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Method of proof in mathematics
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for
Constructive_proof
Philosphical view that existence proofs must be constructive
assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
First article on transfinite set theory
Both constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led
Cantor's first set theory article
Cantor's_first_set_theory_article
About simultaneous modular congruences
n_{1}\cdots n_{k}} is large. The third one uses the existence proof given in § Existence (constructive proof). It is the most convenient when the product n 1 ⋯ n
Chinese_remainder_theorem
Proof in set theory
binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following lemma: If s1, s2, ... , sn, ... is any enumeration
Cantor's_diagonal_argument
Logical principle
Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof: According to Brouwer, a statement that
Law_of_excluded_middle
Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue
Constructive_logic
Unique positive real number which when multiplied by itself gives 2
yielding a direct proof of irrationality in its constructively stronger form, not relying on the law of excluded middle. This proof constructively exhibits an
Square_root_of_2
Probability theorem on no events occurring
60th birthday). Vol. II. North-Holland. pp. 609–627. Moser, Robin A. (2008). "A constructive proof of the Lovasz Local Lemma". arXiv:0810.4812 [cs.DS].
Lovász_local_lemma
Reasoning for mathematical statements
the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that
Mathematical_proof
vast majority of positive results about computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing
Non-constructive algorithm existence proofs
Non-constructive_algorithm_existence_proofs
Branch of mathematical logic
techniques from recursion theory as well as proof theory. Functional interpretations are interpretations of non-constructive theories in functional ones. Functional
Proof_theory
Various systems of symbolic logic
used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the
Intuitionistic_logic
Category of mathematical proof
another fifth power: 275 + 845 + 1105 + 1335 = 1445. Proof by counterexample is a form of constructive proof, in that an object disproving the claim is exhibited
Proof_of_impossibility
Power series with rational exponents
solution of the equation can be expressed as a Puiseux series. Moreover, the proof provides an algorithm for computing these Puiseux series, and, when working
Puiseux_series
Axiomatic set theories based on the principles of mathematical constructivism
statement. The difference is that the constructive proofs are harder to find. In set theory, a restriction to the constructive reading of existence apriori leads
Constructive_set_theory
Number that is not a ratio of integers
integers and therefore a rational number. Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab is rational:
Irrational_number
Proof assistant
automation routines and extraction of a certified program from the constructive proof of its formal specification. Rocq works within the theory of the calculus
Rocq
Theorem in topology
and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's
Brouwer_fixed-point_theorem
Theorem which asserts the existence of an object
theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive, since the
Existence_theorem
Combinatorial analog of the Borsuk-Ulam theorem
but with opposite signs. The first proofs were non-constructive, by way of contradiction. Later, constructive proofs were found, which also supplied algorithms
Tucker's_lemma
On bipartite matching and vertex cover
size of a matching equals the smallest size of a vertex cover. The constructive proof described above provides an algorithm for producing a minimum vertex
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Mathematical use of "there exists"
n=25} . The mathematical proof of an existential statement about "some" object may be achieved either by a constructive proof, which exhibits an object
Existential_quantification
Type of polynomial used in Numerical Analysis
Bernstein. Polynomials in this form were first used by Bernstein in a constructive proof of the Weierstrass approximation theorem. With the advent of computer
Bernstein_polynomial
Theorem in the mathematics of Lie's theory
A different geometric proof was discovered in 2000 by Duistermaat and Kolk. Unlike the previous ones, it is a constructive proof: the integrating Lie group
