The 2D Dependency Pair Framework for Conditional Rewrite Systems—Part II: Advanced Processors and Implementation Techniques

Abstract

Proving termination of programs in ‘real-life’ rewriting-based languages like CafeOBJ, Haskell, Maude, etc., is an important subject of research. To advance this goal, faithfully capturing the impact in the termination behavior of the main language features (e.g., conditions in program rules) is essential. In Part I of this work, we have introduced a 2D Dependency Pair Framework for automatically proving termination properties of Conditional Term Rewriting Systems. Our framework relies on the notion of processor as the main practical device to deal with proofs of termination properties of conditional rewrite systems. Processors are used to decompose and simplify the proofs in a divide and conquer approach. With the basic proof framework defined in Part I, here we introduce new processors to further improve the ability of the 2D Dependency Pair Framework to deal with proofs of termination properties of conditional rewrite systems. We also discuss relevant implementation techniques to use such processors in practice.

Appendices

Use of ${\textsf {P}}_{{ RT}}$ with AGES (Example 23)

AGES  specification

AGES  goal

AGES  output

Proof of Theorem 5

First, we introduce a slight generalization of the notion of O-chain in Definition 1.

Definition 13

Let $\mathcal{P},\mathcal{Q},\mathcal{R}$, and $\mathcal{S}$ be CTRSs. A $(\mathcal{P},\mathcal{Q},\mathcal{R},\mathcal{S})$-O-chain is a finite or infinite sequence of (renamed) rules $u_i\rightarrow v_i\Leftarrow c_i\in \mathcal{P}$, which are viewed as conditional dependency pairs, together with a substitution $\sigma$ satisfying that, for all $i\ge 1$, (i) for all $s\rightarrow t\in c_i$, $\sigma (s)\rightarrow ^*_\mathcal{R}\sigma (t)$ and (ii) $\sigma (v_i)(\rightarrow _{\mathcal{S},\mathcal{R}}\cup {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q},\mathcal{R}})^*\sigma (u_{i+1})$. A $(\mathcal{P},\mathcal{Q},\mathcal{R},\mathcal{S})$-O-chain is called minimal if for all $i\ge 1$, whenever

$$\begin{aligned} \sigma (v_i)=w_{i1}(\rightarrow ^*_{\mathcal{S},\mathcal{R}}\circ {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q},\mathcal{R}}) \cdots (\rightarrow ^*_{\mathcal{S},\mathcal{R}}\circ {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q},\mathcal{R}})\,w_{im_i}\rightarrow _{\mathcal{S},\mathcal{R}}^*\sigma (u_{i+1}), \end{aligned}$$

in the chain, then for all j, $1\le j\le m_i$, $w_{ij}$ is $\mathcal{R}$-operationally terminating.

Clearly, a (minimal) $(\mathcal{P},\mathcal{Q},\mathcal{R})$-O-chain can be viewed as a (minimal) $(\mathcal{P},\mathcal{Q},\mathcal{R},\mathcal{R})$-O-chain. Given a CTRS problem ${\check{\tau }}=(\mathcal{P},\mathcal{Q},\mathcal{R},\varphi )$, we prove that for every infinite minimal $(\mathcal{P},\mathcal{Q},\mathcal{R},\mathcal{R})$-O-chain we can construct an infinite $(\mathcal{P},\mathcal{Q},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R},\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N})$-O-chain, where $\mathcal{N}^c \subseteq \mathcal{R}$ can be different from $\mathcal{N}^\rightarrow \subseteq \mathcal{R}$ is some cases. First, we define the following interpretation:

Definition 14

(Interpretation) Let $\mathcal{R}=({\mathcal{F}},R)$ be a CTRS and $\mathcal{N}\subseteq \mathcal{R}$. Let > be an arbitrary total ordering on terms in ${\mathcal{T}({\mathcal{F}}\cup \{\bot ,\mathsf {c}^\varDelta \},{\mathcal{X}})}$, where $\bot$ is a fresh constant symbol and $\mathsf {c}^\varDelta$ is a fresh binary symbol. The interpretation $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}$ is a mapping from operationally terminating terms in ${\mathcal{T}({\mathcal{F}},{\mathcal{X}})}$ to terms in ${\mathcal{T}({\mathcal{F}}\cup \{\bot ,\mathsf {c}^\varDelta \},{\mathcal{X}})}$ defined as follows:

• 1. $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(x) = x$ if $x \in {\mathcal{X}}$,

• 2. $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(f(t_1,\ldots , t_k)) = f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_k))$ if there is no rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unifies with $\ell$ with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible,

• 3. $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(f(t_1,\ldots , t_k)) = \mathsf {c}^\varDelta (f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_1),\ldots ,\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_k)),t')$ otherwise.

where

$$\begin{aligned}{}\begin{array}[t]{l} t' = order \left( \{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u) \mid t \rightarrow _{\mathcal{R}} u\} \right) \\ order(T) = \left\{ \begin{array}{lll} \bot , &{} \text {if }&{} T = \emptyset \\ \mathsf {c}^\varDelta (t,order(T - \{t\})) &{}\text {if }&{} t\text { is minimal in the totally}\\ &{} &{}\text {ordered set }(T,>) \end{array} \right. \\ \end{array} \end{aligned}$$

The interpretation $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}$ does not generate an infinite term when the input term is operationally terminating.

