Establishment of a theorem using inference from the axioms
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable.[1] If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.[2][3]
The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.
Formal proofs often are constructed with the help of computers in interactive theorem proving (e.g., through the use of proof checker and automated theorem prover).[4] Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable, depending upon the formal system in use.
and mathematics, a formalproof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of...
involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and...
mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formalproofs by human-machine collaboration...
logical truth. Logical consequence is necessary and formal, by way of examples that explain with formalproof and models of interpretation. A sentence is said...
alphabet, formed by a formal grammar (consisting of production rules or formation rules). Deductive system, deductive apparatus, or proof system, which has...
1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of formal languages...
verification of these systems is done by ensuring the existence of a formalproof of a mathematical model of the system. Examples of mathematical objects...
formal verification may be used to produce a program in a more formal manner. For example, proofs of properties or refinement from the specification to a program...
theoretical analysis, this approach is more suited for constructing detailed formalproofs and is generally preferred in the research literature. In this article...
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects,...
the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. In addition to the better readability...
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory...
Mathematical induction is an inference rule used in formalproofs, and is the foundation of most correctness proofs for computer programs. Despite its name, mathematical...
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as:...
of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formalproof was accepted...
distinction between formal and informal words for "you" Formalproof, a fully rigorous proof as is possible only in a formal system Dynamic and formal equivalence...
accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature...
and vice versa. In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional...
mathematical logic, a proof calculus or a proof system is built to prove statements. A proof system includes the components: Formal language: The set L...
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition...
theorem) and independently shortly thereafter by Alan Turing in 1936 (Turing's proof). Church proved that there is no computable function which decides, for...
the consequent and denying the antecedent. See also contraposition and proof by contrapositive. The form of a modus tollens argument is a mixed hypothetical...
{\displaystyle \Sigma } . The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish...
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