Interactive Theorem Proving in Coq and the Curry
... Identifiers, Environments and Context In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identif ...
... Identifiers, Environments and Context In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identif ...
writing and reasoning in math
... 7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the ...
... 7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the ...
CHAPTER 1 The main subject of Mathematical Logic is
... In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, when the free occurrences of x are replaced by E 0 (also expressed by saying that there must be no “clashes of var ...
... In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, when the free occurrences of x are replaced by E 0 (also expressed by saying that there must be no “clashes of var ...
Implication - Abstractmath.org
... in the middle of a mathematical argument. Perhaps the meaning of (b) can be clarified by expanding the wording to read: " x can be greater than 3 only if x>2." Note that sentences of the form " P only if Q" about ordinary everyday things generally do not mean the same thing as "If P then Q"; that is ...
... in the middle of a mathematical argument. Perhaps the meaning of (b) can be clarified by expanding the wording to read: " x can be greater than 3 only if x>2." Note that sentences of the form " P only if Q" about ordinary everyday things generally do not mean the same thing as "If P then Q"; that is ...
Section I(e)
... [Negative] [Negative] is positive therefore from xy 0 we could have both x 0 [Negative] and y 0 [Negative] An example of where the converse is true is the following: x 5 0 x 5 [This is true] x 5 x 5 0 [This is also true] Hence the converse, Q P [Q implies P], of the proposi ...
... [Negative] [Negative] is positive therefore from xy 0 we could have both x 0 [Negative] and y 0 [Negative] An example of where the converse is true is the following: x 5 0 x 5 [This is true] x 5 x 5 0 [This is also true] Hence the converse, Q P [Q implies P], of the proposi ...
A Note on Bootstrapping Intuitionistic Bounded Arithmetic
... BBASIC axioms. Proof (B-1) follows from formula (b) of Proposition 1 and (CU-4). (B-4) is an immediate consequence of (CU-3) and (b) and (c) of Proposition 1. To show CUBASIC+ |= (B-5), first note that x 6= 0 ⊃ 1 ≤ x by (CU-2) and (CU-3); hence x 6= 0 ⊃ 0 6= |2x| by (CU-8) and (e) of Proposition 1 ...
... BBASIC axioms. Proof (B-1) follows from formula (b) of Proposition 1 and (CU-4). (B-4) is an immediate consequence of (CU-3) and (b) and (c) of Proposition 1. To show CUBASIC+ |= (B-5), first note that x 6= 0 ⊃ 1 ≤ x by (CU-2) and (CU-3); hence x 6= 0 ⊃ 0 6= |2x| by (CU-8) and (e) of Proposition 1 ...
Proofs by induction - Australian Mathematical Sciences Institute
... The Tower of Hanoi is a famous puzzle, sometimes known as the End of the World Puzzle. Legend tells that, at the beginning of time, there was a Hindu temple containing three poles. On one of the poles was a stack of 64 gold discs, each one a little smaller than the one beneath it. The monks of the t ...
... The Tower of Hanoi is a famous puzzle, sometimes known as the End of the World Puzzle. Legend tells that, at the beginning of time, there was a Hindu temple containing three poles. On one of the poles was a stack of 64 gold discs, each one a little smaller than the one beneath it. The monks of the t ...
An Independence Result For Intuitionistic Bounded Arithmetic
... For the definition of Kripke models of intuitionistic bounded arithmetic and basic results about them, see [M2] and [B2]. The general results on intuitionistic logic and arithmetic, and also Kripke models, can be found in [TD]. [MM] contains a study of weak fragments of first-order intuitionistic ar ...
... For the definition of Kripke models of intuitionistic bounded arithmetic and basic results about them, see [M2] and [B2]. The general results on intuitionistic logic and arithmetic, and also Kripke models, can be found in [TD]. [MM] contains a study of weak fragments of first-order intuitionistic ar ...
Intuitionistic Logic
... that the truth values of A and B are known before one can settle the status of A → B. Heyting showed that this is asking too much. Consider A = “there occur twenty consecutive 7’s in the decimal expansion of π”, and B = “there occur nineteen consecutive 7’s in the decimal expansion of π”. Then ¬A ∨ ...
... that the truth values of A and B are known before one can settle the status of A → B. Heyting showed that this is asking too much. Consider A = “there occur twenty consecutive 7’s in the decimal expansion of π”, and B = “there occur nineteen consecutive 7’s in the decimal expansion of π”. Then ¬A ∨ ...
Full text
... We need the following lemmas for the proof of our theorem. Lemma 1: Let m > 2 and n > 3 be rational integers and let a ...
... We need the following lemmas for the proof of our theorem. Lemma 1: Let m > 2 and n > 3 be rational integers and let a ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.