Lemma 2

Let $\mathcal{R}=({\mathcal{F}},R)$ be a CTRS, $\mathcal{N}\subseteq \mathcal{R}$ and t in ${{\mathcal{T}({\mathcal{F}},{\mathcal{X}})}}$. If t is $\mathcal{R}$-operationally terminating then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t)$ is a finite term.

Proof

Suppose not. Without loss of generality we choose a t minimal, i.e. $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t)$ is an infinite term but $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t')$ is a finite term for every proper subterm $t'$ of t. By cases:

• 1. If $t = x \in {\mathcal{X}}$, the $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(x) = x$, yielding a contradiction.

• 2. If $t = f(t_1,\ldots ,t_k)$ and there is no rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $\ell$ and $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t) = f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_1), \ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_k))$. By the minimality assumption, for every $1 \le i \le n$, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_i)$ is finite, yielding again a contradiction.

• 3. If $t = f(t_1,\ldots ,t_k)$ and there is $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that the lhs $\ell$ and $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t) = \mathsf {c}^\varDelta (s,s')$, where $s = f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_1), \ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_k))$. By the minimality assumption, for every $1 \le i \le n$, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(t_i)$ is finite and s is finite. We have to analyze $s'$. We have two possibilities:

  • There are infinite rewritings starting from t, which implies t is not operationally terminating, leading to a contradiction.

  • There is a term u such that $t \rightarrow _{\ell \rightarrow r \Leftarrow c} u$, $\ell \rightarrow r \Leftarrow c \in \mathcal{R}$ and $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u)$ is infinite. We know that t is operationally terminating, t is V-terminating (conditions do not generate infinite computations), u is operationally terminating and $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u)$ is infinite. We can find a subterm $u'$ of u such that $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u')$ is infinite and $u'$ is minimal. Since any other option generates a finite term for $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u')$, the only possibility is infinitely iterate over this step, but, if this occur, we can construct an infinite sequence of the form:

    $$\begin{aligned} t \rightarrow _\mathcal{R}u \unrhd u' \rightarrow _{\mathcal{R}} v \unrhd v' \rightarrow _{\mathcal{R}} \cdots \end{aligned}$$

    contradicting the operational termination of t.

$\square$

If we have a substitution $\sigma$ such that for all $x \in {\mathcal{D}}om(\sigma )$, $\sigma (x)$ is operationally terminating, we can construct a substitution $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}$.

Definition 15

Let $\mathcal{R}$ be a CTRS, $\mathcal{N}\subseteq \mathcal{R}$ and $\sigma$ an $\mathcal{R}$-operationally terminating substitution. We denote by $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}$ a substitution that replaces occurrences of $x \in {\mathcal{D}}om(\sigma )$ by $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (x))$.

We can extract the following properties from $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}$.

Lemma 3

(Properties of $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}$) Let $\mathcal{R}=({\mathcal{F}},R)$ be a CTRS, $\mathcal{N}\subseteq \mathcal{R}$, $s,t, \sigma (s), \sigma (t) \in {\mathcal{T}({\mathcal{F}},{\mathcal{X}})}$ be $\mathcal{R}$-operationally terminating terms and $\sigma$ an $\mathcal{R}$-operationally terminating substitution. We have:

• 1. If $\textsf {Needed}_{\mathcal{R}_F}(t) \subseteq \mathcal{N}$ then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t))=\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$.

• 2. $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$.

• 3. Assuming that whenever $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}$, we have $\textsf {Needed}_{\mathcal{R}_F}(r') \subseteq \mathcal{N}$ and $\textsf {Needed}_{\mathcal{R}_F}(s_i') \subseteq \mathcal{N}$, where $s_i' \rightarrow t_i' \in c'$. If $\sigma (s) \rightarrow _{\mathcal{R}}^* \sigma (t)$ and $\textsf {Needed}_{\mathcal{R}_F}(s) \subseteq \mathcal{N}$ then $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow ^*_{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta } \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}} (t)$.

Proof

• 1. Similar to Item 2.

• 2. By structural induction on t:

  • If t is a variable x then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (x)) = \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(x)$.

  • If $t = f(t_1,\ldots ,t_k)$ then:

    • if there is no rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $f(\textsc {ECap}_\mathcal{R}(t_1), \ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unifies with $\ell$ with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, then

      $$\begin{aligned} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) = f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_1)),\ldots ,\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_k))). \end{aligned}$$

      By the I.H., we have $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_i)) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t_i)$, for all $1 \le i \le k$. Hence, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$;

    • if there is a rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unifies with $\ell$ with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, then, for some $t'$, we have

      $$\begin{aligned} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) = \mathsf {c}^\varDelta (f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_1)),\ldots ,\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_k))),t') \end{aligned}$$

      Using the $\mathcal{C}_{\varepsilon }^\varDelta$-rule $\mathsf {c}^\varDelta (x,y) \rightarrow x$, we obtain

      $$\begin{aligned} f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_1)),\ldots ,\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t_k)) \end{aligned}$$

      again and, therefore, the same conclusion $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$.

