To define the conditions on the inequalities, we need the notion of asigned generation tree. Because formally defining such trees seems inevitably cumbersome and not very enlightening, we first give a hopefully clarifying example.
Example 1.2.1. InblmτD, letαbe the formulax∨2(x∧y). Then thepositive generation tree
forαlooks as follows:
x∨2(x∧y) + x + 2(x∧y) + x∧y + x − x + y +
The adjectivepositiverefers to the fact that in the construction of the tree, the top node (theroot
of the tree) was labeled ‘+’. If we put a −at the top node, we get the negative version of the tree. The rule for labelling subsequent nodes with signs is simple: in the coordinates where the order type of the operation at hand is 1, we label the node with the same sign as the node directly above it, and where the order type is∂, we put the opposite sign.
One more example: take τ = ((∂,1),(1, ∂)), and denote the operations by ◦ and •, respectively. Then the negative generation tree ofβ=x◦(x•y) looks as follows:
x◦(x•y) − x + x•y − x − y +
Now for the formal definition.
Definition 1.2.2. Let αbe ablmτ-term. We inductively define the generation tree T(α) of
α, as follows:
• Ifα=xfor some variablex,T(α) is the tree consisting of one node called ‘x’ and no edges.
• Ifα=♥(α1, . . . , αn) for somen-ary operation symbol♥(which may also be one of the binary operations∨and∧, or the nullary operations⊥and>), thenT(α) is the tree consisting of a node called ‘α’ andn disjoint subtreesT(α1), . . . , T(αn), with edges from the node αto
the root of each T(αi). By the operation at the node α in T(α) we mean ♥, and the root ofT(αi) is called theith direct successor of the node α.
We now define the positive and negative generation tree, T+(α) and T−(α), respectively, as follows. LetT+(α) be the tree T(α) with a label + added to the root, andT−(α) the tree with
a label −added to the root. Then, walking down all the paths in the tree from the root to the leaves, label all nodes inT+(α) andT−(α) with either a + or a−according to the following rule:
To theith direct successor ofn, assign thesameoroppositelabel as the node n, according to
whether theithcoordinate of the order type of the operation at the nodenis 1 or∂, respectively.
After this process, all nodes in the treesT+(α) andT−(α) will be labeled with a sign.
We now introduce some useful notation from [7].
Asubtermsof a termαis the content of any node in the generation tree ofα. By a subterm we will always mean a specific occurrence of that subterm. (That is, in Example 1.2.1 above, there are two different subterms of β which have ‘x’ as their content. It is useful to regard them as different, because these two nodes get different signs in the signed generation tree.)
Note that a subtermsgenerates asubtreeofα, namely the one which hassas its root. We will identify subterms with their subtrees, and writes≺αfor “sis a subterm ofα”.
A useful expansion of this notation is to write, for example, “+s≺ −α” ifshas the sign + at its root in the negative generation tree of α. We will also write “is≺+α” to mean thats has the sign that is dictated byi in the positive tree of α.
Given an order type = (1, . . . , n) for all of the variables x1, . . . , xn occurring (possibly more than once) inα, we say that an occurrencexi≺αagrees with ifixi≺+α. We say that the termαagrees with if every occurrence of a variable agrees with, and we abbreviate this as ‘(α)’. A termαis calleduniformif there exists somesuch that(α).
If s≺ αand φ is any blmτ-formula, we write α[φ/s] for the result of substituting the subterm
s by φ. (Note, again, that this substitution occurs only once, at the location of the particular subtermsin the tree ofα.)
Example 1.2.3 (Example 1.2.1, continued). In the example formula β =x◦(x•y) above, we have +x≺ −β, −x≺ −β and +y≺ −β, butnot−y≺ −β.
For an example of the substitution notation, we haveβ[z/(x•y)] =x◦z. We cannot unambiguously write β[z/x], because substitution can happen only once, so we would need to specify explicitly
whichoccurence ofxwe want to replace withz.
Neitherαnorβ is uniform, but the formulaα0 :=α[z/x]isuniform, becauseα0 agrees with the order type= (1,1,1), that is,(α0) holds.
The following lemma is an obvious consequence of these definitions.
Lemma 1.2.4. If a term α agrees with an order type, then the term function αA: A → A is
monotone.
Proof. By induction on the complexity ofα. The base step is the observation that all variables have monotone term functions, and the inductive step holds because each operation symbol3is
interpreted as a monotone functionf:A→A, and monotonicity is a property that is preserved by functional composition.
To finish this section, we give a small glossary which translates the the terminology of the paper [22] to the terms we have defined above.
Term in [22] Corresponding term here
positive agrees with (1, . . . ,1) negative agrees with (∂, . . . , ∂)
-positive agrees with
-negative agrees with∂
uniform agrees with some
It is apparent from this table that the “agreeing with”-relation unifies and simplifies some of the ‘old’ terminology.