Lie's_third_theorem
Concept in computer science
Bisping, Benjamin; et al. (2016), "Mechanical Verification of a Constructive Proof for FLP", in Blanchette, Jasmin Christian; Merz, Stephan (eds.), Interactive
Consensus_(computer_science)
Theorem in Euclidean geometry
Theorem until Mohr's work was rediscovered. Several proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection
Mohr–Mascheroni_theorem
Unsolved problem in computer science
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive
P_versus_NP_problem
Universality of construction using just a straightedge and a single circle with center
straightedge alone by describing their constructive steps in terms of the five basic constructions. Alternative proofs do exist for the Poncelet–Steiner theorem
Poncelet–Steiner_theorem
Mathematical theory of data types
foundations are constructive, and this includes most of the ones used by proof assistants.[citation needed] It is possible to add non-constructive features to
Type_theory
Every polynomial has a real or complex root
(1821). It contained Argand's proof, although Argand is not credited for it. None of the proofs mentioned so far is constructive. It was Weierstrass who raised
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Mathematical methods
logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas
Realizability
Computer science award
ACM. 54 (3): 12–es. doi:10.1145/1236457.1236459. S2CID 53244523. "A constructive proof of the general Lovász Local Lemma". Journal of the ACM. 57 (2). 2010
Gödel_Prize
Form of mathematical proof
to a log-n-step loop. Because of that, proofs using prefix induction are "more feasibly constructive" than proofs using predecessor induction. Predecessor
Mathematical_induction
Technique for proving sets have equal size
prescribed vertex degrees" – by Gilles Schaeffer. "Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials" – by Doron Zeilberger
Bijective_proof
rejecting non-constructive proofs such as those involving the law of excluded middle in its full generality. constructive proof A proof that demonstrates
Glossary_of_logic
On the remainder of division by x – r
this formula, we obtain: f ( r ) = R . {\displaystyle f(r)=R.} A constructive proof—that does not involve the existence theorem of Euclidean division—uses
Polynomial_remainder_theorem
Mathematical logic concept
arguments be of a constructive character, allowing us to deal with more general forms of inference. Gentzen's first version of his consistency proof was not published
Gentzen's_consistency_proof
Proof by Alan Turing
his use of the reductio ad absurdum form of proof. We must emphasize the "constructive" nature of this proof. Turing describes what could be a real machine
Turing's_proof
Branch of mathematical logic
its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many
Reverse_mathematics
Mathematical construction of a set with an equivalence relation
relation, or the equality on the quotient set). In proof theory, particularly the proof theory of constructive mathematics based on the Curry–Howard correspondence
Setoid
Theorem in topology
full exposition.) Freund, Robert M.; Todd, Michael J. (1982). "A constructive proof of Tucker's combinatorial lemma". Journal of Combinatorial Theory
Borsuk–Ulam_theorem
German mathematician (1862–1943)
existence of such a set, it was not a constructive proof—it did not display "an object"—but rather, it was an existence proof and relied on use of the law of
David_Hilbert
Logical principles
used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the
Law_of_thought
Class of algorithms that find approximate solutions to optimization problems
of approximation algorithm providing an additive guarantee is the constructive proof of Vizing’s theorem. It shows how to color the edges of an undirected
Approximation_algorithm
Arrangement of numbers
unique up to reflections and rotations. Meng gives a less intricate constructive proof. The order-3 magic hexagon with numbers 1 through 19 and magic sum
Magic_hexagon
and a sketch of its proof appears in work of H. J. Kusher (appendix 3, page 106) published in 1964. a detailed constructive proof appears in work of Dario
Probability distribution of extreme points of a Wiener stochastic process
Probability_distribution_of_extreme_points_of_a_Wiener_stochastic_process
Theorem in mathematical logic
and a positive occurrence in some formula of T. We present here a constructive proof of the Craig interpolation theorem for propositional logic. Theorem—