• 3. By induction on the length of the sequence, we have:

  • if $\sigma (s) = \sigma (t)$ then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t))$. By hypothesis, $\textsf {Needed}_{\mathcal{R}_F}(s) \subseteq \mathcal{N}$ and, therefore, $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))$. By Lemma 3(2), $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (t)) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$. Therefore, we get $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$ and, since $\mathcal{C}_{\varepsilon }^\varDelta \subseteq \mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta$, $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$.

  • if $\sigma (s) \rightarrow _{\mathcal{R}} \sigma (s') \rightarrow _{\mathcal{R}}^* \sigma (t)$, where p is the position of redex $\sigma (s)|_p$ in $\sigma (s) {\mathop {\rightarrow }\limits ^{p}}_{\{\ell \rightarrow r \Leftarrow c\},\mathcal{R}} \sigma (s')$. Without loss of generality, we can make $s' = y$, where y is a fresh new variable:

    • First assume that there is no position q such that $q \le p$, $\sigma (s)|_q = f(s_1,\ldots ,s_k)$ and there is no rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $f(\textsc {ECap}_\mathcal{R}(s_1),\ldots ,\textsc {ECap}_\mathcal{R}(s_k))$ unifies with $\ell$ with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible. Applying Definition 14,

      $$\begin{aligned} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)[\sigma (s)|_p]_p) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_p)]_p. \end{aligned}$$

      Then, $\ell \rightarrow r \Leftarrow c \in \mathcal{N}$, $\sigma (s)|_p = \sigma (\ell )$, $\sigma (s)|_p$ is V-terminating and $\sigma (s')|_p = \sigma (r)$ for some substitution $\sigma$. By definition, $\textsf {Needed}_{\mathcal{R}_F}(r) \subseteq \mathcal{N}$ and for each $s_i \rightarrow t_i \in c$, $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}$. Since $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_p) =\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (\ell ))$, by Lemma 3 (2), we get $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (\ell )) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(\ell )$. Now, for every $s_i \rightarrow t_i \in c$, we can apply the I.H. to $\sigma (s_j) \rightarrow _{\mathcal{R}}^* \sigma (t_j)$ ($\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}$ because $\ell \rightarrow r \Leftarrow c \in \mathcal{N}$ and, by hypothesis, if $s_i \rightarrow t_i \in c$, $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}$), obtaining $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s_j) \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t_j)$. Furthermore, we have $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(r) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (r))$ (by hypothesis $\textsf {Needed}_{\mathcal{R}_F}(r) \subseteq \mathcal{N}$ because $\ell \rightarrow r \Leftarrow c \in \mathcal{N}$). We have:

      $$\begin{aligned} \begin{array}{l} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_p)]_p =\\ \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (\ell ))]_p \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(\ell )]_p \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^+\\ \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(r)]_p = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (r))]_p =\\ \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)[\sigma (r)]_p) \end{array} \end{aligned}$$

      and $\sigma (s') = \sigma (s)[\sigma (r)]_p$. Since $\textsf {Needed}_{\mathcal{R}_F}(s) \subseteq \mathcal{N}$, by Lemma 3 (1), we have $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s)$ and since $s' \in {\mathcal{X}}$, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s')) = \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s')$. Therefore $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^+ \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s')$.

    • Now consider the case where there is a position q such that $q \le p$, $\sigma (s)|_q = f(s_1,\ldots ,s_k)$ and there is a rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}$ such that $\ell$ and $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible. Applying Definition 14, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)[\sigma (s)|_q]_q) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_q)]_q$. By Definition 14,

      $$\begin{aligned} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s')|_q) \in order \left( \{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(u') \mid \sigma (s)|_q \rightarrow _{\mathcal{R}} u'\} \right) . \end{aligned}$$

      By applying $\mathcal{C}_{\varepsilon }^\varDelta$ rules, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_q) \rightarrow ^+_{\mathcal{C}_{\varepsilon }^\varDelta } \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s')|_q)$. We have:

      $$\begin{aligned} \begin{array}{l} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)|_q)]_q \rightarrow ^+_{\mathcal{C}_{\varepsilon }^\varDelta }\\ \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s))[\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s')|_q)]_q = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)[\sigma (s')|_q]_q) \end{array} \end{aligned}$$

      and $\sigma (s') = \sigma (s)[\sigma (s')|_q]_q$. Since $\textsf {Needed}_{\mathcal{R}_F}(s) \subseteq \mathcal{N}$, by Lemma 3 (1), we have $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s)) = \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s)$ and since $s' \in {\mathcal{X}}$, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}(\sigma (s')) = \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s')$. Therefore $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow _{\mathcal{C}_{\varepsilon }^\varDelta }^+ \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s')$, but since $\mathcal{C}_{\varepsilon }^\varDelta \subseteq \mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta$ then we can also write $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^+ \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s')$.

