1 Introduction
Dershowitz and Tzameret extended the FriedmanKřížGordeev’s tree embedding theorem with gap conditions by relaxing the well orderedness condition for the labels of treenodes to a well quasi orderedness condition. The first purpose of our paper is to generalize OkadaTakeuti’s quasi ordinal diagram systems ([14, 13]) using this DershowitzTzameret’s result. The second purpose is to analyze to which extent such a nonsimplification ordering could be used as an extension of the typical termination proof method based on simplification orderings. We especially consider a “secondorder (patternmatchingbased) rewrite rule” version of Buchholz’s hydra game.
A typical termination proof method for firstorder term rewriting systems is to show the termination of a term rewriting system by verifying that for each rewrite rule of , holds, where is a strictly orderpreserving mapping and is a well founded ordering with the substitution property and the monotonicity property. Here, the substitution property and the monotonicity property mean (i) for any substitution (for the list of variables) , if holds then holds, and (ii) for any context , if holds then holds, respectively. The properties (i) and (ii) guarantee the termination of the whole because any application of (firstorder) rewrite rule has a form for some context and some substitution . In this note, we restrict our attention to the identity for our basic argument.
The method has been widely used for termination proofs as well as a tool for KnuthBendix completion. The method itself would be attractive not only for the traditional firstorder rewriting but also for higherorder or graphicpatternmatchingbased rewriting. One could expect that strong and general ordering structures in proof theory would be useful for this termination proof method of higherorder patternmatchingbased rewrite systems.
However, the use of strong orderings such as on Takeuti’s ordinal diagram systems, which is a nonsimplification ordering, cannot satisfy the two basic properties (i) and (ii). Because of this difficulty, instead of the traditional termination proof method, various different techniques for the termination of higherorder rewriting systems have been utilized; for example, Jouannaud and Okada ([10]) introduced a generalized form of TaitGirard’s reducibility candidates method (cf. also [2, 3] with Blanqui).
Hence at a first look, it seems hard to adapt ordinal diagram systems to the traditional termination method. It is a natural question how we could adapt them to the termination proof method especially for higherorder rewriting systems. We aim to answer this question in the present paper.
This paper is structured as follows. We first define our generalized quasi ordinal diagram systems (§2.1), then prove the well quasi orderedness of these systems as a corollary of the DershowitzTzameret’s version of tree embedding theorem (§2.2). Next, we propose a termination proof method induced by the monotonicity property and a restricted substitution property of (§3.1). Finally, we take a version of Buchholz game as an example of patternmatchingbased secondorder rewrite systems and show its termination by another termination proof method in terms of (§3.2). Note that the termination of the original version and its variants of KirbyParis’s hydra game could be proved in the traditional termination method of simplification orderings (cf. [9]).
2 Well quasi ordered systems of generalized quasi ordinal diagrams
In this section, we first generalize quasi ordinal diagram systems of OkadaTakeuti ([13, 14]) by using an arbitrary well partial ordering as the inner node labels (§2.1). Next, we prove the well quasi orderedness of these generalized systems as a direct corollary of the DershowitzTzameret’s version ([7]) of tree embedding theorem with gap condition (§2.2).
2.1 Formulation of generalized quasi ordinal diagram systems
A quasi ordering is a pair of a set and a binary relation such that for any , holds (reflexivity) and for any , if and hold then holds (transitivity). A partial ordering is a quasi ordering with the antisymmetry: For any , if and hold then holds. A linear ordering is a partial ordering with the linearity: For any , or holds. A well quasi ordering is a quasi ordering such that for any infinite sequence from , there are numbers and such that and holds. A well partial ordering is a partial ordering that is a well quasi ordering. For a quasi ordering , we use abbreviation “” for “ and .” Note that the well quasi orderedness has a weaker definition saying that for any infinite decreasing sequence from , there are numbers and such that and holds. A weak well quasi ordering is a quasi ordering satisfying this condition. In this paper, we show the stronger version of well quasi orderedness of our quasi ordinal diagram systems.
Let be a well partial ordering and be a well quasi ordering. We define the set of constants and the set of unary function symbols as follows: , . Let be a signature defined as the union of , and with a varyadic function symbol .
The predomain of generalized quasi ordinal diagrams on and is the set of all terms constructed from symbols in . To follow the notation in [13, 14], we denote terms of by and adopt the following abbreviations: , . We call a term that belongs to or has the form of a connected term of , and a term that has the form an unconnected term of . When an unconnected term is indicated as , we assume that all of are connected. For a connected term , a term is a component of if and only if holds. For an unconnected term , a term is a component of if and only if for some with , holds.
Definition 2.1 (Labeled Finite Trees).
Labeled finite trees are defined as follows.

