2 )) → □ (□ (¬𝑌3 )) ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1398 (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) T1507: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 )))

→

^ (^ (¬ (□𝑌3 )))

⇝

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→

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Cluster 1399 (^ (^𝑌1 ) ∨ ^ (^𝑌2 )) T1509: □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1400 T1510: (^ (𝑌1 ∨ 𝑌2 ) ∨ □ (𝑌1 ∨ 𝑌2 )) → 𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1401 T1511: (^ (𝑌1 ∨ 𝑌2 ) ∨ ^ (^ (𝑌1 ∨ 𝑌2 ))) → □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1402 T1514: (¬𝑌1 ∨ ¬𝑌2 ) → □𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1403 T1515: (¬ (□𝑌1 ) ∨ ¬ (□𝑌2 )) → 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1404 T1517: ¬𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1405 T1520: ¬ (𝑌1 ∧ 𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1406 T1524: ¬ (𝑌1 ∧ □𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1407 T1525: ¬ (𝑌1 ∧ □𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1408 T1526: ¬ (𝑌1 ∧ □ (¬ (𝑌1 ∧ ^𝑌2 ))) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1409 T1529: ¬ (^𝑌1 ∧ □𝑌2 ) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1410 T1534: ¬ (□𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 1411 Continued on next page 156

Transformations

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T1537: ¬ (□ (𝑌1 ∧ 𝑌2 )) ⇝ ^ (¬𝑌2 )

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Cluster 1412 T1538: ¬ (□ (𝑌1 ∧ ¬𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1413 T1539: ¬ (□ ((𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) ∧ ¬𝑌4 )) ⇝ ^ (¬𝑌4 ) Cluster 1414 T1540: ¬ (□ (^𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1415 T1541: ¬ (□ (𝑌1 ↔ 𝑌2 )) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1416 T1548: ¬ (□ (𝑌1 ∨ 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1417 T1553: ¬ (^𝑌1 ) ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 1418 T1554: ¬ (^ (𝑌1 ∧ ^𝑌2 )) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1419 T1555: ¬ (^ (𝑌1 ∧ ¬𝑌2 )) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1420 T1556: ¬ (^ (□𝑌1 )) ⇝ □ (¬𝑌1 ) Cluster 1421 T1557: ¬ (^ (□𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1422 T1561: ¬ (^ (𝑌1 → 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1423 T1563: ¬ (^ (𝑌1 ∨ 𝑌2 )) ⇝ □ (¬𝑌1 ∨ ¬𝑌2 ) Cluster 1424 T1564: ¬ (^𝑌1 ↔ ^𝑌2 ) ⇝ ^ (𝑌1 ↔ ¬𝑌2 ) Cluster 1425 T1565: ¬ (^𝑌1 ↔ ^𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1426 T1567: ¬ (𝑌1 → ¬𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1427 T1568: ¬ (□𝑌1 → 𝑌2 ) ⇝ □ (¬𝑌2 ∨ ¬𝑌1 ) Cluster 1428 T1569: ¬ (□𝑌1 → (𝑌2 ∨ 𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1429 T1570: ¬ (^𝑌1 → 𝑌2 ) ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1430 Continued on next page 157

Transformations

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T1571: ¬ (𝑌1 ∨ 𝑌2 ∨ ^𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 )

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Cluster 1431 T1572: ¬ (𝑌1 ∨ 𝑌2 ∨ ¬ (□𝑌3 )) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1432 T1574: ¬ (^𝑌1 ∨ ^𝑌2 ) ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1433 T1575: ¬ (^𝑌1 ∨ ^𝑌2 ) ⇝ □ (𝑌1 ∧ 𝑌2 ) Cluster 1434 T1576: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ ¬ (□𝑌2 ) Cluster 1435 T1577: 𝑌1 ∨ 𝑌2 ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1436 T1579: 𝑌1 ∨ 𝑌2 ∨ □𝑌3 ⇝ □ (𝑌3 ∨ 𝑌1 ∨ 𝑌2 ) Cluster 1437 T1581: 𝑌1 ∨ 𝑌2 ⇝ 𝑌1 ∧ ^𝑌2 Cluster 1438 T1582: 𝑌1 ∨ 𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1439 T1591: 𝑌1 ∨ □𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1440 T1595: 𝑌1 ∨ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1441 T1596: 𝑌1 ∨ □ (¬𝑌2 ) ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 1442 T1597: 𝑌1 ∨ □ (¬𝑌2 ) ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1443 T1600: 𝑌1 ∨ □ (𝑌1 ∨ 𝑌2 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1444 T1601: 𝑌1 ∨ ^𝑌1 ∨ ^𝑌2 ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1445 T1605: 𝑌1 ∨ ^𝑌2 ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1446 T1606: 𝑌1 ∨ ^𝑌2 ⇝ 𝑌1 ∧ ¬ (^𝑌2 ) Cluster 1447 T1608: 𝑌1 ∨ ^𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1448 T1613: 𝑌1 ∨ ^ (𝑌2 ∧ 𝑌1 ) ⇝ ^𝑌1 ∨ ^ (^𝑌1 ) Cluster 1449 Continued on next page 158

Transformations

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T1617: 𝑌1 ∨ ^ (□ (¬𝑌1 )) ∨ □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

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Cluster 1450 T1619: 𝑌1 ∨ ^ (^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1451 T1623: 𝑌1 ∨ ^ (¬ (𝑌2 ∧ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1452 T1624: 𝑌1 ∨ ¬𝑌1 ∨ □ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1453 T1625: 𝑌1 ∨ ¬𝑌1 ∨ ^ (¬𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1454 T1626: 𝑌1 ∨ ¬𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1455 T1628: 𝑌1 ∨ ¬ (𝑌2 ∧ 𝑌3 ) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1456 T1629: 𝑌1 ∨ ¬ (□𝑌1 ) ∨ ¬ (□ (□𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1457 T1631: 𝑌1 ∨ ¬ (^𝑌1 ) ∨ ^𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1458 T1633: 𝑌1 ∨ ¬ (^ (𝑌2 ∧ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1459 T1634: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌1 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 ) Cluster 1460 T1636: (𝑌1 ∧ 𝑌2 ) ∨ 𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1461 T1638: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌1 ∧ 𝑌2 ∧ 𝑌3 ) ∨ 𝑌2 ⇝ □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 ) Cluster 1462 T1642: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ □𝑌1 ) ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1463 T1644: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ ^𝑌1 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1464 T1645: (𝑌1 ∧ 𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1465 T1646: (𝑌1 ∧ 𝑌2 ) ∨ (^𝑌2 ∧ ^ (□𝑌2 )) ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1466 T1647: (𝑌1 ∧ 𝑌2 ) ∨ (¬𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ↔ 𝑌2 Cluster 1467 T1649: (𝑌1 ∧ 𝑌2 ) ∨ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1468 Continued on next page 159