    Finally, applying again the I.H. to $\sigma (s') \rightarrow _{\mathcal{R}}^* \sigma (t)$ (note that $s' \in {\mathcal{X}}$, therefore $\textsf {Needed}_{\mathcal{R}_F}(s') \subseteq \mathcal{N}$ is vacuously true), we get

    $$\begin{aligned} \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow ^+_{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta } \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s') \rightarrow _{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta }^* \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t) \end{aligned}$$

    and, therefore, $\sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(s) \rightarrow ^*_{\mathcal{N}\cup \mathcal{C}_{\varepsilon }^\varDelta } \sigma _{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}}}(t)$.

$\square$

Lemma 4

(Compatibility between needed rules) Let $\mathcal{R}=({\mathcal{F}},R)$ be a CTRS, $\mathcal{N}^c, \mathcal{N}^\rightarrow \subseteq \mathcal{R}$ and $t \in {\mathcal{T}({\mathcal{F}},{\mathcal{X}})}$ is $\mathcal{R}$-operationally terminating. If $\mathcal{N}^\rightarrow \subseteq \mathcal{N}^c$, then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t)$.

Proof

By structural induction on t:

• if $t \in {\mathcal{X}}$ then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t)$.

• if $t = f(t_1,\ldots ,t_k)$ then:

  • if there is no rule $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}^\rightarrow$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ and $\ell$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, then we have $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_k))$. By the I.H., $f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_k)) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t_k)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t)$

  • if there is no rule $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}^c$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ and $\ell$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, but there is $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}^\rightarrow$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ and $\ell$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, then $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_k))$. This is because, by Definition 14,

    $$\begin{aligned} \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(f(t_1,\ldots ,t_n)) = c(f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_k)),t'). \end{aligned}$$

    By I.H., $f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t_k)) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } f(\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t_1),\ldots , \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t_k)) = \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t)$.

  • if there is a rule $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}^c$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ and $\ell$ unify with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible, $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = order(\{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(u) \mid t \rightarrow _{\mathcal{R}} u\})$ and $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t) = order(\{\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(u) \mid t \rightarrow _{\mathcal{R}} u\})$. By applying the I.H. to the $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(u)$ terms, we have $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(u) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(u)$ and, therefore, we can conclude $\varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\varDelta } \varPhi ^{\varDelta }_{\mathcal{R},\mathcal{N}^c}(t)$.

$\square$

Lemma 5

(Properties of $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }$ and $\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}$) Let $\mathcal{R}=({\mathcal{F}},R)$ be a CTRS, $\mathcal{N}^c, \mathcal{N}^\rightarrow \subseteq \mathcal{R}$, $t, u \in {\mathcal{T}({\mathcal{F}},{\mathcal{X}})}$ be $\mathcal{R}$-operationally terminating. Furthermore, we assume that

• if a rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}^\rightarrow$, then

  • 1. $\textsf {Needed}^\rightarrow _{\mathcal{R}_F}(r') \subseteq \mathcal{N}^\rightarrow$,

  • 2. $\textsf {Needed}^\rightarrow _{\mathcal{R}_F}(s_i') \subseteq \mathcal{N}^\rightarrow$ if $s_i' \rightarrow t_i' \in c'$ and $fv (t_i')$ does not hold,

  • 3. $\textsf {Needed}_{\mathcal{R}_F}(s_i') \subseteq \mathcal{N}^c$ if $s_i' \rightarrow t_i' \in c'$ and $fv (t_i')$ holds;

• if a rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}^c$, then

  • 1. $\textsf {Needed}_{\mathcal{R}_F}(r') \subseteq \mathcal{N}^c$,

  • 2. $\textsf {Needed}_{\mathcal{R}_F}(s_i') \subseteq \mathcal{N}^c$;

• if there exists a a rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}^\rightarrow$, $s_i' \rightarrow t_i' \in c$ and $sfv (t_i')$ does not hold then $\mathcal{N}^\rightarrow \subseteq \mathcal{N}^c$; and,

• if there exists a a rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}^\rightarrow$, $s_i' \rightarrow t_i' \in c$ and $fv (t_i')$ does not hold then $\mathcal{C}_{\varepsilon }$ is used instead of $\mathcal{C}_{\varepsilon }^\mathcal{N}$ and $\mathcal{C}_{\varepsilon }^\mathcal{R}$ in the following result (and is (93) therefore removed).

If $t \rightarrow ^*_{\mathcal{R}} u$ then $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}\cup \{(93)\}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(u)$.

Proof

We proceed by induction on the length of the sequence $t \rightarrow ^*_{\mathcal{R}} u$.

• if $t = u$ then $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(u)$.

