( 1993 ) " Equational characterization of all varieties of MV-algebras ," Journal of Algebra 221: 123 – 131.

Equational logic was common before Principia Mathematica ( e. g., Peirce ,< sup > 1, 2, 3 </ sup > Johnson 1892 ), and has present-day advocates ( Gries and Schneider 1993 ).

An Equational Formulation of LF.

* Cooper, D. C., 1972, " Theorem Proving in Arithmetic without Multiplication " in B. Meltzer and D. Michie, eds., Machine Intelligence.

A few years later, in his seminal 1971 paper " The Complexity of Theorem Proving Procedures ", Cook formalized the notions of polynomial-time reduction ( a. k. a. Cook reduction ) and NP-completeness, and proved the existence of an NP-complete problem by showing that the Boolean satisfiability problem ( usually known as SAT ) is NP-complete.

His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-Completeness.

These include Studies in Logic, Grammar and Rhetoric, Intelligent Computer Mathematics, Interactive Theorem Proving, Journal of Automated Reasoning and the Journal of Formalized Reasoning.

Proving Carnot's Theorem

* Theorem Proving System, an automated theorem proving system for first-order and higher-order logic

* Interactive Theorem Proving ( conference ), an annual international academic conference

* Laura I. Meikle and Jacques D. Fleuriot ( 2003 ), Formalizing Hilbert's Grundlagen in Isabelle / Isar, Theorem Proving in Higher Order Logics, Lecture Notes in Computer Science, Volume 2758 / 2003, 319-334,

* Interactive Theorem Proving for Agda Users

* Theorem Proving and Automated Reasoning Systems

* International Workshop on First-Order Theorem Proving

* Paramodulation-Based Theorem Proving, Robert Nieuwenhuis and Alberto Rubio, Handbook of Automated Reasoning I ( 7 ), Elsevier Science and MIT Press, 2001.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Problem II. 8 in the Arithmetica ( edition of 1670 ), annotated with Fermat's comment which became Fermat's Last Theorem.

This result is now known as Church's Theorem or the Church – Turing Theorem ( not to be confused with the Church – Turing thesis ).

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem.

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

* A space elevator is also constructed in the course of Clarke's final novel ( co-written with Frederik Pohl ), The Last Theorem.

:: Boone-Rogers Theorem: There is no uniform partial algorithm which solves the word problem in all finitely presented groups with solvable word problem.

Strangely, this approach is often used for cases where Theorem I applies, which creates problems with the basic model assumptions.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

Remembrance of the story of the bull's hide and the foundation of Carthage is preserved in mathematics in connection with the Isoperimetric problem which is sometimes called Dido's Problem ( and similarly the Isoperimetric theorem is sometimes called Dido's Theorem ).

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

, with some help from Richard Taylor, proved the Taniyama – Shimura – Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

* Zenkov, DV, AM Bloch, and JE Marsden The Lyapunov-Malkin Theorem and Stabilization of the Unicycle with Rider.

Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization.

The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7. 4 in the preprint ' Ricci Flow with surgery on three-manifolds '.

Under Desargues ' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically.

Gentzen's so-called " Main Theorem " ( Hauptsatz ) about LK and LJ was the cut-elimination theorem, a result with far-reaching meta-theoretic consequences, including consistency.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble ( i. e., the total or convective time derivative is zero ).

Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:

Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.

# The Fundamental Theorem of Natural Selection

1989 An interpretation and proof of the Fundamental Theorem of Natural Selection.