Transformations

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T1651: (𝑌1 ∧ 𝑌2 ) ∨ ¬ (^ (¬𝑌2 )) ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 )

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Cluster 1469 T1652: (𝑌1 ∧ 𝑌2 ) ∨ ¬ (𝑌1 ∨ 𝑌2 ) ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1470 T1653: (𝑌1 ∧ □𝑌1 ) ∨ (^𝑌1 ∧ □𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1471 T1654: (𝑌1 ∧ □𝑌2 ) ∨ 𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1472 T1655: (𝑌1 ∧ □𝑌2 ) ∨ (𝑌1 ∧ □𝑌3 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1473 T1656: (𝑌1 ∧ □𝑌2 ) ∨ (𝑌1 ∧ □ (□𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1474 T1659: (𝑌1 ∧ □𝑌2 ) ∨ ^𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1475 T1660: (𝑌1 ∧ □𝑌2 ) ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1476 T1661: (𝑌1 ∧ □ (¬𝑌1 )) ∨ ^ (𝑌1 ∧ □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1477 T1662: (𝑌1 ∧ ^𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1478 T1663: (𝑌1 ∧ ^𝑌2 ) ∨ 𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1479 T1664: (𝑌1 ∧ ^𝑌2 ) ∨ (𝑌1 ∧ □𝑌3 ) ⇝ 𝑌1 → (𝑌3 ∨ □𝑌1 ) Cluster 1480 T1665: (𝑌1 ∧ ^𝑌2 ) ∨ (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 1481 T1666: (𝑌1 ∧ ^𝑌2 ) ∨ (𝑌2 ∧ ^𝑌1 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1482 T1667: (𝑌1 ∧ ^𝑌2 ) ∨ (^𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1483 T1668: (𝑌1 ∧ ^𝑌2 ) ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1484 T1669: (𝑌1 ∧ ^𝑌2 ) ∨ ^𝑌2 ⇝ ^𝑌2 ∨ ^ (^𝑌2 ) Cluster 1485 T1670: (𝑌1 ∧ ^𝑌2 ) ∨ ¬𝑌2 ⇝ 𝑌2 → 𝑌1 Cluster 1486 T1671: (𝑌1 ∧ ^ (𝑌1 ∧ ^𝑌2 )) ∨ (𝑌1 ∧ ^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1487 Continued on next page 160

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Cluster 1488 T1673: (𝑌1 ∧ ^ (□𝑌2 )) ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1489 T1674: (𝑌1 ∧ ^ (¬𝑌1 )) ∨ 𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1490 T1675: (𝑌1 ∧ ^ (¬ (𝑌1 ∧ 𝑌2 ))) ∨ (¬𝑌1 ∧ ^ (𝑌1 ∧ 𝑌2 )) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ 𝑌2 ) Cluster 1491 T1676: (𝑌1 ∧ ¬𝑌2 ∧ ¬𝑌3 ) □ (𝑌1 ∨ 𝑌2 ∨ 𝑌3 )

∨

(𝑌2 ∧ ¬𝑌1 ∧ ¬𝑌3 )

∨

(𝑌3 ∧ ¬𝑌1 ∧ ¬𝑌2 )

⇝

Cluster 1492 T1677: (𝑌1 ∧ ¬𝑌2 ) ∨ 𝑌2 ⇝ □ (¬ (𝑌2 ∧ 𝑌1 )) Cluster 1493 T1680: (𝑌1 ∧ ¬𝑌2 ) ∨ (𝑌2 ∧ ¬𝑌1 ) ⇝ ¬ (𝑌1 ∨ 𝑌2 ) Cluster 1494 T1682: (𝑌1 ∧ ¬𝑌2 ) ∨ ^𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1495 T1683: (𝑌1 ∧ ¬𝑌2 ) ∨ ^ (𝑌2 ∧ ¬𝑌1 ) ⇝ ^ (𝑌1 ↔ ¬𝑌2 ) Cluster 1496 T1685: (𝑌1 ∧ ¬𝑌4 ) ∨ (𝑌2 ∧ ¬𝑌4 ) ∨ (𝑌3 ∧ ¬𝑌4 ) ⇝ ^ (¬𝑌4 ) Cluster 1497 T1687: (𝑌1 ∧ ¬ (^𝑌1 ) ∧ ¬ (^ (□𝑌2 ))) ∨ (¬𝑌1 ∧ ^𝑌1 ∧ ^ (□𝑌2 )) ^ (𝑌1 ∧ □𝑌2 )

⇝ ¬𝑌1 ↔

Cluster 1498 T1688: (𝑌1 ∧ ¬ (^𝑌1 )) ∨ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1499 T1689: (𝑌1 ∧ ¬ (^ (𝑌1 ∧ □𝑌2 ))) ∨ ¬𝑌1 ∨ ^ (𝑌1 ∧ □𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1500 T1690: (𝑌1 ∧ ¬ (𝑌1 ∨ 𝑌2 )) ∨ □𝑌3 ⇝ 𝑌3 → (𝑌1 ∨ □𝑌3 ) Cluster 1501 T1692: (^𝑌1 ∧ ^ (□𝑌2 ) ∧ ¬𝑌1 ) ∨ (𝑌1 ∧ ¬ (^ (𝑌1 ∧ □𝑌2 ))) ^ (𝑌1 ∧ □𝑌2 )

⇝

¬𝑌1

↔

Cluster 1502 T1697: (^𝑌1 ∧ □𝑌2 ) ∨ □𝑌1 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1503 T1698: (^𝑌1 ∧ □𝑌2 ) ∨ ^ (𝑌2 ∧ ^𝑌1 ) ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 1504 T1699: (^𝑌1 ∧ □𝑌2 ) ∨ ^ (^𝑌1 ) ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 1505 Continued on next page 161

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Cluster 1506 T1701: (^𝑌1 ∧ ^𝑌2 ) ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1507 T1702: (^𝑌1 ∧ ^ (^𝑌1 )) ∨ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 )) Cluster 1508 T1703: (^𝑌1 ∧ ^ (^𝑌2 )) ∨ 𝑌1 ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1509 T1704: (^ (¬𝑌1 ) ∧ □𝑌1 ) ∨ ¬𝑌1 ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1510 T1705: ((^𝑌1 → ^ (□𝑌2 )) ∧ ¬𝑌1 ) ∨ (𝑌1 ∧ ¬ (^𝑌1 → ^ (□𝑌2 ))) ^ (𝑌1 ∧ □𝑌2 )