Craig_interpolation
Topological space with a dense countable subset
numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable
Separable_space
Axiom of set theory
Bridges, Constructive analysis, Springer-Verlag, 1985. Fred Richman, "Constructive mathematics without choice", in: Reuniting the Antipodes—Constructive and
Axiom_of_choice
Multivariate functions can be written using univariate functions and summing
Jürgen; Griebel, Michael (2009). "On a constructive proof of Kolmogorov's superposition theorem". Constructive Approximation. 30 (3): 653–675. doi:10
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Theorem in quantum mechanics
a proof that relies upon this principle will not be a constructive proof. However, the theorem can be reformulated in such a way that a constructive proof
Gleason's_theorem
Real-valued function
{\displaystyle H^{1}} , in the noted paper Fefferman & Stein 1972: a constructive proof of this result, introducing new methods and starting a further development
Bounded_mean_oscillation
1247 Chinese text by Qin Jiushao
describing Chinese Remainder Theorem for the first time and providing a constructive proof for it, the text investigated: Indeterminate equations "Linglong method"
Mathematical Treatise in Nine Sections
Mathematical_Treatise_in_Nine_Sections
Mathematical model of the physical space
nonconstructive proofs just as sound as constructive ones, they are often considered less elegant, intuitive, or practically useful. Euclid's constructive proofs often
Euclidean_geometry
Legal concept regarding termination of employment
to another employee without reason or explanation The burden of proof in constructive dismissal cases lies with the employee. The Equal Employment Opportunity
Constructive_dismissal
Polynomial ideals are finitely generated
use of non-constructive methods. For example, the basis theorem asserts that every ideal has a finite generator set, but the original proof does not provide
Hilbert's_basis_theorem
Relation between algebraic varieties and polynomial ideals
following constructive proof of the weak form is one of the oldest proofs (the strong form results from the Rabinowitsch trick, which is also constructive). The
Hilbert's_Nullstellensatz
Description of degree sequences of graphs
Amitabha; Venugopalan, Sushmita; West, Douglas B. (2010), "A short constructive proof of the Erdős–Gallai characterization of graphic lists", Discrete Mathematics
Erdős–Gallai_theorem
Form of interpolation
value p(a) with complexity O(n2). The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by Bernstein and has gained
Polynomial_interpolation
Mathematical theorem in the study of analysis
supremum norm ‖f − p‖ < ε. The page for Bernstein polynomials outlines a constructive proof of the above theorem. For differentiable functions, Jackson's inequality
Stone–Weierstrass_theorem
American-Canadian computer scientist, contributor to complexity theory
activities in computer science for the last decade. In his "Feasibly Constructive Proofs and the Propositional Calculus" paper published in 1975, he introduced
Stephen_Cook
Argument that leads to a logical absurdity
Bishop, Errett 1967. Foundations of Constructive Analysis, New York: Academic Press. ISBN 4-87187-714-0 "Proof By Contradiction". www2.edc.org. Retrieved
Reductio_ad_absurdum
Mathematical models of strategic interactions
numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games
Game_theory
original proofs of Malgrange and Ehrenpreis did not use explicit constructions as they used the Hahn–Banach theorem. Since then several constructive proofs have
Malgrange–Ehrenpreis_theorem
Game whose outcome can be correctly predicted
initial position, given perfect play on both sides . This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually
Solved_game
Decomposition of periodic functions
pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In 1922, Andrey Kolmogorov published an article titled
Fourier_series
Theorem on triangulation graph colorings
2. The proof of the general case was first given by de Loera, Peterson, and Su in 2002. They provide two proofs: the first is non-constructive and uses
Sperner's_lemma
Establishment of a theorem using inference from the axioms
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language)
Formal_proof
Israeli mathematician
Retrieved 29 September 2025. Zeilberger, Doron (1989). "Kathy O'Hara's constructive proof of the unimodality of the Gaussian polynomials". Amer. Math. Monthly
Doron_Zeilberger