A finite tree is a partial ordering such that is a finite set with the least element called the root, and for any , the set is linearly ordered with respect to .

Let be a quasi ordering. A labeled finite tree is a pair of a finite tree and a mapping .
Let be a finite tree. For any , if the set is nonempty, then we call its greatest element the immediate lower node of . We call maximal elements of leaves of . The greatest lower bound of is denoted by .
Example 2.2.
Let and be the following well partial ordering and well quasi ordering, respectively, where the arrow means and means .
In addition, we stipulate that for any and any , hold. Here we use the symbol since is similar to the ordinal . An example of forest representation are as follows.
Note that is represented by tree’s sum and branching. By the definition of the identity below, denotes the associativecommutative sum of connected terms, which is called “natural sum.” In term rewrite orderings such as the recursive path ordering, is often represented as .
Definition 2.3 (Identity on ).
For any two terms and of , the identity relation holds if and only if either (1) both of and are elements of , and is identical with in the sense of , or (2) , and hold, or (3) and hold with and there is a permutation of such that holds for any .
Definition 2.4 (sections).
For any two terms and of and any element of , the relation , which we call “ is an section of ”, is defined as follows.

If is an element of , then never holds.

If holds, then

when holds, if and only if or ,

when holds, if and only if ,

when holds, never holds.


If holds with , then if and only if for some , .
Example 2.5 (An example of an section).
Consider the domain defined in Example 2.2.
(0,0) *=0∙*+!U5 =”A”, (3,3) *=0∙*+!Lω’ =”B”, (0,6) *=0∙*+!R3 =”C”, (3,9) *=0∙*+!LU2 =”D”, (3,14) *α =”F”, (23,6) *β:= =”G”, (18,18) =”E1”, (6,18) =”E2”, (18,18) =”E3”, (12,18) =”E4”, (70, 11) *Here appears as labels, in this order, below . =”E5”, (70, 6) *Since are greater than or equal to , holds by definition, =”E6”, (70, 1) *where β≡(5, ⋯# (ω’ , ⋯# (3, ⋯# (2, α) # ⋯) # ⋯) # ⋯). @”A”;”B” @”B”;”C” @”C”;”D” @”A”;”E1” @”D”;”E2” @”A”;”E3” @”D”;”E4” @”E1”;”E3”
An element of is an index of if and only if there is a such that is an section of . Set . For any and any finite set of terms, define . We denote the cardinality of by . For any finite set of terms, the total number of all occurrences of and in is denoted by .
Definition 2.6 (Ordering on ).
For any element of , the relation on is defined by double induction on (1) and (2) , in this order.

If both of and are elements of , then for any element of , if and only if .

If is an element of and is not, then for any element of , both and hold.

If and hold with , then for any element of , if and only if one of the following conditions holds:

there is a such that for any , and hold,

there is a such that , and if then the following holds.


If , and hold, then if and only if either holds or both and hold.

If and hold and is an element of , then if and only if either

there is a such that , or

for any , both of and hold, and if holds then holds for any minimal element of , otherwise holds.
We say holds by condition, when the condition () above holds. Similarly, we say holds by condition, when the condition () above holds. Note that if is a finite set, in the condition () can be taken as a maximal element of , and the resulting ordering becomes the same as the one defined above.

Remark 2.7.
Consider defined in Example 2.2. For the readers familiar to the recursive path ordering (cf. [5]), the table below suggests some similarity of to and the richness of in the sense that for some depends on for another . When we stipulate that holds for any , we have the following table:
Recursive path ordering  The ordering 

if for some with ,  if for some , 
or and for any with ,  or and for any , 
or and .  or and for any and 
. 
Lemma 2.8.
For any well partial ordering , any well quasi ordering and any element of ,
is a quasi ordering.
Proof.
Prove the following sublemmas (1) and (2) by double induction on and double induction on , respectively. (1) (Transitivity) For every and every , if and hold then holds. (2) (Reflexivity) For every and every , the following two hold: holds, and if holds then for any with , and hold. ∎
Remark 2.9.
Let us comment on the definitions of sections (Definition 2.4) and (Definition 2.6).