⇝ ¬𝑌1 ↔

Cluster 1511 T1706: (¬ (□𝑌1 ) ∧ □𝑌2 ) ∨ (¬ (□𝑌1 ) ∧ □𝑌3 ) ⇝ □ (^ ((𝑌2 ∨ 𝑌3 ) ∧ ¬ (^𝑌1 ))) Cluster 1512 T1710: □𝑌1 ∨ □𝑌2 ∨ □𝑌3 ∨ □𝑌4 ∨ □𝑌5 ⇝ □ (𝑌2 ∨ 𝑌3 ∨ 𝑌4 ) Cluster 1513 T1716: □𝑌1 ∨ □ (□𝑌1 ) ⇝ □𝑌1 Cluster 1514 T1718: □𝑌1 ∨ □ (¬ (𝑌2 ∨ 𝑌3 )) ⇝ □ (¬ (𝑌2 ∨ 𝑌3 ) → 𝑌1 ) Cluster 1515 T1719: □ ((𝑌1 ∨ 𝑌2 ) ∧ □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1516 T1720: □ ((𝑌1 ∨ 𝑌2 ) ∧ ¬𝑌3 ) ∨ □ (𝑌3 ∧ ¬ (𝑌1 ∨ 𝑌2 )) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1517 T1722: □ (𝑌1 → (𝑌1 ∨ 𝑌2 )) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1518 T1723: □ (^ (𝑌1 ∨ 𝑌2 ) → 𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1519 T1726: □ (𝑌1 ∨ 𝑌2 ) ∨ □ (¬𝑌3 ) ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1520 T1727: □ (𝑌1 ∨ □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1521 T1729: ^𝑌1 ∨ □𝑌2 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1522 T1731: ^𝑌1 ∨ □ (□ (□𝑌1 )) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1523 T1734: ^𝑌1 ∨ ^𝑌2 ⇝ 𝑌1 ∧ □ (𝑌1 ∧ 𝑌2 )

Continued on next page 162

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Cluster 1524 T1737: ^𝑌1 ∨ ^ (^𝑌1 ) ⇝ □𝑌1 Cluster 1525 T1739: ^𝑌1 ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1526 T1740: ^𝑌1 ∨ ^ (^ (□𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1527 T1741: ^ (𝑌1 ∧ 𝑌2 ) ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1528 T1742: ^ (𝑌1 ∧ 𝑌2 ) ∨ ^𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1529 T1743: ^ (𝑌1 ∧ ^𝑌2 ) ∨ ^𝑌2 ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1530 T1744: ^ (𝑌1 ∧ ^𝑌2 ) ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1531 T1745: ^ (^𝑌1 ∧ □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1532 T1746: ^ (¬𝑌1 ∧ ¬𝑌2 ) ∨ □ (¬𝑌3 ) ⇝ ^ (¬𝑌1 ∧ ¬𝑌2 ∧ □𝑌3 ) Cluster 1533 T1750: ^ (^𝑌1 ) ∨ □ (□𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1534 T1751: ^ (^𝑌1 ) ∨ ^ (^ (^𝑌1 )) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1535 T1753: ^ (𝑌1 → (𝑌1 ∨ 𝑌2 )) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1536 T1754: ^ ((𝑌1 ∨ 𝑌2 ) → 𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1537 T1755: ^ (¬𝑌1 ) ∨ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1538 T1756: ^ (¬𝑌1 ) ∨ □ (^𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1539 T1757: ^ (¬𝑌1 ) ∨ □ (^ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1540 T1761: ^ (¬ (𝑌1 ∨ 𝑌2 )) ∨ □𝑌3 ⇝ □ (¬ (𝑌1 ∨ 𝑌2 ) → 𝑌3 ) Cluster 1541 T1763: ^ (𝑌1 ∨ 𝑌2 ) ∨ ^𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1542 T1766: (𝑌1 ↔ 𝑌3 ) ∨ (𝑌2 ↔ 𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 )

Continued on next page 163

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Cluster 1543 T1767: (□𝑌2 ↔ □𝑌1 ) ∨ (□𝑌3 ↔ □𝑌1 ) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1544 T1772: (𝑌1 → 𝑌2 ) ∨ ^𝑌2 ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1545 T1774: (𝑌1 → 𝑌2 ) ∨ (𝑌1 → □𝑌2 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1546 T1777: (𝑌1 → □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1547 T1778: (𝑌1 → □𝑌1 ) ∨ (^𝑌1 → 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1548 T1779: (𝑌1 → □𝑌2 ) ∨ ^ (^ (^𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1549 T1782: (𝑌1 → □ (□𝑌3 )) ∨ (𝑌2 → □ (□𝑌3 )) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1550 (𝑌1 → □ (¬𝑌1 )) T1783: (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

∨

(𝑌1 → □ (□ (¬𝑌1 )))

⇝

𝑌1

→

Cluster 1551 T1784: (𝑌1 → ^𝑌1 ) ∨ (𝑌1 → □𝑌1 ) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1552 T1786: (𝑌1 → ^𝑌2 ) ∨ (𝑌1 → ^ (¬𝑌2 )) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1553 T1787: (𝑌1 → ^𝑌2 ) ∨ (𝑌3 → ^𝑌2 ) ⇝ □𝑌2 → ¬ (𝑌1 ↔ 𝑌3 ) Cluster 1554 T1788: (𝑌1 → ^ (□𝑌2 )) ∨ (𝑌1 → □ (^𝑌3 )) ⇝ □𝑌1 → ¬ (𝑌2 ↔ 𝑌3 ) Cluster 1555 T1796: (□𝑌1 → ¬𝑌1 ) ∨ (□𝑌1 → □ (¬𝑌1 )) ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1556 T1797: (□ (𝑌1 ∨ 𝑌2 ) → □𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1557 T1798: (^𝑌1 → 𝑌1 ) ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1558 T1799: (^𝑌1 → 𝑌1 ) ∨ (□𝑌1 → 𝑌1 ) ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1559 T1800: (^𝑌1 → 𝑌2 ) ∨ □ (^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1560 T1801: (^𝑌1 → 𝑌2 ) ∨ (□𝑌1 → 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1561 Continued on next page 164

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Cluster 1562 T1803: (^𝑌1 → □𝑌2 ) ∨ □ (^𝑌1 ) ⇝ 𝑌1 → (𝑌2 ∨ □𝑌1 ) Cluster 1563 T1804: (^𝑌1 → ((𝑌1 ∨ 𝑌2 ) → 𝑌2 )) ∨ (□𝑌1 → 𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1564 T1805: (^ (^𝑌1 ) → ^ (^ (□ (¬𝑌2 )))) ∨ (^ (^𝑌3 ) → ^ (^ (□ (¬𝑌2 )))) □ (^ ((𝑌1 ∨ 𝑌3 ) ∧ ¬ (^𝑌2 )))