• if $t {\mathop {\rightarrow }\limits ^{p}}_{\mathcal{R}} t' \rightarrow _{\mathcal{R}}^* u$, where p is the position p of the redex $t|_p$ in $t {\mathop {\rightarrow }\limits ^{p}}_{\ell \rightarrow r \Leftarrow c} t'$:

  • First assume that there is no position q with $q \le p$, $t|_q = f(t_1,\ldots ,t_k)$ and rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{R}- \mathcal{N}^\rightarrow$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ and $\ell '$ unify with mgu $\theta$ and $\theta (c')$ is $\mathcal{R}$-feasible. Applying Definition 14, $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t[t|_p]_p) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t|_p)]_p$. We have $\ell \rightarrow r \Leftarrow c \in \mathcal{N}^\rightarrow$, $t|_p = \sigma (\ell )$, $t|_p$ is V-terminating and $t'|_p = \sigma (r)$ for some substitution $\sigma$. Moreover, by definition we have $\textsf {Needed}^\rightarrow _{\mathcal{R}_F}(r) \subseteq \mathcal{N}^\rightarrow$ and for every $s_i \rightarrow t_i \in c$ we have $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^c$ if $fv (t_i)$ and $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^\rightarrow$, otherwise. Starting with $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t|_p) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\ell ))$, by Lemma 3 (2), we get $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\ell )) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(\ell )$. Now, for every $s_i \rightarrow t_i \in c$, we consider three cases:

    • 1. If $sfv (t_i)$ holds ($fv (t_i)$ also holds), then $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i) = \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^c}}(s_i) = \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(s_i) = s_i$ because $s_i$ is ground. Since $sfv (t_i)$ implies $fv (t_i)$, by hypothesis $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^c$ (if $\ell \rightarrow r \Leftarrow c \in \mathcal{N}^\rightarrow$ then $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^c$ if $fv (t_i)$ holds). Since $\mathcal{N}^c$ satisfies the conditions of $\mathcal{N}$ in Lemma 3 (3) (thanks to the second item in the lemma), we can apply Lemma 3 (3), where $\mathcal{N}= \mathcal{N}^c$, getting $s_i \rightarrow ^*_{\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(t_i)$. Note that $t_i$ and r share no variables because $fv (t_i)$ holds.

    • 2. If $fv (t_i)$ holds but $sfv (t_i)$ does not, by hypothesis $\mathcal{N}^\rightarrow \subseteq \mathcal{N}^c$ (there is a $s_i \rightarrow t_i \in c$ such that $sfv (t_i)$ does not hold). Therefore, by applying Rule (93) repeatedly to transform every occurrence of $\mathsf {c}^\mathcal{N}$ by $\mathsf {c}^\mathcal{R}$ (only when $\mathcal{C}_{\varepsilon }^\mathcal{N}\ne \mathcal{C}_{\varepsilon }^\mathcal{R}$, i.e. when there is no rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{N}^\rightarrow$, $s_i' \rightarrow t_i' \in c$ and $fv (t_i')$ does not hold) and Lemma 4, we obtain $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i) \rightarrow ^*_{\{(93)\}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^{\mathcal{R}}_{\mathcal{R},\mathcal{N}^c}}(s_i)$. Furthermore, $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^c$ (if $\ell \rightarrow r \Leftarrow c \in \mathcal{N}^\rightarrow$, then $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^c$ if $fv (t_i)$ holds). Since $\mathcal{N}^c$ satisfies the conditions of $\mathcal{N}$ in Lemma 3 (3) (thanks to the second item in the lemma), we can apply Lemma 3 (3), where $\mathcal{N}= \mathcal{N}^c$, getting $\sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(s_i) \rightarrow ^*_{\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(t_i)$. Note that $t_i$ and r share no variables because $fv (t_i)$ holds.

    • 3. If $fv (t_i)$ (and, therefore, $sfv (t_i)$) does not hold, $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^\rightarrow$ (if $\ell \rightarrow r \Leftarrow c \in \mathcal{N}^\rightarrow$ then $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^\rightarrow$ if $fv (t_i)$ does not holds) and $\mathcal{C}_{\varepsilon }^\mathcal{N}= \mathcal{C}_{\varepsilon }^\mathcal{R}= \mathcal{C}_{\varepsilon }$. Since we have $\textsf {Needed}_{\mathcal{R}_F}(s_i) \subseteq \mathcal{N}^\rightarrow$, then $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (s_i)) =\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i)$ by Lemma 3(1) and by the I.H., we have $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (s_i)) \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (t_i))$. Finally, by Lemma 3 (2), $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (t_i)) \rightarrow _{\mathcal{C}_{\varepsilon }^\mathcal{N}}^* \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(t_i)$. Thus, $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i) \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(t_i)$. But, since $\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N}\subseteq \mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{N}$ by hypothesis (there is a $s_i \rightarrow t_i \in c$ such that $sfv (t_i)$ does not hold), we can conclude that $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s_i) \rightarrow ^*_{\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{N}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(t_i)$. Note that $t_i$ and r can share variables, but they are instances of interpreted terms using $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }$, as desired.

    Now, consider r. For every variable x in r, we know that $x \in \ell$ or $x \in t_i$ such that $s_i \rightarrow t_i \in c$ and $fv (t_i)$ (and also $sfv (t_i)$) does not hold. We obtain $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(r)$ and, by hypothesis ($\textsf {Needed}_{\mathcal{R}_F}(r) \in \mathcal{N}^\rightarrow$), $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(r) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (r)) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t'|_p)$. We have:

    $$\begin{aligned} \begin{array}{l} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t|_p)]_p =\\ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\ell ))]_p \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(\ell )]_p \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^+\\ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(r)]_p = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (r))]_p =\\ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t[\sigma (r)]_p) \end{array} \end{aligned}$$

    and $t' = t[\sigma (r)]_p$. Therefore, we have that $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^+ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t')$.