Mathematical proof expressed visually
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident
Proof_without_words
via the probabilistic method. They are particularly used for non-constructive proofs. Normal numbers exist. Moreover, computable normal numbers exist
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves
Mathematical_object
Rule of inference of propositional logic
P\lor R} " appear on lines of a proof, " Q ∨ S {\displaystyle Q\lor S} " can be placed on a subsequent line. The constructive dilemma rule may be written
Constructive_dilemma
Hungarian and American mathematician and physicist (1903–1957)
are congruent by translation). His next paper dealt with giving a constructive proof without the axiom of choice that 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}}
John_von_Neumann
Mathematical logic concept
its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its
Contraposition
American mathematician (1928–1983)
{\displaystyle \mathbb {\mathbb {C} } ^{n}} , and a new proof of Remmert's proper mapping theorem. Constructive mathematics. Bishop became interested in foundational
Errett_Bishop
its original proof Mathematical induction and a proof Proof that 0.999... equals 1 Proof that 22/7 exceeds π Proof that e is irrational Proof that π is irrational
List_of_mathematical_proofs
Type whose definition depends on a value
user can supply a constructive proof that a type is inhabited (i.e., that a value of that type exists) then a compiler can check the proof and convert it
Dependent_type
argument was not improved upon for many years. Also, it is not a constructive proof: it does not exhibit a concrete position that needs this many moves
Optimal solutions for the Rubik's Cube
Optimal_solutions_for_the_Rubik's_Cube
Non-contradiction of a theory
complete. A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven
Consistency
Field in logic and theoretical computer science
Theory of Computing. pp. 517–526. Cook, Stephen (1975). "Feasibly constructive proofs and the propositiona calculus". Proceedings of the 7th Annual ACM
Proof_complexity
Concept in algebraic geometry
S2CID 122056627 Bierstone, Edward; Milman, Pierre D. (1991), "A simple constructive proof of Canonical Resolution of Singularities", in Mora, T.; Traverso,
Resolution_of_singularities
Subfield of mathematics
into four areas: set theory model theory recursion theory, and proof theory and constructive mathematics (considered as parts of a single area). Additionally
Mathematical_logic
Puzzle of disappearance of information in a black hole
Masaki; Warner, Nicholas P. (May 2015). "Habemus Superstratum! A constructive proof of the existence of superstrata". Journal of High Energy Physics.
Black hole information paradox
Black_hole_information_paradox
On coloring the edges of graphs
1016/j.dam.2010.06.019, MR 2679785 Misra, J.; Gries, David (1992), "A constructive proof of Vizing's Theorem", Information Processing Letters, 41 (3): 131–133
Vizing's_theorem
Limitative results in mathematical logic
undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
be found in randomized polynomial time. Moser, Robin A. (2009), "A constructive proof of the Lovász local lemma", STOC'09—Proceedings of the 2009 ACM International
Entropy_compression
Proof assistant program
directly from proof terms. This also applies to non-constructive proofs, using a refined A-translation. The system is supported by automatic proof search and
MINLOG
Axiomatization of arithmetic
result means that all Peano arithmetic theorems have a proof that consists of a constructive proof followed by a classical logical rewriting. Roughly, the
Heyting_arithmetic
Sparse graph with strong connectivity
alternative construction of bipartite Ramanujan graphs. The original non-constructive proof was turned into an algorithm by Michael B. Cohen. Later the method
Expander_graph
Matrix factorisation in mathematics
1 {\displaystyle H_{i}=QA_{i}Q^{-1}} is upper quasi-triangular. A constructive proof for the Schur decomposition is as follows: every operator A on a complex
Schur_decomposition
Problem in computer science
[Hilbert] was only angry and frustrated, but then he began to try to deal constructively with the problem... Gödel himself felt—and expressed the thought in
Halting_problem
Left-invariant (or right-invariant) measure on locally compact topological group
Sciences de Paris, 211: 759–762 Alfsen, E.M. (1963), "A simplified constructive proof of existence and uniqueness of Haar measure", Math. Scand., 12: 106–116
Haar_measure