⇝

Cluster 1565 T1807: ((𝑌1 ∨ ¬𝑌2 ) → □𝑌3 ) ∨ ((𝑌2 ∨ ¬𝑌1 ) → □𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1566 T1808: ¬𝑌1 ∨ □𝑌1 ⇝ 𝑌1 ∨ □ (¬𝑌1 ) Cluster 1567 T1809: ¬𝑌1 ∨ □ (𝑌1 ∧ ^𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ □𝑌2 ) Cluster 1568 T1811: ¬𝑌1 ∨ □ (¬𝑌1 ) ⇝ 𝑌1 → ^ (¬𝑌1 ) Cluster 1569 T1815: ¬𝑌1 ∨ ^ (𝑌1 ∧ 𝑌2 ) ⇝ ¬𝑌1 ↔ ^ (𝑌1 ∧ 𝑌2 ) Cluster 1570 T1816: ¬𝑌1 ∨ ^ (^𝑌2 ) ⇝ 𝑌1 → (^𝑌2 ∨ ^ (^𝑌2 )) Cluster 1571 T1817: ¬𝑌1 ∨ ¬𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1572 T1819: ¬𝑌1 ∨ ¬𝑌2 ⇝ 𝑌1 ∨ 𝑌2 Cluster 1573 T1820: ¬ (𝑌1 ∧ 𝑌2 ) ∨ □𝑌2 ⇝ 𝑌2 → (𝑌1 ∨ □𝑌2 ) Cluster 1574 T1821: ¬ (□𝑌1 ) ∨ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1575 T1823: ¬ (^𝑌1 ) ∨ □ (^𝑌2 ) ⇝ □ (^ (𝑌2 ∧ ¬ (^𝑌1 ))) Cluster 1576 T1825: ¬ (^𝑌1 ) ∨ ^𝑌2 ⇝ □ (¬ (𝑌1 ∧ 𝑌2 )) Cluster 1577 T1828: 𝑌1 ∨ 𝑌2 ∨ 𝑌3 ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1578 T1830: 𝑌1 ∨ 𝑌2 ∨ □ (¬𝑌3 ) ⇝ □𝑌3 → ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1579 T1831: 𝑌1 ∨ ^𝑌1 ∨ 𝑌2 ⇝ 𝑌2 → (^𝑌1 ∨ ^ (^𝑌1 ))

Continued on next page 165

Transformations

#dst

%dst

0.01%

0.03%

0.01%

0.03%

0.01%

0.03%

0.01%

0.03%

0.01%

0.03%

0.01%

0.03%

Cluster 1580 T1832: 𝑌1 ∨ ^𝑌1 ∨ □𝑌1 ⇝ □𝑌1 ∧ □ (□𝑌1 ) Cluster 1581 T1833: 𝑌1 ∨ ¬𝑌2 ∨ 𝑌2 ∨ ¬𝑌1 ⇝ ¬ (𝑌1 ↔ 𝑌2 ) Cluster 1582 T1834: □ (¬𝑌1 ) ∨ □ (□ (¬𝑌1 )) ∨ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1583 T1835: ^ (^𝑌1 ) ∨ ^ (^𝑌2 ) ∨ □𝑌3 ⇝ □ (^ ((𝑌1 ∨ 𝑌2 ) ∧ ¬ (^𝑌3 ))) Cluster 1584 T1836: ^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )) ∨ □𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 ))) Cluster 1585 T1837: ¬𝑌1 ∨ □ (¬𝑌1 ) ∨ 𝑌1 ⇝ 𝑌1 → (^ (¬𝑌1 ) ∨ ^ (^ (¬𝑌1 )))

166

B.5

CTL: Hierarchical Clustering of Transformations Table 40: Transformation hierarchy clusters Transformations Whole data set ANY transformation explained

#dst

%dst

7210 2588

100.00% 35.89%

2695 409

100.00% 15.18%

215 36 28 27 23

2.98% 0.50% 0.39% 0.37% 0.32%

9 2 1 2 1

0.33% 0.07% 0.04% 0.07% 0.04%

179 18 6

2.48% 0.25% 0.08%

6 1 1

0.22% 0.04% 0.04%

119 25 24 16 2

1.65% 0.35% 0.33% 0.22% 0.03%

10 2 1 2 1

0.37% 0.07% 0.04% 0.07% 0.04%

107

1.48%

0.04%

1.37%

0.07%

96 8 5

1.33% 0.11% 0.07%

8 2 1

0.30% 0.07% 0.04%

94 84 6 4

1.30% 1.17% 0.08% 0.06%

3 1 1 1

0.11% 0.04% 0.04% 0.04%

90 69 8 1

1.25% 0.96% 0.11% 0.01%

6 2 1 1

0.22% 0.07% 0.04% 0.04%

71 58 5

0.98% 0.80% 0.07%

3 1 1

0.11% 0.04% 0.04%

Cluster 1 T2: 𝑌1 ⇝ AG𝑌1 T304: ¬𝑌1 ⇝ AG (¬𝑌1 ) T222: EF (¬𝑌1 ) ⇝ EF (AG (¬𝑌1 )) T199: EF𝑌1 ⇝ AG (EF𝑌1 ) T262: EX𝑌1 ⇝ AG (EX𝑌1 ) Cluster 2 T145: AX𝑌1 ⇝ AG𝑌1 T174: AX (EF𝑌1 ) ⇝ AG (EF𝑌1 ) T192: AX (¬𝑌1 ) ⇝ AG (¬𝑌1 ) Cluster 3 T11: AF𝑌1 ⇝ AG𝑌1 T46: AF (¬𝑌1 ) ⇝ AG (¬𝑌1 ) T205: EF (AF (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) T36: AF (EF𝑌1 ) ⇝ AG (EF𝑌1 ) T17: AF (AF𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 4 T148: AX𝑌1 ⇝ AG (EX𝑌1 ) Cluster 5 T12: AF𝑌1 ⇝ AG (EF𝑌1 ) Cluster 6 T265: EX𝑌1 ⇝ EF𝑌1 T112: AG (EX𝑌1 ) ⇝ AG (EF𝑌1 ) T270: EX (AG (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 7 T151: AX𝑌1 ⇝ EX𝑌1 T86: AG (AX𝑌1 ) ⇝ AG (EX𝑌1 ) T314: ¬ (AX𝑌1 ) ⇝ ¬ (EX𝑌1 ) T299: EF𝑌1 → AX𝑌2 ⇝ EF𝑌1 → EX𝑌2 Cluster 8 T7: 𝑌1 ⇝ EF𝑌1 T53: AG𝑌1 ⇝ AG (EF𝑌1 ) T124: AG (¬𝑌1 ) ⇝ EF (AG (¬𝑌1 )) T129: 𝑌1 ∧ 𝑌2 ⇝ EF (𝑌1 ∧ 𝑌2 ) Cluster 9 T1: 𝑌1 ⇝ AF𝑌1 T52: AG𝑌1 ⇝ AG (AF𝑌1 ) T51: AG𝑌1 ⇝ AF (AG𝑌1 ) Cluster 10 Continued on next page 167