  • Now consider the case where there is a position q such that $q \le p$ and $t|_q = f(t_1,\ldots ,t_k)$ and there is a rule in $\ell \rightarrow r \Leftarrow c \in \mathcal{R}- \mathcal{N}^\rightarrow$ such that $f(\textsc {ECap}_\mathcal{R}(t_1),\ldots ,\textsc {ECap}_\mathcal{R}(t_k))$ unifies with $\ell$ with mgu $\theta$ and $\theta (c)$ is $\mathcal{R}$-feasible. By Definition 14, $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}}(t) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}}(t[t|_q]_q) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}}(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}}(t|_q)]_q$. By Definition 14, we have $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t'|_q) \in order \left( \{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(u') \mid t|_q \rightarrow _{\mathcal{R}} u'\} \right)$. By applying $\mathcal{C}_{\varepsilon }^\mathcal{N}$ rules, we have $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t|_q) \rightarrow ^+_{\mathcal{C}_{\varepsilon }^\mathcal{N}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t'|_q)$. Thus:

    $$\begin{aligned} \begin{array}{l} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t|_q)]_q \rightarrow ^+_{\mathcal{C}_{\varepsilon }^\mathcal{N}}\\ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t)[\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t'|_q)]_q = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t[t'|_q]_q) \end{array} \end{aligned}$$

    and $t' = t[t'|_q]_q$. Therefore $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow _{\mathcal{C}_{\varepsilon }^\mathcal{N}}^+ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t')$, but since $\mathcal{C}_{\varepsilon }^\mathcal{N}\subseteq \mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N}$ and $\mathcal{C}_{\varepsilon }^\mathcal{N}\subseteq \mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{N}$, then $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^+ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t')$.

  Finally, applying the I.H. to $t' \rightarrow _{\mathcal{R}}^* u$ ($\mathcal{N}^\rightarrow$ and $\mathcal{N}^c$ do not change),

  $$\begin{aligned} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^+ \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t') \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^* \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(u) \end{aligned}$$

  and, therefore, $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(t) \rightarrow _{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}^* \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(u)$.

$\square$

Theorem 5. ${\textsf {P}}_{{ RTN}}$ is $\textsf {m}$-sound and complete.

Proof

Completeness follows by Corollary 1. Regarding soundness, we proceed by contradiction. Assume that A is an infinite minimal $(\mathcal{P},\mathcal{Q},\mathcal{R},\mathcal{R})$-O-chain, but there is no infinite minimal $(\mathcal{P}[\emptyset ]_\alpha ,\mathcal{Q}[\emptyset ]_\alpha ,\mathcal{R},\mathcal{R})$-O-chain. By finiteness of $\mathcal{P}$ and $\mathcal{Q}$, there are $\mathcal{P}'\subseteq \mathcal{P}$ and $\mathcal{Q}'\subseteq \mathcal{Q}$ such that A has a tail B that we can write as a reduction sequence as follows:

$$\begin{aligned} \sigma (u^1)\rightarrow _{\mathcal{P}',\mathcal{R}}\sigma (v^1)(\rightarrow _\mathcal{R}\cup {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q}',\mathcal{R}})^* \sigma (u^2)\rightarrow _{\mathcal{P}',\mathcal{R}} \sigma (v^2)(\rightarrow _\mathcal{R}\cup {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q}',\mathcal{R}})^*\sigma (u^3)\rightarrow _{\mathcal{P}',\mathcal{R}}\cdots \end{aligned}$$

for some substitution $\sigma$, where all pairs in $\mathcal{P}'$ and $\mathcal{Q}'$ are used infinitely often and terms $u^i$, $v^i$, $i \ge 1$, are operationally terminating on $\mathcal{R}$.

Now, we consider the different steps with the different sets in the infinite sequence.

• 1. If $\sigma (u^i)\rightarrow _{\mathcal{P}',\mathcal{R}}\sigma (v^i)$ for $\alpha _i:u^i \rightarrow v^i \Leftarrow c^i \in \mathcal{P}'$, we have $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^i)) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(u^i)$, by Lemma 3 (2). For all $s^i_j \rightarrow t^i_j \in c^i$, we must have $\sigma (s^i_j) \rightarrow ^*_{\mathcal{R}} \sigma (t^i_j)$. We consider three cases:

  • (a) If $sfv _{\alpha _i}(t^i_j)$ holds ($fv (t_i)$ also holds), then $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) = \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^c}}(s^i_j) = \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(s^i_j) = s^i_j$ because $s^i_j$ is ground. Since $sfv (t^i_j)$ implies $fv (t^i_j)$, by definition of $\textsf {NeedR}^{c}$ (see (69)), we have that $\textsf {Needed}_{\mathcal{R}_F}(s^i_j)\subseteq \textsf {NeedR}^{c}({\check{\tau }})$. By item 1 in Definition 11, $\textsf {NeedR}^{c}({\check{\tau }})\subseteq \mathcal{N}^c$. Then, $\textsf {Needed}_{\mathcal{R}_F}(s^i_j) \subseteq \mathcal{N}^c$. By Lemma 3(3), where $\mathcal{N}= \mathcal{N}^c$, $s^i_j \rightarrow ^*_{\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(t^i_j)$. Note that $t^i_j$ and r share no variables because $fv (t^i_j)$ holds.

