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isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Fold_ZF.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Below Then we use that sequence of tranformations to define the sequence of partial folds (called FoldSeq. By means of InductiveSeqVarFN. Defined in InductiveSeq ZF. Theory) which implements the inductive sequence determined by a starting point and a sequence of transformations. Finally, we define the fold of a sequence as the last element of the sequence of the partial folds. Definition of a sequence of partial folds. Apply...

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Order_ZF_1.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Order ZF 1 imports. We define some notions related to order relations based on the nonstrict orders ( ( leq ) type). Some people however prefer to talk about these notions in terms of the strict order relation ( (. And also for Metamath. In this theory file we repeat some developments from Order ZF. Definitions and basic properties. The next definition is the strict version of the linear order. What we write as R Orders A.

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Fol1.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Isabelle/ZF builds on the first order logic. Almost everything one would like to have in this area is covered in the standard Isabelle libraries. The material in this theory provides some lemmas that are missing or allow for a more readable proof style. Notions and lemmas in FOL. This section contains mostly shortcuts and workarounds that allow to use more readable coding style. Results up Isabelle in an infinite loop).

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Order_ZF_1a.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Order ZF 1a imports. This theory is a continuation of Order ZF. And talks about maximuma and minimum of a set, supremum and infimum and strict (not reflexive) versions of order relations. Maximum and minimum of a set. That every finite set has well-defined maximum and minimum. For antisymmetric relations maximum of a set is unique if it exists. Order ZF 4 L1:. A1: ( text{antisym}(r) ) and. A2: ( text{HasAmaximum}(r,A) ).

sciencehouse.blogspot.com

Scientific Clearing House: Formal Proof

http://sciencehouse.blogspot.com/2008/11/automated-theorem-proving.html

In the past, scholars could gather in coffee houses and talk about the latest scientific or mathematical discoveries. The goal of Scientific Clearing House is to provide a venue for such discourse. Saturday, November 22, 2008. This month's Notices of the American Mathematical Society. Is highlighting formal proof. That imply the truth (or more accurately the provability) of another term called the conclusion. Can we define a topology on the space of all terms? I suppose a metric could be the fewest numbe...

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Finite_ZF_1.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Finite ZF 1 imports. Finite1 Order ZF 1a. This theory is based on Finite1. Theory and is obsolete. It contains properties of finite sets related to order relations. See the FinOrd. Theory for a better approach. Finite vs. bounded sets. The goal of this section is to show that finite sets are bounded and have maxima and minima. Finite set has a maximum - induction step. Finite ZF 1 1 L1:. A1: ( r text{ is total on } X ) and.

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/FinOrd_ZF.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Finite ZF func ZF 1. This theory file contains properties of finite sets related to order relations. Part of this is similar to what is done in Finite ZF 1. Except that the development is based on the notion of finite powerset defined in Finite ZF. Rather the one defined in standard Isabelle Finite. Finite vs. bounded sets. The goal of this section is to show that finite sets are bounded and have maxima and minima. For linea...

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Finite1.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. This theory extends Isabelle standard Finite. Theory. It is obsolete and should not be used for new development. Use the Finite ZF. In this section we consider various properties of Fin. Datatype (even though there are no datatypes in ZF set theory). Lemma from Finite.thy. First we define a property of finite sets that we want to show. Lemma are those needed by the main induction step in lemma Fin induct. A1: ( W in TA ) and.

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Introduction.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. This theory does not contain any formalized mathematics used in other theories, but is an introduction to IsarMathLib project. How to read IsarMathLib proofs - a tutorial. The first thing that mathematicians typically do is to define notions. In Isar this is done with the definition. Is used to denote both zero (of natural numbers) and the empty set, which is not surprising as those two things are the same in set theory.

isarmathlib.org

IsarMathLib

http://www.isarmathlib.org/Nat_ZF_IML.html

A library of formalized mathematics for Isabelle/ZF theorem proving environment. Nat ZF IML imports. The induction lemmas in the standard Isabelle's Nat.thy file like for example nat induct. The next theorem is a version of induction on natural numbers that I was thought in school. A1: ( n in nat ) and. A2: ( P(0) ) and. A3: ( forall k in nat. P(k) longrightarrow P(succ(k) ). X in nat ), ( P(x) ). P(n) ) by (rule. A nonzero natural number has a predecessor. Nat ZF 1 L3:. A1: ( n in nat ) and. A2: ( foral...