Transformations

#dst

%dst

T285: EX (EX (EF𝑌1 )) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1

0.97%

0.04%

55 5

0.76% 0.07%

7 2

0.26% 0.07%

54 54

0.75% 0.75%

1 1

0.04% 0.04%

0.67%

0.04%

0.62%

0.04%

44 36 1

0.61% 0.50% 0.01%

7 2 1

0.26% 0.07% 0.04%

40 17

0.55% 0.24%

3 1

0.11% 0.04%

0.50%

0.04%

0.47%

0.04%

0.46%

0.07%

32 29

0.44% 0.40%

3 1

0.11% 0.04%

0.42%

0.07%

0.37%

0.07%

0.37%

0.04%

26 6 2 1

0.36% 0.08% 0.03% 0.01%

8 2 1 1

0.30% 0.07% 0.04% 0.04%

25 3

0.35% 0.04%

7 1

0.26% 0.04%

Cluster 11 T231: EG𝑌1 ⇝ EF𝑌1 T107: AG (EG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 12 ⌈ = T8: 𝑌1 ⇝ EX𝑌1 ⌊ = T54: AG𝑌1 ⇝ AG (EX𝑌1 ) Cluster 13 T282: EX (EX𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 14 T59: AG (AF𝑌1 ) ⇝ AF (AG𝑌1 ) Cluster 15 T15: AF𝑌1 ⇝ EF𝑌1 T62: AG (AF𝑌1 ) ⇝ AG (EF𝑌1 ) T24: AF (AG𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 16 T196: EF𝑌1 ⇝ AF𝑌1 T93: AG (EF𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 17 T138: ¬ (EX𝑌1 ) ∧ 𝑌2 ⇝ 𝑌2 → AX (¬𝑌1 ) Cluster 18 T90: AG (EF𝑌1 ) ⇝ AF (AG𝑌1 ) Cluster 19 T14: AF𝑌1 ⇝ AX𝑌1 Cluster 20 T6: 𝑌1 ⇝ AX𝑌1 T115: AG (𝑌1 → 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 21 T263: EX𝑌1 ⇝ AX𝑌1 Cluster 22 T142: A (𝑌1 U𝑌2 ) ⇝ E (𝑌1 U𝑌2 ) Cluster 23 T13: AF𝑌1 ⇝ AG (EX𝑌1 ) Cluster 24 T143: AX𝑌1 ⇝ 𝑌1 T83: AG (AX𝑌1 ) ⇝ AG𝑌1 T156: AX (AG𝑌1 ) ⇝ AG𝑌1 T176: AX (EF𝑌1 ) ⇝ EF𝑌1 Cluster 25 T258: EX𝑌1 ⇝ 𝑌1 T110: AG (EX𝑌1 ) ⇝ AG𝑌1

Continued on next page 168

Transformations ⌈ = T284: EX (EX𝑌1 ) ⇝ EX𝑌1 | = T321: ¬ (EX𝑌1 ) ⇝ ¬𝑌1 ⌊ = T322: ¬ (EX (EX𝑌1 )) ⇝ ¬ (EX𝑌1 )

#dst

%dst

1 1 1

0.01% 0.01% 0.01%

1 1 1

0.04% 0.04% 0.04%

24 17

0.33% 0.24%

3 1

0.11% 0.04%

23 2 1

0.32% 0.03% 0.01%

5 1 1

0.19% 0.04% 0.04%

0.32%

0.07%

0.28%

0.11%

0.28%

0.04%

0.28%

0.04%

0.26%

0.07%

18 8 1

0.25% 0.11% 0.01%

7 2 1

0.26% 0.07% 0.04%

0.25%

0.04%

16 1

0.22% 0.01%

7 1

0.26% 0.04%

16 3

0.22% 0.04%

5 1

0.19% 0.04%

0.22%

0.07%

0.22%

0.04%

0.22%

0.04%

0.21%

0.19%

Cluster 26 T50: AG𝑌1 ⇝ AF𝑌1 T72: AG (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 27 T49: AG𝑌1 ⇝ 𝑌1 T87: AG (AX𝑌1 ) ⇝ AX𝑌1 T96: AG (EF𝑌1 ) ⇝ EF𝑌1 Cluster 28 T147: AX𝑌1 ⇝ AG (EF𝑌1 ) Cluster 29 T56: AG𝑌1 ⇝ AX𝑌1 Cluster 30 T169: AX (AX𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 31 T246: EG (¬𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 32 T152: AX (AF𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 33 T150: AX𝑌1 ⇝ EF𝑌1 T85: AG (AX𝑌1 ) ⇝ AG (EF𝑌1 ) T160: AX (AG𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 34 T210: EF (𝑌1 ∧ 𝑌2 ) ⇝ E (𝑌2 U𝑌1 ) Cluster 35 T197: EF𝑌1 ⇝ AG𝑌1 T217: EF (EX𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 36 T226: EG𝑌1 ⇝ AG𝑌1 T241: EG (EF𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 37 T201: EF𝑌1 ⇝ AX𝑌1 Cluster 38 T33: AF (AX𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 39 T218: EF (EX (EX𝑌1 )) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 40 T259: EX𝑌1 ⇝ AG𝑌1