  • (b) If $fv (t^i_j)$ holds but $sfv (t^i_j)$ does not hold, then, by definition of $\textsf {NeedR}^{c}$ (see (69)), we have that $\textsf {Needed}_{\mathcal{R}_F}(s^i_j)\subseteq \textsf {NeedR}^{c}({\check{\tau }})$. By Definition 11, item 1, $\textsf {NeedR}^{c}({\check{\tau }})\subseteq \mathcal{N}^c$. Therefore, $\textsf {Needed}_{\mathcal{R}_F}(s^i_j) \subseteq \mathcal{N}^c$. If $\mathcal{C}_{\varepsilon }^\mathcal{N}\ne \mathcal{C}_{\varepsilon }^\mathcal{R}\ne \mathcal{C}_{\varepsilon }$, i.e. when there is no rule $\ell ' \rightarrow r' \Leftarrow c' \in \mathcal{P}\cup \mathcal{Q}\cup \mathcal{N}^\rightarrow$, $s_i' \rightarrow t_i' \in c$ and $fv (t_i')$ does not hold, applying Rule (93) repeatedly we transform every occurrence of $\mathsf {c}^\mathcal{N}$ by $\mathsf {c}^\mathcal{R}$, obtaining $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) \rightarrow ^*_{\{(93)\}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j)$; otherwise $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) = \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j)$. Furthermore, $\mathcal{N}^\rightarrow \subseteq \mathcal{N}^c$ because there is a $s_i \rightarrow t_i \in c$ such that $sfv (t_i)$ does not hold. Since $\mathcal{N}^\rightarrow \subseteq \mathcal{N}^c$, applying Lemma 4, we get $\sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(s^i_j)$. Since $\textsf {Needed}_{\mathcal{R}_F}(s^i_j) \subseteq \mathcal{N}^c$, by Lemma 3 (3), where $\mathcal{N}= \mathcal{N}^c$, we obtain $\sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(s^i_j) \rightarrow ^*_{\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{R}_{\mathcal{R},\mathcal{N}^c}}(t^i_j)$. Note that $t^i_j$ and r share no variables because $fv (t^i_j)$ holds.

  • (c) If $fv (t^i_j)$ (and, therefore, $sfv (t_i)$) does not hold, by definition of $\textsf {NeedR}^{\rightarrow }$ (see (68)), we have that $\textsf {Needed}_{\mathcal{R}_F}(s^i_j)\subseteq \textsf {NeedR}^{\rightarrow }({\check{\tau }})$. By item 5 in Definition 11, we have $\mathcal{C}_{\varepsilon }^\mathcal{N}= \mathcal{C}_{\varepsilon }^\mathcal{R}= \mathcal{C}_{\varepsilon }$. By item 2 in Definition 11, $\textsf {NeedR}^{\rightarrow }({\check{\tau }})\subseteq \mathcal{N}^\rightarrow$. Therefore, $\textsf {Needed}_{\mathcal{R}_F}(s^i_j) \subseteq \mathcal{N}^\rightarrow$. Note that the conditions of Lemma 5 are fulfilled by item 3 in Definition 11. Then, by Lemma 5, $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (s^i_j)) = \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (t^i_j))$ and, by Lemma 3 (2),

    $$\begin{aligned} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (t^i_j)) \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(t^i_j), \end{aligned}$$

    and therefore $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(s^i_j) \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(t^i_j)$. Note that $t^i_j$ and $v^i_j$ can share variables, but they are instances of interpreted terms using $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }$, as we want.

  Now, consider $v^i_j$. For every variable x in $v^i_j$, we know that $x \in u^i$ or $x \in t^i_j$ such that $s^i_j \rightarrow t^i_j \in c^i_j$ and $fv (t^i_j)$ (and also $sfv (t^i_j)$) does not hold. Therefore, we obtain $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(v^i_j)$ and, by Definition 11, ($\textsf {Needed}_{\mathcal{R}_F}(v^i_j)\subseteq \textsf {NeedR}^{\rightarrow }({\check{\tau }})$), $\sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(v^i_j) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^i_j))$.

• 2. If $\sigma (u^i_j)\rightarrow _{\mathcal{Q}',\mathcal{R}}\sigma (v^i_j)$ for $u^i_j \rightarrow v^i_j\Leftarrow c^i_j \in \mathcal{Q}'$, we can follow the exact same reasoning.

• 3. If $w^i_j \rightarrow ^*_{\mathcal{R}} w^i_{j+1}$, by Lemma 5 (conditions of Lemma 5 are fulfilled by item 3 in Definition 11) we have $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(w^i_j) \rightarrow ^+_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(w^i_{j + 1})$.

