# Tagged Questions

For questions regarding the different ways to generate and verify theorems via specialized computer languages, algorithms, and other computer-aided tools.

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### Could mathematical reasoning be non-axiomatic?

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, ...
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### If finitely many algebraic identities do not imply some identity, is there always a finite counterexample?

This question just popped up while experimenting with Prover9 and Mace4. Say we have a finite signature and some finite set of identities $E_i$ in the sense of universal algebra, like the axioms for ...
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### Constructive proof of pigeonhole principle as stated in Software Foundations book

I'm trying to prove the pigeonhole principle from Pierce et al. Software Foundations book and I'm stuck with trying to do so without use of the principle of excluded middle. Here Coq formulation of ...
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### The right way of defining a predicate

My theory contains a definition of lists: L(H,T) is a list, H is the first element (head), T is the list of remaining elements (tail), nil is empty list. So [A,B,C] = L(A,L(B,L(C,nil))). I defined ...
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### Unary predicate for finite number of values

I am working with automated prover. I am creating a theory, where an unary predicate PR should be true just for several constants, false otherwise. I made following axioms: ...
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### Is this a well defined problem in terms of Euclidean Geometry?

I am trying to construct an example of a geometric problem, stated in terms of Euclidean Geometry, that is not Machine Provable (or in an equivalent definition Automatically Provable)-i.e no computer ...
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### Deductive logic counter-intuitive result

I am working on a small proof in deductive logic. Here is what must be proved: $(\exists x \in T \mid A \implies P(x)) \implies A \implies (\forall x \in T \mid P(x))$ To me that looks unprovable ...
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### Equivalence classes of $\{ij\ |\ i, j ∈ \{a, b\}^* , i \neq j\}$

I want to find the equivalence classes (Nerode-relation) of this language: $L = \{ij\ |\ i,j \in \{a,b\}^*,\ i \neq j\}$ It says that this language is regular and that it has 2 equivalence classes, ...
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### From a set of rules, derive the implications?

I've only just become interested in this domain, so sorry if I'm not using the correct terminologies. What I want is the following: Say I have a set of rules (or constraints), I want to derive some ...
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### Prime numbers qns

Show that for any prime number $p$, $q$, $r$, one has $p^2+q^2$ does not equal to $r^2$. I have no idea how to start and prove it. One stumbling part is that we cannot deduce with certainty that r ...
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### Help with Prover9 for weak propositional systems

I am trying to get Prover9 to work, but apparently am not using exactly the correct commands. Can someone give me a hint, please? This is just a test case, but ...
117 views

### Proofs about theorem-provers in ZFC, in ZFC

Is the following statement provable in ZFC for some $A$: "$A$ is an algorithm which, when given as input a proposition $p$ in the language of ZFC, outputs 'yes' only if $p$ is provable in ZFC, 'no' ...
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### Has any previously unknown result been proven by an automated theorem prover?

The Wikipedia page on automated theorem proving states: Despite these theoretical limits, in practice, theorem provers can solve many hard problems... However it is not clear whether these 'hard ...
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### Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
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### define the “optimal” automatic theorem prover

my question is : is it possible to define in some way what should do an "optimal automatic mathematician" ? There are two points of view of an automatic theorem prover / automatic mathematician : ...
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### Distance between theorems

In automated proving one can define the best proof of a theorem as the one which minimizes the length of the proof. Given a set of known statements one could define the difficulty of a theorem as the ...
282 views

### A simple, yet non-superficial explanation of what “paramodulation” means in the context of automated theorem proving?

Modern automated theorem provers seem to be paramodulation-based. I only have a superficial understanding of what this means: we derive a proposition whose truth is implied from the truth of [two?] ...
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### An endless loop in a program that search for mathematical theorems and proofs − a milestone? [closed]

I don't know if there exist computer programs working on its own, trying to find and prove theorems, delivering proofs and go on searching for new theorems. But if (when) there are such programs, ...
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### What paradigm of automated theorem proving is appropriate for Principia Mathematica-style formalization?

I am in possession of a book, which, inspired by Russell's Principia Mathematica (PM) and logical positivism, attempts to formalize a specific domain by determining axioms and deducing theorems from ...
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### Could computers someday discover theorems or find demonstrations?

Cloud computing and quantum computers bring computers to what seems like a limitless calculation power? If one sees all the mathematical operations and theorems as a toolset that a computer can use, ...
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### Automata Language regularity proof by construction.

I've been trying to prove or disprove a question that popped during our last session in Uni, we've been using automaton constructing to prove regularity for a while now and I really do have the handle ...
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### Status of declarative proof languages in proof assistants

I'm interested in formalising mathematics and logics in a proof assistant, both to get to know a proof assistant and to make an archive of proofs for myself (nothing too fancy, mainly first order ...
137 views

### Algorithm to find a proof of every provable theorem.

I found this pdf while searching on automated theorem provers: https://www.math.ucdavis.edu/~greg/145/notproof.pdf It says: "Proof by rote algorithm Non-proof courses in mathematics generally ...
81 views

### Programming First-Order Logic

So I recently started reading about logic, and I have decided to try to implement the subject in my final project for a mathematical programming class I am taking. I wasn't going to try to make ...
287 views

### Automatic theorem prover for proving simple theorems?

Is there a simple software that I could use to practice proving theorems in my course of mathematical logic? Basically what I need is ability to 1) define what axioms and laws I am allowed to use in ...
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### An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
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### natural language proof assistant

I was wondering whether there has been any attempt to create a proof assistant that you write in it, in english, I mean you write your proof the usual way in TeX(maybe use a 'simpler english') then ...
138 views

### Equivalent statements

If one looks at the directed graph of all theorems proved, let there be a vertex between statement A and B directional if A implies B, ideally it will connect results across different fields. Is there ...
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### Would it be possible to concoct a “harmful” axiom?

Suppose I run an automated theorem prover. It begins with the axioms of ZFC, and using a random number generator, it proves more theorems, and it runs for two days. At the end of the second day, it ...
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### A question about a projection of a variety

Let $\mathbb K$ be an algebraically closed field (of characteristic zero) and $H$ an irreducible variety in $\mathbb K ^n$. Let $t \in \mathbb K [x_1,\ldots,x_n]$ and let $T:= \mathsf V ( t )$ be the ...
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### What does Equational Theorem Prover do?

http://www.cs.unm.edu/~mccune/eqp/ What does EQP do? Is there any paper that explains what it does? README and other read files do not provide such information - it only talks of how to use it and ...
298 views

### Inequalities of quotients of elementary symmetric polynomials

Many inequalities regarding symmetric polynomials such as this are posed as problems http://www.artofproblemsolving.com/Wiki/index.php/1997_USAMO_Problems/Problem_5 (a^3+b^3+abc)^{-1}+(b^3+c^3+abc)...
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### Coq transparency issues with type class fields

I am having some issues with, I suspect, transparency of fields in type classes. Consider a type class such as ...
339 views

### In a monoid, does $x \cdot y=e$ imply $y \cdot x=e$?

A monoid is a set $S$ together with a binary operation $\cdot:S \times S \rightarrow S$ such that: The binary operation $\cdot$ is associative, that is, $(a\cdot b) \cdot c=a\cdot (b \cdot c)$ for ...
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### Knuth-Bendix ordering and the one unary function

I'm looking at Knuth-Bendix ordering in the context of theorem proving with the superposition calculus. The explanations of KBO that I've been able to find say among other things that each function ...