Continued on next page

169

Transformations

#dst

%dst

0.21%

0.04%

0.18%

0.15%

13 13

0.18% 0.18%

1 1

0.04% 0.04%

0.15%

0.07%

0.15%

0.04%

10 1

0.14% 0.01%

4 1

0.15% 0.04%

0.14%

0.04%

0.14%

0.04%

9 2 1 1

0.12% 0.03% 0.01% 0.01%

6 1 1 1

0.22% 0.04% 0.04% 0.04%

0.12%

0.04%

0.12%

0.04%

0.12%

0.04%

0.12%

0.04%

8 6

0.11% 0.08%

2 1

0.07% 0.04%

0.11%

0.04%

0.11%

0.04%

0.11%

0.04%

Cluster 41 T127: AG (¬ (EF𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 42 T57: AG𝑌1 ⇝ EF𝑌1 Cluster 43 ⌈ = T223: EF (¬ (AG𝑌1 )) ⇝ EF (AG (¬𝑌1 )) ⌊ = T308: ¬ (AG𝑌1 ) ⇝ AG (¬𝑌1 ) Cluster 44 T306: ¬ (AF𝑌1 ) ⇝ AG (¬𝑌1 ) Cluster 45 T109: AG (EG (¬𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 46 T195: EF𝑌1 ⇝ 𝑌1 T92: AG (EF𝑌1 ) ⇝ AG𝑌1 Cluster 47 T42: AF (𝑌1 → 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 48 T135: ¬𝑌1 ∧ EX (¬𝑌1 ∧ EF𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 49 T303: ¬𝑌1 ⇝ 𝑌1 T323: ¬ (EX (¬𝑌1 )) ⇝ ¬ (EX𝑌1 ) T123: AG (¬𝑌1 ) ⇝ AG𝑌1 T311: ¬ (AX𝑌1 ) ⇝ AX𝑌1 Cluster 50 T21: AF (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 51 T29: AF (AX𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 52 T122: AG (EX𝑌1 → EX𝑌2 ) ⇝ EF𝑌1 → EX𝑌2 Cluster 53 T228: EG𝑌1 ⇝ AG (EF𝑌1 ) Cluster 54 T144: AX𝑌1 ⇝ AF𝑌1 T84: AG (AX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 55 T89: AG (EF𝑌1 ) ⇝ AF𝑌1 Cluster 56 T234: EG (AF𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 57 T276: EX (AX𝑌1 ) ⇝ AG (EX𝑌1 )

Continued on next page 170

Transformations

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%dst

0.10%

0.15%

7 5

0.10% 0.07%

2 1

0.07% 0.04%

0.10%

0.04%

0.10%

0.04%

0.10%

0.04%

6 2 1

0.08% 0.03% 0.01%

6 2 1

0.22% 0.07% 0.04%

0.08%

0.07%

0.08%

0.04%

0.08%

0.04%

0.08%

0.04%

0.08%

0.04%

0.08%

0.04%

0.08%

0.04%

0.07%

0.11%

0.07%

0.04%

0.07%

0.04%

0.07%

0.04%

0.07%

0.04%

Cluster 58 T130: 𝑌1 ∧ AX𝑌2 ⇝ AG (EX𝑌2 ) Cluster 59 T225: EG𝑌1 ⇝ AF𝑌1 T106: AG (EG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 60 T18: AF (AF𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 61 T43: AF (𝑌1 → EX𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 62 T261: EX𝑌1 ⇝ AG (EF𝑌1 ) Cluster 63 T9: AF𝑌1 ⇝ 𝑌1 T61: AG (AF𝑌1 ) ⇝ AG𝑌1 T20: AF (AG𝑌1 ) ⇝ AG𝑌1 Cluster 64 T22: AF (AG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 65 T111: AG (EX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 66 T158: AX (AG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 67 T229: EG𝑌1 ⇝ AG (EX𝑌1 ) Cluster 68 T230: EG𝑌1 ⇝ AX𝑌1 Cluster 69 T266: EX (AF𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 70 T312: ¬ (AX𝑌1 ) ⇝ AX (¬𝑌1 ) Cluster 71 T75: AG (𝑌1 ∧ 𝑌2 ) ⇝ AG𝑌1 Cluster 72 T4: 𝑌1 ⇝ AG (EX𝑌1 ) Cluster 73 T88: AG (AX (¬𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 74 T146: AX𝑌1 ⇝ AG (AF𝑌1 ) Cluster 75 T219: EF (𝑌1 → 𝑌2 ) ⇝ EF (𝑌1 ∧ 𝑌2 )

Continued on next page 171

Transformations

#dst

%dst

0.07%

0.04%

0.07%

0.04%

0.07%

0.04%

4 1

0.06% 0.01%

3 1

0.11% 0.04%

0.06%

0.07%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.06%

0.04%

0.11%

0.04%

Cluster 76 T283: EX (EX𝑌1 ) ⇝ EF𝑌1 Cluster 77 T300: EG𝑌1 → EF𝑌2 ⇝ 𝑌1 → AF𝑌2 Cluster 78 T302: ¬ (EX𝑌1 ) → ¬ (EX𝑌2 ) ⇝ EF𝑌2 → EX𝑌1 Cluster 79 T252: 𝑌1 ↔ 𝑌2 ⇝ 𝑌1 → 𝑌2 T255: EF𝑌1 ↔ EX𝑌2 ⇝ EF𝑌1 → EX𝑌2 Cluster 80 T168: AX (AX𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 81 T16: AF𝑌1 ⇝ EF (AG𝑌1 ) Cluster 82 T27: AF (AF𝑌1 ∧ 𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 83 T125: AG (¬ (AF𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 84 T139: ¬ (EX (EX𝑌1 )) ∧ EF𝑌1 ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 85 T141: ¬ (EX (EX𝑌1 )) ∧ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 86 T163: AX (𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 87 T206: EF (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 88 T213: EF (AX𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 89 T227: EG𝑌1 ⇝ AG (AF𝑌1 ) Cluster 90 T289: EX (¬𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 91 T293: 𝑌1 → 𝑌2 ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 92 T114: AG (EX𝑌1 ) ⇝ EF𝑌1 Cluster 93 T38: AF (EG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 94 Continued on next page 172

Transformations

#dst

%dst

T76: AG (𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌1 ∨ 𝑌2 )

0.04%

3 3

0.04% 0.04%

1 1

0.04% 0.04%

0.04%

0.03%

0.07%

Cluster 95 T91: AG (EF𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 96 ⌈ = T97: AG (EF𝑌1 ) ⇝ EF (AG𝑌1 ) ⌊ = T102: AG (EF (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 97 T101: AG (EF (EG𝑌1 )) ⇝ AF (AG𝑌1 ) Cluster 98 T103: AG (EF (¬ (EF𝑌1 ))) ⇝ AF (AG (¬𝑌1 )) Cluster 99 T121: AG (EX𝑌1 → EX𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 100 T131: EF𝑌1 ∧ EX𝑌2 ⇝ EF𝑌1 → EX𝑌2 Cluster 101 T159: AX (AG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 102 T162: AX (AG (AX𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 103 T198: EF𝑌1 ⇝ AG (AF𝑌1 ) Cluster 104 T221: EF (¬𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 105 T236: EG (AG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 106 T238: EG (AX𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 107 T242: EG (EF𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 108 T248: EG (¬ (AG𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 109 T264: EX𝑌1 ⇝ AX (¬𝑌1 ) Cluster 110 T292: 𝑌1 → 𝑌2 ⇝ AG (𝑌1 → AF𝑌2 ) Cluster 111 T318: ¬ (EF𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 112 T32: AF (AX𝑌1 ) ⇝ AG (EF𝑌1 )

