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AboutThisSite
Introduction
ZF1
Nat_ZF
func_ZF
EquivClass1
Finite_ZF
FinSupp_ZF
Topology_ZF
Topology_ZF_1
Topology_ZF_1b
Topology_ZF_2
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Group_ZF
Group_ZF_1
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GettingStarted

FinSupp_ZF

Thu, 03 Jul 2008 00:09:00 GMT

theory FinSupp_ZF imports Finite_ZF Group_ZF_2

begin

Functions with finite support are those functions valued in a monoid that are equal to the neutral element everywhere except a finite number of points. They form a submonoid of the space of all functions valued in the monoid with the natural pointwise operation (or a subgroup if functions are valued in a group). Finitely supported functions show up in the IsarMathLib's construction of real numbers. Namely (for an abelian group (G,A)) we say that f: G\rightarrow G is an almost homomorphism if the map (n,m) \mapsto f(n+m) - f(n) - f(m) is finitely supported. The (additive group of) real numbers is really constructed as the quotient group: (almost homomorphism)/(finitely supported functions), where the group G we start with is the group of integers. Also polynomials can be viewed as ring valued sequences that have finite support.

Functions with finite support

In this section we provide the definition and set up notation for formalizing the notion of finitely supported functions.

Spupport of a function is the subset of its domain where the values are not zero.

Definition
Supp(f,G,A) \equiv \{x \in domain(f).\ f(x) \neq TheNeutralElement(G,A)\}

A finitely supported function is such that its support is in the finite powerset of its domain.

Definition
FinSupp(X,G,A) \equiv \{f \in X\rightarrow G.\ Supp(f,G,A) \in FinPow(X)\}

We will use the additive notation writing about finitely supported functions. In the finsupp context defined below we assume that (M,A) is a monoid and X is some arbitrary set. We denote \mathcal{A} to be the pointwise operation on M-valued functions on X corresponding to the monoid operation A, (denoted as + ). 0 is the neutral element of the monoid.

Locale finsupp
fixes M, A,
assumes monoidAsssum: \text{IsAmonoid}(M,A)
fixes monoper,
defines a + b \equiv A\langle a,b\rangle
fixes X,
fixes pointewiseoper,
defines \mathcal{A} \equiv A \text{ lifted to function space over } X
fixes funoper,
defines a \oplus b \equiv \mathcal{A} \langle a,b\rangle
fixes neut,
defines 0 \equiv TheNeutralElement(M,A)
fixes supp,
defines supp(f) \equiv Supp(f,M,A)

We can use theorems proven in the monoid0 context.

lemma (in finsupp ) monoid0_valid:
shows monoid0(M,A) using monoidAsssum , monoid0_def

The sum of monoid valued functions is a monoid valued function.

lemma (in finsupp ) lifted_op_closed:
assumes f:X \rightarrow M, g:X \rightarrow M shows f\oplus g : X\rightarrow Mproof

What is the value of a sum of monoid-valued functions?

lemma (in finsupp ) sum_val:
assumes f:X \rightarrow M, g:X \rightarrow M and x \in X shows (f\oplus g)(x) = f(x) + g(x) using assms , monoid0_valid , monoid0.lifted_val

The support of the sum of functions is contained in the union of supports.

lemma (in finsupp ) supp_sum_union:
assumes f:X \rightarrow M, g:X \rightarrow M shows supp(f\oplus g) \subseteq supp(f) \cup supp(g)proof

The sum of finitely supported functions is finitely supported.

lemma (in finsupp ) sum_finsupp:
assumes f \in FinSupp(X,M,A), g \in FinSupp(X,M,A) shows f\oplus g \in FinSupp(X,M,A)proof

The neutral element of the lifted (pointwise) operation is the function equal zero everywhere. In the next lemma we show that this is a finitely supported function.

lemma (in finsupp ) const_zero_fin_supp:
shows TheNeutralElement(X\rightarrow M, \mathcal{A} ) \in FinSupp(X,M,A) using monoidAsssum , Group_ZF_2_1_L2 , monoid0_valid , unit_is_neutral , func1_3_L1 , func1_3_L2 , func1_1_L1 , Supp_def , empty_in_finpow , FinSupp_def

end

Comments on FinSupp_ZF

Group_ZF_1b

Wed, 18 Jun 2008 01:04:00 GMT

theory Group_ZF_1b imports Group_ZF

begin

In a typical textbook a group is defined as a set G with an associative operation such that two conditions hold: A: there is an element e\in G such that for all g\in G we have e\cdot g = g and g\cdot e =g. We call this element a "unit" or a "neutral element" of the group.
B: for every a\in G there exists a b\in G such that a\cdot b = e, where e is the element of G whose existence is guaranteed by A. The validity of this definition is rather dubious to me, as condition A does not define any specific element e that can be referred to in condition B - it merely states that a set of such units e is not empty. Of course it does work in the end as we can prove that the set of such neutral elements has exactly one element, but still the definition by itself is not valid. You just can't reference a variable bound by a quantifier outside of the scope of that quantifier.
One way around this is to first use condition A to define the notion of a monoid, then prove the uniqueness of e and then use the condition B to define groups. Another way is to write conditions A and B together as follows:

\exists_{e \in G} \ (\forall_{g \in G} \ e\cdot g = g \wedge g\cdot e = g) \wedge (\forall_{a\in G}\exists_{b\in G}\ a\cdot b = e).

This is rather ugly. What I want to talk about is an amusing way to define groups directly without any reference to the neutral elements. Namely, we can define a group as a non-empty set G with an associative operation "\cdot " such that
C: for every a,b\in G the equations a\cdot x = b and y\cdot a = b can be solved in G.
This theory file aims at proving the equivalence of this alternative definition with the usual definition of the group, as formulated in Group_ZF.thy. The informal proofs come from an Aug. 14, 2005 post by buli on the matematyka.org forum.

An alternative definition of group

First we will define notation for writing about groups.

We will use the multiplicative notation for the group operation. To do this, we define a context (locale) that tells Isabelle to interpret a\cdot b as the value of function P on the pair \langle a,b \rangle.

Locale group2
fixes P,
fixes dot,
defines a \cdot b \equiv P\langle a,b\rangle

The next theorem states that a set G with an associative operation that satisfies condition C is a group, as defined in IsarMathLib Group_ZF theory.

theorem (in group2 ) altgroup_is_group:
assumes A1: G\neq 0 and A2: P \text{ is associative on } G and A3: \forall a\in G.\ \forall b\in G.\ \exists x\in G.\ a\cdot x = b and A4: \forall a\in G.\ \forall b\in G.\ \exists y\in G.\ y\cdot a = b shows \text{IsAgroup}(G,P)proof

The converse of altgroup_is_group: in every (classically defined) group condition C holds. In informal mathematics we can say "Obviously condition C holds in any group." In formalized mathematics the word "obviously" is not in the language. The next theorem is proven in the context called group0 defined in the theory Group_ZF.thy. Similarly to the group2 that context defines a\cdot b as P\langle a,b\rangle It also defines notation related to the group inverse and adds an assumption that the pair (G,P) is a group to all its theorems. This is why in the next theorem we don't explicitely assume that (G,P) is a group - this assumption is implicit in the context.

theorem (in group0 ) group_is_altgroup:
shows \forall a\in G.\ \forall b\in G.\ \exists x\in G.\ a\cdot x = b and \forall a\in G.\ \forall b\in G.\ \exists y\in G.\ y\cdot a = bproof

end