(* This file is a part of IsarMathLib - a library of formalized mathematics written for Isabelle/Isar. Copyright (C) 2005, 2006 Slawomir Kolodynski This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Groups - and alternative definition› theory Group_ZF_1b imports Group_ZF begin text{* 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 @{text "Group_ZF.thy"}. The informal proofs come from an Aug. 14, 2005 post by buli on the matematyka.org forum. *} subsection{*An alternative definition of group*} text{*First we will define notation for writing about groups.*} text{*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 (infixl "⋅" 70) defines dot_def [simp]: "a ⋅ b ≡ P`⟨a,b⟩" text{*The next theorem states that a set $G$ with an associative operation that satisfies condition C is a group, as defined in IsarMathLib @{text "Group_ZF"} theory.*} theorem (in group2) altgroup_is_group: assumes A1: "G≠0" and A2: "P {is associative on} G" and A3: "∀a∈G.∀b∈G. ∃x∈G. a⋅x = b" and A4: "∀a∈G.∀b∈G. ∃y∈G. y⋅a = b" shows "IsAgroup(G,P)" proof - from A1 obtain a where "a∈G" by auto with A3 obtain x where "x∈G" and "a⋅x = a" by auto from A4 `a∈G` obtain y where "y∈G" and "y⋅a = a" by auto have I: "∀b∈G. b = b⋅x ∧ b = y⋅b" proof fix b assume "b∈G" with A4 `a∈G` obtain y⇩_{b}where "y⇩_{b}∈G" and "y⇩_{b}⋅a = b" by auto from A3 `a∈G` `b∈G` obtain x⇩_{b}where "x⇩_{b}∈G" and "a⋅x⇩_{b}= b" by auto from `a⋅x = a` `y⋅a = a` `y⇩_{b}⋅a = b` `a⋅x⇩_{b}= b` have "b = y⇩_{b}⋅(a⋅x)" and "b = (y⋅a)⋅x⇩_{b}" by auto moreover from A2 `a∈G` `x∈G` `y∈G` `x⇩_{b}∈G` `y⇩_{b}∈G` have "(y⋅a)⋅x⇩_{b}= y⋅(a⋅x⇩_{b})" "y⇩_{b}⋅(a⋅x) = (y⇩_{b}⋅a)⋅x" using IsAssociative_def by auto moreover from `y⇩_{b}⋅a = b` `a⋅x⇩_{b}= b` have "(y⇩_{b}⋅a)⋅x = b⋅x" "y⋅(a⋅x⇩_{b}) = y⋅b" by auto ultimately show "b = b⋅x ∧ b = y⋅b" by simp qed moreover have "x = y" proof - from `x∈G` I have "x = y⋅x" by simp also from `y∈G` I have "y⋅x = y" by simp finally show "x = y" by simp qed ultimately have "∀b∈G. b⋅x = b ∧ x⋅b = b" by simp with A2 `x∈G` have "IsAmonoid(G,P)" using IsAmonoid_def by auto with A3 show "IsAgroup(G,P)" using monoid0_def monoid0.unit_is_neutral IsAgroup_def by simp qed text{* The converse of @{text "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 @{text "group0"} defined in the theory @{text "Group_ZF.thy"}. Similarly to the @{text "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 "∀a∈G.∀b∈G. ∃x∈G. a⋅x = b" and "∀a∈G.∀b∈G. ∃y∈G. y⋅a = b" proof - { fix a b assume "a∈G" "b∈G" let ?x = "a¯⋅ b" let ?y = "b⋅a¯" from `a∈G` `b∈G` have "?x ∈ G" "?y ∈ G" and "a⋅?x = b" "?y⋅a = b" using inverse_in_group group_op_closed inv_cancel_two by auto hence "∃x∈G. a⋅x = b" and "∃y∈G. y⋅a = b" by auto } thus "∀a∈G.∀b∈G. ∃x∈G. a⋅x = b" and "∀a∈G.∀b∈G. ∃y∈G. y⋅a = b" by auto qed end