Continued on next page 173

Transformations

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%dst

0.03%

0.07%

0.03%

0.07%

2 1

0.03% 0.01%

2 1

0.07% 0.04%

0.03%

0.07%

0.03%

0.07%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

0.04%

0.03%

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Cluster 113 T65: AG (AF𝑌1 ) ⇝ EF𝑌1 Cluster 114 T194: AX (𝑌1 ∨ 𝑌2 ) ⇝ AG𝑌1 Cluster 115 T202: EF𝑌1 ⇝ EX𝑌1 T94: AG (EF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 116 T203: EF (AF𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 117 T256: E (𝑌1 U𝑌2 ) ⇝ AG (𝑌1 → 𝑌2 ) Cluster 118 T3: 𝑌1 ⇝ AG (AF𝑌1 ) Cluster 119 T26: AF (𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌1 ∨ 𝑌2 ) Cluster 120 T30: AF (EF𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AF𝑌1 ) Cluster 121 T31: AF (AX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 122 T39: AF (EG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 123 T48: AF (¬ (EF𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 124 T55: AG𝑌1 ⇝ AG (¬𝑌1 ) Cluster 125 T60: AG (AF𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 126 T64: AG (AF𝑌1 ) ⇝ AX𝑌1 Cluster 127 T69: AG (AF (𝑌1 → 𝑌2 )) ⇝ AG (𝑌1 → AF𝑌2 ) Cluster 128 T78: AG (EF𝑌1 ∧ 𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 129 T79: AG (EF𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AF𝑌1 ) Cluster 130 T81: AG (¬𝑌1 ∧ AF𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 131 Continued on next page 174

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Cluster 132 T95: AG (EF𝑌1 ) ⇝ AX𝑌1 Cluster 133 T134: ¬𝑌1 ∧ AX (¬𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 134 T155: AX (AF (¬𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 135 T157: AX (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 136 T164: AX (EX𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 137 T175: AX (EF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 138 ⌈ = T182: AX (EX𝑌1 ) ⇝ AG𝑌1 ⌊ = T186: AX (EX (EX𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 139 T183: AX (EX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 140 T187: AX (𝑌1 → 𝑌2 ) ⇝ EF (𝑌2 ∧ 𝑌1 ) Cluster 141 T191: AX (EX𝑌1 → EX𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 142 T200: EF𝑌1 ⇝ AG (EX𝑌1 ) Cluster 143 T207: EF (AG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 144 T216: EF (E (𝑌2 U𝑌1 )) ⇝ EF (𝑌1 ∧ 𝑌2 ) Cluster 145 T240: EG (EF𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 146 T250: EG (¬ (EF𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 147 T253: 𝑌1 ↔ EF𝑌2 ⇝ AG (𝑌1 → AF𝑌2 ) Cluster 148 T267: EX (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 149 T268: EX (AG𝑌1 ) ⇝ AG (EF𝑌1 )

Continued on next page 175

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Cluster 150 T278: EX (AX (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 151 T279: EX (EG𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 152 T291: EX (𝑌1 ∨ 𝑌2 ) ⇝ EF (𝑌1 ∧ 𝑌2 ) Cluster 153 T296: 𝑌1 → EG𝑌2 ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 154 T297: 𝑌1 → EX𝑌2 ⇝ 𝑌1 → AF𝑌2 Cluster 155 T309: ¬ (AG𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 156 T317: ¬ (EF𝑌1 ) ⇝ AX (¬𝑌1 ) Cluster 157 T320: ¬ (EG (¬𝑌1 )) ⇝ AG (AF𝑌1 ) Cluster 158 T324: (¬𝑌1 ∧ ¬𝑌2 ) ∨ (¬𝑌1 ∧ 𝑌2 ) ∨ (¬𝑌2 ∧ 𝑌1 ) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 159 T5: 𝑌1 ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 160 T10: AF𝑌1 ⇝ AF (AG (¬𝑌1 )) Cluster 161 T19: AF (AF (𝑌1 → 𝑌2 )) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 162 T23: AF (AG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 163 T25: AF (AG (𝑌1 → 𝑌2 )) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 164 T28: AF (AG𝑌1 ∧ 𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 165 T34: AF (AX𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 166 T35: AF (EF𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 167 T37: AF (EF𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 168 T40: AF (EG (AX𝑌1 )) ⇝ AG (EX𝑌1 )

Continued on next page 176

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Cluster 169 T41: AF (EX𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 170 T44: AF (¬𝑌1 → 𝑌2 ) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 171 T45: AF (¬𝑌1 ) ⇝ AG𝑌1 Cluster 172 T47: AF (¬ (𝑌1 ∧ 𝑌2 )) ⇝ AG (EF (𝑌1 ∨ 𝑌2 )) Cluster 173 T58: AG𝑌1 ⇝ EF (AG (¬𝑌1 )) Cluster 174 T63: AG (AF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 175 ⌈ = T66: AG (AF𝑌1 ) ⇝ EF (AG𝑌1 ) ⌊ = T71: AG (AF (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 176 T67: AG (AF (AF𝑌1 )) ⇝ AF (AG𝑌1 ) Cluster 177 T68: AG (AF (AX𝑌1 )) ⇝ AF (AG𝑌1 ) Cluster 178 T70: AG (AF (¬𝑌1 )) ⇝ AG (EF𝑌1 ) Cluster 179 T73: AG (AG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 180 T74: AG (AG (EF𝑌1 )) ⇝ AF (AG𝑌1 ) Cluster 181 T77: AG (AF𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AF𝑌1 ) Cluster 182 T80: AG (EG𝑌1 ∧ 𝑌2 ) ⇝ AX (EF𝑌1 ∧ 𝑌2 ) Cluster 183 T98: AG (EF𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 184 T99: AG (EF (AF𝑌1 )) ⇝ AF (AG𝑌1 ) Cluster 185 T100: AG (EF (AX𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 186 T104: AG (EF (¬ (EX𝑌1 ))) ⇝ AF (AG (¬𝑌1 )) Cluster 187 Continued on next page 177

