Volume 24 (2016): Issue 4 (December 2016)

The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12].

Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18].

In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17].

Then we show that the projective space induced (in the sense defined in [9]) by ℝ³ is a projective plane (in the sense defined in [10]).

Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer [12] observes

“It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest – mortgages, annuities, etc.”

In this article we prove the Leibniz series for π which states that π4=∑n=0∞(−1)n2⋅n+1.$${\pi \over 4} = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)^n } \over {2 \cdot n + 1}}.} $$

The formalization follows K. Knopp [8], [1] and [6]. Leibniz’s Series for Pi is item #26 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes

α ⇒ (β ⇒ α),

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.

This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].