Theorem in complex analysis
same problem. The method of the proof suggested by Mergelyan is constructive, and remains the only known constructive proof of the result.[citation needed]
Mergelyan's_theorem
French mathematician (born 1947)
5802/aif.1287 "Six Lectures on Transseries, Analytical Functions and the Constructive Proof of Dulac's Conjecture", in D. Schlomiuk's Bifurcations and Periodic
Jean_Écalle
Symbolic logic system
B.} This can be read as follows: Given a constructive proof of A ∨ B {\displaystyle A\lor B} and constructive rejection of A {\displaystyle A} , one unconditionally
Minimal_logic
In addition, some adherents of these schools reject non-constructive proofs, such as using proof by contradiction when showing the existence of an object
Philosophy_of_mathematics
Theorem in set theory
König's proof uses the principle of excluded middle to draw a conclusion through case analysis. As such, the above proof is not a constructive one. In
Schröder–Bernstein_theorem
Programming paradigm based on applying and composing functions
intuitionistic type theory (also called constructive type theory), which associated functional programs with constructive proofs expressed as dependent types. This
Functional_programming
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
Girl/Female
Indian
Built; Construction; Creative Art; All Creation
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Creation; Evolution; Construction
Girl/Female
Hindu, Indian, Marathi
Produce; New Construction
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Tamil, Telugu
Construction; Arrangement; Creative Art; All Creation
Boy/Male
Indian
Proof
Girl/Female
Hindu
Creation, Construction, Arrangement
Boy/Male
Indian
Proof
Girl/Female
Indian
Many signs & proofs, Verses in the Quran, Royal
Boy/Male
Muslim
Proof
Girl/Female
Hindu
Creation, Construction, Arrangement
Boy/Male
Muslim
Proof
Boy/Male
Arabic, Muslim
A Persian Construction Probably from the Arabic Mawla (Master; Leader; Lord)
Boy/Male
Muslim
Argument, Reasoning, Proof
Girl/Female
Hindu
Light, Beauty, Prosperity, Rank, Power, Steel construction company
Girl/Female
Indian
Many signs & proofs, Verses in the Quran, Royal
Girl/Female
Tamil
Creation, Construction, Arrangement
Boy/Male
Indian
Argument, Reasoning, Proof
Girl/Female
Muslim
Guide, Proof
Girl/Female
Tamil
Light, Beauty, Prosperity, Rank, Power, Steel construction company
Girl/Female
Tamil
Creation, Construction, Arrangement
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
Boy/Male
Hindu, Indian, Kannada, Marathi, Telugu
Gold
Girl/Female
Hindu
Lover of lotus
Male
Iranian/Persian
Variant spelling of Persian Jamshid, possibly JAMSHAD means "shining river."
Boy/Male
Indian
Developer
Boy/Male
Indian
Thought, Imagination, Ecstasy, Mirth, Devotion
Girl/Female
Tamil
Thirsty, Silver
Girl/Female
Tamil
Goddess Durga
Girl/Female
Hindu, Indian
Name of a Hindu Goddess
Female
Swedish
Danish and Swedish variant form of Scandinavian Gunhild, GUNILLA means "war-battle."
Male
Hebrew
(עַמִּי×ָסָף) Hebrew name AMIASAF means "my people have been gathered in (reunited)."
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
CONSTRUCTIVE PROOF
adv.
In a constructive manner; by construction or inference.
n.
The act of constructing; construction.
a.
Obstructive.
n.
The method of construing, interpreting, or explaining a declaration or fact; an attributed sense or meaning; understanding; explanation; interpretation; sense.
n.
The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition.
a.
Serving or tending to bind or constrict.
a.
Reconstructing; tending to reconstruct; as, a reconstructive policy.
n.
The act of constructing vaults; a vaulted construction.
a.
Conveying knowledge; serving to instruct or inform; as, experience furnishes very instructive lessons.
a.
Derived from, or depending on, construction or interpretation; not directly expressed, but inferred.
a.
Pertaining to a master builder, or to architecture; evincing skill in designing or construction; constructive.
a.
Building; constructing.
a.
Having ability to construct or form; employed in construction; as, to exhibit constructive power.
n.
The act of fabricating, framing, or constructing; construction; manufacture; as, the fabrication of a bridge, a church, or a government.
a.
Constructive.
n.
Instructive discourse.
n.
That which is constructed or formed; an edifice; a fabric.
a.
Building up; constructive; -- opposed to destructive.
n.
An obstructive person or thing.
a.
According to interpretation; constructive.