Therefore, we can construct an infinite sequence of the form:

$$\begin{aligned} \begin{array}{rcl} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^1)) &{} \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} &{} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(u^1) \\ &{} \rightarrow _{\mathcal{P}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} &{} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(v^1) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^1))\\ &{} (\rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\cup {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}})^* &{} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^2)) \\ &{} \rightarrow ^*_{\mathcal{C}_{\varepsilon }^\mathcal{N}} &{} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(u^2)\\ &{} \rightarrow _{\mathcal{P}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}} &{} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(v^2) = \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^2)) \\ &{} (\rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\cup {\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}})^* &{} \sigma _{\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }}(u^3)\\ &{} \rightarrow _{\mathcal{P}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}&{} \cdots \end{array} \end{aligned}$$

Although $\mathcal{Q}'$ could be empty (if no pair in $\mathcal{Q}'$ is used to connect pairs in $\mathcal{P}'$) $\mathcal{P}'$ is not empty. By Definition 6.2, for all $i\ge 1$ and $u^i\rightarrow v^i\Leftarrow c^i\in \mathcal{P}'$,

$$\begin{aligned} \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^i)))&( > rsim \cup \succeq \cup \sqsupset )&\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }\sigma (v^i))) \end{aligned}$$

(104)

Note that $\alpha \notin \mathcal{P}'\cup \mathcal{Q}'$. Otherwise, we get a contradiction as follows. Since

$$\begin{aligned} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^i))(\rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\circ {{\;{\mathop {\longrightarrow }\limits ^{{\varLambda }}}^=_{\mathcal{Q}',\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\,}})^*\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{i+1})), \end{aligned}$$

there are pairs $\overline{u}^k_i\rightarrow \overline{v}^k_i\Leftarrow \bigwedge _{j=1}^{n_{ik}} \overline{u}^k_{ij}\rightarrow \overline{v}^k_{ij}\in \mathcal{Q}'$ for k, $1\le k\le \kappa _i$ ($\kappa _i=0$ indicates that no pair in $\mathcal{Q}'$ is necessary to connect $\sigma (v^i)$ and $\sigma (u^{i+1})$; if $\mathcal{Q}'=\emptyset$, then $\kappa _i=0$ for all $i\ge 0$) such that $\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{u}^k_{ij}))\rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{v}^k_{ij}))$ for all j, $1\le j\le n_{ik}$ and k, $1\le k\le \kappa _i$, and

$$\begin{aligned} \begin{array}{l} \varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^i))\rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{u}^{1}_i)){\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{v}^{1}_{i}))\\ \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\cdots \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{u}^{\kappa _i}_{i})){\mathop {\rightarrow }\limits ^{{\varLambda }}}_{\mathcal{Q},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{v}^{\kappa _i}_{i}))\\ \rightarrow ^*_{\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N},\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R}}\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{i+1})) \end{array} \end{aligned}$$

for $i\ge 1$. By Definition 6.1, for all k, $1\le k< \kappa _i$ we have

$$\begin{aligned} \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^i))) > rsim ^*\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{u}^1_{i+1})))&\text { and }\nonumber \\ \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{v}^k_i))) > rsim ^* \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{u}^{k+1}_{i}))) \end{aligned}$$

(105)

and $\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (\overline{v}^{\kappa _i}_i))) > rsim ^*\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{i+1})))$ (or $\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^i))) > rsim ^* \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u_{i+1})))$ if $\kappa _i=0$). Also,

$$\begin{aligned} \pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^k_i)))&( > rsim \cup \succeq \cup \sqsupset )&\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (v^k_i))) \end{aligned}$$

(106)

for all k, $1\le k< \kappa _i$. By compatibility among $> rsim$, $\succeq$, and $\sqsupset$, from (104) and (106) we conclude that $\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^i)))\,( > rsim \cup \succeq \cup \sqsupset )^+\,\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{i+1})))$. Since $\alpha$ occurs infinitely often in B,

there is an infinite set $\mathcal{J}\subseteq {\mathbb {N}}$ such that $\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^j)))\,\sqsupset \,\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{j+1})))$ for all $j\in \mathcal{I}$. And we have $\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^i)))\,( > rsim \cup \succeq \cup \sqsupset )\,\pi (\varPhi ^\mathcal{N}_{\mathcal{R},\mathcal{N}^\rightarrow }(\sigma (u^{i+1})))$ for all other $u^i\rightarrow v^i\Leftarrow c^i\in \mathcal{P}'$ with $i\in {\mathbb {N}}-\mathcal{J}$. Thus, by using the compatibility conditions of the removal triple, we obtain an infinite decreasing $\sqsupset$-sequence that contradicts the well-foundedness of $\sqsupset$.

Thus, $\mathcal{P}'\subseteq \mathcal{P}[\emptyset ]_\alpha$ and $\mathcal{Q}'\subseteq \mathcal{Q}[\emptyset ]_\alpha$, i.e., B is transformed into an infinite (minimal) $(\mathcal{P}[\emptyset ]_\alpha ,\mathcal{Q}[\emptyset ]_\alpha ,\mathcal{N}^c\cup \mathcal{C}_{\varepsilon }^\mathcal{R},\mathcal{N}^\rightarrow \cup \mathcal{C}_{\varepsilon }^\mathcal{N})$-O-chain, leading to a contradiction. $\square$