We defined by taking as not a well quasi ordering but a well partial ordering. The antisymmetry of is needed to verify the transitivity of . In addition, note that the proof for the lemma above neither depends on the well quasi orderedness of nor the one of . If we take as a well ordering, then in Definition 2.6.5.() there always exists the least element of if it is nonempty. This is the only difference of our orderings from the ones in [13, 14].
The central quasi ordering in this paper is a generalization of the linear ordering appeared in [12], which satisfies the monotonicity property. We will show in Section 3.1 that our ordering satisfies the monotonicity property as well (cf. Lemma 3.1).
Definition 2.10 (Ordering ).
For any two terms of , holds if and only if holds for any element of .
Let be the reflexive closure of . It is obvious that is a quasi ordering and that for any and , holds if and only if both and holds.
Next, we restrict the predomains to the path comparable treedomain , since there is a counterexample for the well quasi orderedness of . In fact, we have a counterexample for its weak well quasi orderedness.
Example 2.11.
A counterexample for the weak well quasi orderedness of is as follows: Let be and be , where , and for any and , neither nor holds. Then, we have the following infinite sequence.
This kind of counterexamples is blocked if we restrict the predomains to the path comparable treedomains . In the rest of this subsection and the next subsection, we assume that predomains satisfy the following conditions: and holds for any and any . In addition, we identify a connected term of with a labeled finite tree in the manner of Example 2.2. For any quasi ordering and any two elements of , we say is comparable with if and only if or holds.
Definition 2.12 (Path comparable treedomains of , cf. [7]).
The path comparable treedomain of is defined as follows:

If holds, then is a connected term of .

Let be connected terms of with . If is comparable with for any and any , then is a connected term of .

If are connected terms of , then is an unconnected term of .
A generalized quasi ordinal diagram system is a pair of a path comparable treedomain and the quasi ordering on this domain. We often abbreviate as . Hereafter, we call a connected term (resp. an unconnected term) of a connected gqod (resp. an unconnected gqod). For , we have the following lemma by induction on .
Lemma 2.13.
For any element of , holds for any element of with .
2.2 Well quasi ordering proof for via Dershowitz and Tzameret’s tree embedding theorem
We first recall DershowitzTzameret’s tree embedding with gap condition (cf. [7, Definition 3.3]).
Definition 2.14 (Tree Embedding ).
For any two connected gqod’s and
, holds if and only if there is an injection such that

(Node condition 1) for any , holds,

(Node condition 2) for any , holds,

(Edge condition) if is an element of with its immediate lower node in , then for any with , holds, and

(Root condition) if is the root of , then for any with , holds.
Example 2.15.
The following is an example of DershowitzTzameret’s tree embedding (cf. [7, p.87]).
Theorem 2.16 (Theorem 3.1 in [7]).
Let be the set of all connected gqod’s from . Then, is a well quasi ordering.
Next, we extend this tree embedding to forests, namely, unconnected gqod’s and show that holds whenever is embedded into in the sense of this forest embedding.
Definition 2.17 (Forest Embedding ).
For any two terms and of with and (), holds if and only if and there is a permutation of such that for any with .
Proposition 2.18.
For every , if holds, then holds.
Proof.
By Lemma 2.13, one can prove the following sublemma (1) and (2). For every and every with , (1) if is connected and holds, then holds, and (2) if and for some , hold, then holds.
We prove Proposition 2.18 by induction on . The base case is obvious. If is unconnected, then the proposition immediately follows from IH. Suppose that holds. By the definition of , it suffices to prove the proposition when is a connected gqod . In this case, the proposition follows from the following two claims: For any and any , holds, and holds.
First, we show that for any and any , holds. Consider a gqod with and put . We show that for any with . If holds then obviously holds, so we assume that is of the form . Let us denote this outermost occurrence of in by . Suppose that the embedding maps to the outermost occurrence of in a subgqod of . We denote this outermost occurrence of in by . Since holds, we have by IH for any . On the other hand, has a subgqod because holds. Denote this occurrence of in by and suppose that the embedding maps to the outermost occurrence of in a subgqod of (), as the following figure.
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