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Cluster 188 T108: AG (EG𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 189 T113: AG (EX𝑌1 ) ⇝ AX𝑌1 Cluster 190 T116: AG (𝑌1 → AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 191 T117: AG (EF𝑌1 → 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 192 T118: AG (EF𝑌1 → AG𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 193 T119: AG (EG𝑌1 → 𝑌2 ) ⇝ EF𝑌1 → 𝑌2 Cluster 194 T120: AG (EX𝑌1 → 𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 195 T126: AG (¬ (AG𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 196 T128: AG (¬𝑌1 ∨ 𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 197 T132: ¬𝑌1 ∧ 𝑌2 ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 198 T133: ¬𝑌1 ∧ AF𝑌1 ∧ AX (¬𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 199 T136: ¬ (𝑌1 ∧ 𝑌2 ) ∧ AF (¬ (𝑌1 ∧ 𝑌2 )) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 200 T137: ¬ (EF𝑌1 ) ∧ AG𝑌2 ⇝ AX (EF𝑌1 ∧ 𝑌2 ) Cluster 201 T140: ¬ (EX (EX𝑌1 )) ∧ ¬ (EX𝑌1 ) ∧ AF𝑌1 ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 202 T149: AX𝑌1 ⇝ AX (¬𝑌1 ) Cluster 203 T153: AX (AF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 204 T154: AX (AF𝑌1 ) ⇝ EF𝑌1 Cluster 205 T161: AX (AG (AF𝑌1 )) ⇝ AG (EF𝑌1 ) Cluster 206 Continued on next page 178

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T165: AX (¬𝑌1 ∧ AF𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1

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Cluster 207 T166: AX (¬𝑌1 ∧ EF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 208 T167: AX (AX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 209 T170: AX (AX (AG𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 210 T171: AX (AX (AX𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 211 T172: AX (EF𝑌1 ) ⇝ AF𝑌1 Cluster 212 T173: AX (EF𝑌1 ) ⇝ AF (AG𝑌1 ) Cluster 213 T177: AX (EF (𝑌1 ↔ ¬𝑌2 )) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 214 T178: AX (EG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 215 T179: AX (EG𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 216 T180: AX (EG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 217 T181: AX (EX𝑌1 ) ⇝ AF (AG𝑌1 ) Cluster 218 T184: AX (EX𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 219 T185: AX (EX (EG𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 220 T188: AX (𝑌1 → AF𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 221 T189: AX (𝑌1 → EF𝑌2 ) ⇝ AX (EF𝑌2 ∧ 𝑌1 ) Cluster 222 T190: AX (𝑌1 → ¬𝑌2 ) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 ) Cluster 223 T193: AX (¬𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 224 T204: EF (AF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 225 Continued on next page 179

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Cluster 226 T209: EF (𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 227 T211: EF (EF𝑌1 ∧ 𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 228 T212: EF (¬𝑌1 ∧ EX𝑌1 ) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 229 T214: EF (EF (EF𝑌1 )) ⇝ AG (AF𝑌1 ) Cluster 230 T215: EF (EG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 231 T220: EF (¬𝑌1 → 𝑌2 ) ⇝ EF (AG (¬𝑌1 )) Cluster 232 T224: EF (¬ (EF (¬𝑌1 ))) ⇝ EF (AG (¬𝑌1 )) Cluster 233 T232: EG (AF𝑌1 ) ⇝ AG (EF𝑌1 ) Cluster 234 T233: EG (AF𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 235 T235: EG (AG𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 236 T237: EG (AG𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 237 T239: EG (AX (¬𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 238 T243: EG (EG𝑌1 ) ⇝ EF (AG𝑌1 ) Cluster 239 T244: EG (𝑌1 → EF𝑌2 ) ⇝ AG (𝑌1 → AF𝑌2 ) Cluster 240 T245: EG (¬𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 241 T247: EG (¬ (AF𝑌1 )) ⇝ AF (AG (¬𝑌1 )) Cluster 242 T249: EG (¬ (AG (¬𝑌1 ))) ⇝ EF (AG (¬𝑌1 )) Cluster 243 T251: EG (EF𝑌1 ∨ EF𝑌2 ) ⇝ AG (EF (𝑌1 ∨ 𝑌2 )) Cluster 244 Continued on next page 180

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Cluster 245 T257: E (𝑌1 UEF𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 246 T260: EX𝑌1 ⇝ AG (AF𝑌1 ) Cluster 247 T269: EX (AG𝑌1 ) ⇝ AG (EX𝑌1 ) Cluster 248 T271: EX (AF𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 249 T272: EX (EF𝑌1 ∧ 𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 250 T273: EX (EX𝑌1 ∧ 𝑌2 ) ⇝ AX (EF𝑌1 ∧ 𝑌2 ) Cluster 251 T274: EX (EX (EF𝑌1 ) ∧ 𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 252 T275: EX (¬𝑌1 ∧ EX𝑌1 ) ⇝ EF𝑌1 Cluster 253 T277: EX (AX (EF𝑌1 )) ⇝ ¬ (EX𝑌1 ) ∧ ¬𝑌1 ∧ EF𝑌1 Cluster 254 T280: EX (𝑌1 ↔ 𝑌2 ) ⇝ EF (𝑌1 ∧ 𝑌2 ) Cluster 255 T281: EX (EX𝑌1 ) ⇝ AG (AF𝑌1 ) Cluster 256 T286: EX (𝑌1 → 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 257 T287: EX (𝑌1 → EX𝑌2 ) ⇝ AG (𝑌2 → AX𝑌1 ) Cluster 258 T288: EX (EX𝑌1 → 𝑌2 ) ⇝ AG (𝑌1 → AX𝑌2 ) Cluster 259 T290: EX (¬ (AG𝑌1 )) ⇝ EF (AG (¬𝑌1 )) Cluster 260 T294: 𝑌1 → AF (¬𝑌1 ∧ 𝑌2 ) ⇝ AG (𝑌1 → AF𝑌2 ) Cluster 261 T295: 𝑌1 → AX (¬𝑌2 ∧ ¬𝑌1 ∧ 𝑌3 ) ⇝ AG (𝑌1 → AX𝑌3 ) Cluster 262 T298: (¬𝑌1 ∧ 𝑌2 ) → AF𝑌1 ⇝ AG (𝑌2 → AF𝑌1 ) Cluster 263 Continued on next page 181

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T301: (𝑌1 → EX𝑌2 ) → (𝑌1 → EX𝑌3 ) ⇝ EF𝑌2 → EX𝑌3

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Cluster 264 T305: ¬ (AF𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 265 T307: ¬ (AF𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 266 T310: ¬ (AG (EX𝑌1 )) ⇝ AX (¬𝑌1 ) Cluster 267 T313: ¬ (AX𝑌1 ) ⇝ EF (AG (¬𝑌1 )) Cluster 268 T315: ¬ (AX (¬𝑌1 )) ⇝ AG (EX𝑌1 ) Cluster 269 T316: ¬ (EF𝑌1 ) ⇝ AF (AG (¬𝑌1 )) Cluster 270 T319: ¬ (EG𝑌1 ) ⇝ AG (¬𝑌1 ) Cluster 271 T325: AX (¬𝑌1 ∧ 𝑌2 ) ∨ AX (¬𝑌2 ∧ 𝑌1 ) ⇝ AG (¬𝑌1 ∨ ¬𝑌2 )

182

Record · ID 919452 · SHA-256 d9a32210c8df620d
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