* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Proofs in computability theory often invoke the Church – Turing thesis in an informal way to establish the computability of functions while avoiding the ( often very long ) details which would be involved in a rigorous, formal proof.

* Natural deduction, an approach to proof theory that attempts to provide a formal model of logical reasoning as it " naturally " occurs

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof.

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.

It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proved in particular formal systems.

The completeness theorem says that if a formula is logically valid then there is a finite deduction ( a formal proof ) of the formula.

It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

James Margetson ( 2004 ) developed a computerized formal proof using the Isabelle theorem prover.

Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a " functional substitution " rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.

The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In the case of seL4, complete formal verification of the implementation has been achieved, i. e. a mathematical proof that the kernel's implementation is consistent with its formal specification.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed.

The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

Logic, especially in the field of proof theory, considers theorems as statements ( called formulas or well formed formulas ) of a formal language.

Corporations should pay added taxes, to be used for educational purposes ( not necessarily of the formal type ).

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

The result was that I found myself in the ridiculous position of having made a formal engagement by letter for the next week, only two days before my departure from London.

When he had given the call a few moments thought, he went into the kitchen to ask Mrs. Yamata to prepare tea and sushi for the visitors, using the formal English china and the silver tea service which had been donated to the mission, then he went outside to inspect the grounds.

The Commission shall certify to the Secretary of State, upon his request, copies of the formal submissions of claims filed pursuant to subsection ( B ) of Section 4 of this Act for transmission to the foreign government concerned.

As seen in the B. family, there must be an attempt to help the client develop conscious awareness of the problem, especially in the absence of a formal request for assistance.

A second major point of this essay is to examine the formal arrangements for the elections.

Thus the Congress marks a formal recognition of the political system that was central to world politics for a century.

On the one hand, the major European nations had to maintain vis-a-vis each other an emphasis upon sovereignty, independence, formal equality -- thus insuring for themselves individually an optimal freedom of action to maintain the `` flexibility of alignment '' that the system required and to avoid anything approaching a repetition of the disastrous Napoleonic experience.

Most manufacturers were now disposed to heed a proposal for the formal interchange of patents.

Fall foliage and flowers will decorate Los Angeles Country Club for the annual formal party Saturday evening.

Whereas Bultmann's `` center '' position is structurally inconsistent and is therefore indefensible on formal grounds alone, the general position of the `` right '', as represented, say, by Karl Barth, involves the rejection or at least qualification of the demand for demythologization and so is invalidated on the material grounds we have just considered.

His interest in the formal study of religion waned when he was sixteen and he substituted for it an interest in Asian affairs.

Besides the power analysis, there are less formal methods for selecting the number of experimental units.

A trial de novo is usually available for review of informal proceedings conducted by some minor judicial tribunals in proceedings that do not provide all the procedural attributes of a formal judicial trial.

That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules.

Standardized screening tools such as Zung Self-Rating Anxiety Scale, Beck Anxiety Inventory, Taylor Manifest Anxiety Scale and HAM-A ( Hamilton Anxiety Scale ) can be used to detect anxiety symptoms and suggest the need for a formal diagnostic assessment of anxiety disorder.

Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

The formal study of architecture in academic institutions played a pivotal role in the development of the profession as a whole, serving as a focal point for advances in architectural technology and theory.

A similar French influence is seen in classroom ASL in francophone West Africa, where ASL was introduced along with formal education for the deaf by the deaf American missionary Andrew Foster.

On the other hand, some historians have speculated that Charlemagne opposed formal marriages for his daughters out of concern for political rivalries from their potential husbands ; none of Charlemagne's daughters were married, despite political offers of arranged marriages.

" After a vigorous debate, a formal vote for impeachment was held in the House of Representatives on December 5, 1867, and failed, 57 – 108.

It is not implied that formal principles and procedures are so firmly entrenched within the public order of the world community or even of free commonwealths that they will control in all circumstances involving Jews and Gentiles during coming years.

Above all, he had learned to write formal Latin prose and verse to a remarkable degree of artistry.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

The formal ratification by all 13 states was completed in early 1781.

( In less formal treatments, all three systems are commonly called alphabets.

Eventually he gained formal adoption and inclusion of the Twelve Traditions in all future editions of the Big Book.

Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in principle opening up the process to automatisation.

Chiefs and other leaders preside over customary, traditional courts, though all persons have the right to request that their case be considered under the formal British-based legal system.

Euskara Batua was created so that Basque language could be used — and easily understood by all Basque speakers — in formal situations ( education, mass media, literature ), and this is its main use nowadays.

The film is ultimately significant insofar as it displays the first enunciation of De Palma's style in all its major traits – voyeurism, guilt, and a hyper-consciousness of the medium are all on full display, not just as hallmarks, but built into this formal, material apparatus itself.

* Semantics encoding of formal language A in formal language B is a method of representing all terms ( e. g. programs or descriptions ) of language A using language B.

It constituted a formal agreement between Muhammad and all of the significant tribes and families of Yathrib ( later known as Medina ), including Muslims, Jews, and pagans.

In all, Cuba continues to have formal relations with 160 nations, and provided civilian assistance workers – principally medical – in more than 20 nations.

Such a grammar defines the formal language: all words consisting solely of terminal symbols which can be reached by a derivation from the start symbol.

* Type-0 grammars ( unrestricted grammars ) include all formal grammars.

* 8th November 2003 ( The ' of ' and ' the ' may be included in speech ; they are omitted in all but the most formal writing.

Endorsed by all provincial governments except that of Quebec ( led by René Lévesque ), this was the formal Canadian Act of Parliament that achieved full and final political independence from the United Kingdom.

For example, while in a conversion to Judaism a convert must accept basic Jewish principles of faith, and renounce all other religions, the process is more like a form of adoption, or changing national citizenship ( i. e. becoming a formal member of the people, or tribe ), with the convert becoming a " child of Abraham and Sarah.

In formal language theory, a context-free grammar is said to be in Chomsky normal form if all of its production rules are of the form:

A formal grammar is in Chomsky reduced form if all of its production rules are of the form: