2.1 Programs

The syntax of mini-gringo with counting is defined as follows.Footnote ¹ We assume that three countably infinite sets of symbols are selected: numerals, symbolic constants, and variables. We assume that a 1-1 correspondence between numerals and integers is chosen; the numeral corresponding to an integer $n$ is denoted by $\overline n$ . (In examples of programs, we sometimes drop overlines in numerals.)

The set of precomputed terms is assumed to be a totally ordered set containing numerals, symbolic constants, and possibly other symbols, such that numerals are contiguous (every precomputed term between two numerals is a numeral) and are ordered in the standard way. mgc terms are formed from precomputed terms and variables using the unary operation symbol $|\ |$ and the binary operation symbols:

\begin{equation*}+\quad -\quad \times \quad /\quad \backslash \quad ..\end{equation*}

An mgc atom is a symbolic constant optionally followed by a tuple of terms in parentheses. A literal is a mgc atom possibly preceded by one or two occurrences of not. A comparison is an expression of the form $t_1\prec t_2$ , where $t_1$ , $t_2$ are mini-gringo terms, and $\prec$ is $=$ or one of the comparison symbols:

(1) \begin{equation} \neq \quad \lt \quad \gt \quad \leq \quad \geq \end{equation}

An aggregate element is a pair $\textbf {X}:{\textbf {L}},$ where $\textbf {X}$ is a tuple of distinct variables, and $\textbf {L}$ is a conjunction of literals and comparisons such that every member of $\textbf {X}$ occurs in $\textbf {L}$ . An aggregate atom is an expression of one of the forms

(2) \begin{equation} {{\mathit{count}}} \{E\} \geq t,\ {{\mathit{count}}} \{E\} \leq t,\ \end{equation}

where $E$ is an aggregate element and $t$ is a term that does not contain the interval symbol ( $..$ ). The conjunction of aggregate atoms (2) can be written as ${{\mathit{count}}}\{E\}=t$ .

A rule is an expression of the form

(3) \begin{equation} {\mathit{Head}}\leftarrow {\mathit{Body}}, \end{equation}

where

• • $\mathit{Body}$ is a conjunction (possibly empty) of literals, comparisons, and aggregate atoms, and

• • $\mathit{Head}$ is either an atom (then (3) is a basic rule), or an atom in braces (then (3) is a choice rule), or empty (then (3) is a constraint).

A variable that occurs in a rule $R$ is local in $R$ if each of its occurrences is within an aggregate element, and global otherwise. A rule is pure if, for every aggregate element $\textbf {X}:\textbf {L}$ in its body, all variables in the tuple $\textbf {X}$ are local. For example, the rule

\begin{equation*} q(Y) \leftarrow {{\mathit{count}}}\{X:p(X)\} = Y \land X\gt 0 \end{equation*}

is not pure, because $X$ is global.

In mini-gringo with counting, a program is a finite set of pure rules.

2.2 Stable models and strong equivalence

An atom $p(\textbf {t})$ is precomputed if all members of the tuple $\textbf {t}$ are precomputed terms. The semantics of mgc is based on an operator, called $\tau$ , which transforms pure rules into infinitary propositional formulas formed from precomputed atoms (Lifschitz Reference Lifschitz2022, Section 5). For example, the rule

\begin{equation*}q \leftarrow {{\mathit{count}}}\{X:p(X)\} \leq 5\end{equation*}

is transformed by $\tau$ into the formula

\begin{equation*}\left ( \bigwedge _{\Delta \,:\,|\Delta | \gt 5}\neg \bigwedge _{x\in \Delta }\;p(x) \right )\to q,\end{equation*}

where $\Delta$ ranges over finite sets of precomputed terms and $|\Delta |$ stands for the cardinality of $\Delta$ . The antecedent of this implication expresses that a set of more than 5 elements cannot be a subset of $\{X:p(X)\}$ . The result of applying $\tau$ to a program $\Pi$ is defined as the conjunction of formulas $\tau R$ for all rules $R$ of $\Pi$ .

Stable models of an mgc program $\Pi$ are defined as stable models of $\tau \Pi$ in the sense of the work by (Truszczynski Reference Truszczynski2012). Thus stable models of programs are sets of precomputed atoms.

About programs $\Pi _1$ and $\Pi _2$ we say that they are strongly equivalent to each other if $\tau \Pi _1$ is strongly equivalent to $\tau \Pi _2$ ; in other words, if for every set $\Phi$ of infinitary propositional formulas formed from precomputed atoms, $\{\tau \Pi _1\}\cup \Phi$ and $\{\tau \Pi _2\}\cup \Phi$ have the same stable models. It is clear that if $\Pi _1$ is strongly equivalent to $\Pi _2$ then, for any program $\Pi$ , $\Pi _1\cup \Pi$ has the same stable models as $\Pi _2\cup \Pi$ (take $\Phi$ to be $\{\tau \Pi \}$ ).

2.3 Representing MGC terms and atoms by formulas

In first-order formulas used to represent programs, we distinguish between terms of two sorts: the sort general and its subsort integer. General variables are meant to range over arbitrary precomputed terms, and we assume them to be the same as variables used in mgc programs. Integer variables are meant to range over numerals (or, equivalently, integers). In this paper, integer variables are represented by letters from the middle of the alphabet ( $I,\ldots ,N$ ).

The two-sorted signature $\sigma _0$ includes

• • all numerals as object constants of the sort integer;

• • other precomputed terms as object constants of the sort general;

• • the symbol $|\ |$ as a unary function constant; its argument and value have the sort integer;

• • the symbols $+$ , $-$ and $\times$ as binary function constants; their arguments and values have the sort integer;

• • pairs $p/n$ , where $p$ is a symbolic constant and $n$ is a nonnegative integer, as $n$ -ary predicate constants; their arguments have the sort general;

• • symbols (1) as binary predicate constants; their arguments have the sort general.

Note that the definition of $\sigma _0$ does not allow terms that contain a general variable in the scope of an arithmetic operation. For example, the mgc term $Y-\overline 1$ is not a term over $\sigma _0$ .

A formula of the form $(p/n)(\textbf {t})$ , where $\textbf {t}$ is a tuple of terms, can be written also as $p(\textbf {t})$ . Thus precomputed atoms can be viewed as sentences over $\sigma _0$ .

The set of values of an mgc termFootnote ² $t$ can be described by a formula over the signature $\sigma _0$ that contains a variable $Z$ that does not occur in $t$ (Lifschitz et al. Reference Lifschitz, Lühne and Schaub2019; Fandinno and Lifschitz Reference Fandinno and Lifschitz2023a). This formula, “ $Z$ is a value of $t$ ,” is denoted by $\textit {val}_{t}(Z)$ . Its definition is recursive, and we reproduce here two of its clauses:

• • if $t$ is a precomputed term or a variable then $\textit {val}_{t}(Z)$ is $Z=t$ ,

• • if $t$ is $t_1\;\textit {op}\;t_2$ , where op is $+$ , $-$ , or $\times$ then $\textit {val}_{t}(Z)$ is

  \begin{equation*}\exists I J (\textit {val}_{t_1}(I) \land \textit {val}_{t_2}(J) \land Z=I\;\textit {op}\;J),\end{equation*}
  where $I$ and $J$ are integer variables that do not occur in $t$ .

For example, $\textit {val}_{Y-\overline 1}(Z)$ is

\begin{equation*}\exists I J(I=Y \land J=\overline 1\land Z=I-J).\end{equation*}

The translation $\tau ^B$ transforms mgc atoms, literals, and comparisons into formulas over the signature $\sigma _0$ . (The superscript $B$ conveys the idea that this translation expresses the meaning of expressions in bodies of rules). For example, $\tau ^B$ transforms $p(t)$ into the formula $\exists Z(\textit {val}_{t}(Z) \land p(Z))$ . The complete definition of $\tau ^B$ can be found in earlier publications (Lifschitz et al. Reference Lifschitz, Lühne and Schaub2019; Fandinno and Lifschitz Reference Fandinno and Lifschitz2023a) and is reproduced in Appendix A for convenience.

2.4 Representing aggregate expressions and rules

To represent aggregate expressions by first-order formulas, we need to extend the signature $\sigma _0$ (Lifschitz Reference Lifschitz2022, Section 7). The signature $\sigma _1$ is obtained from $\sigma _0$ by adding all predicate constants of the forms

(4) \begin{equation} {\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F\hbox{ and }{\mathit{Atmost}}^{\textbf {X};\textbf {V}}_F \end{equation}

where $\textbf {X}$ and $\textbf {V}$ are disjoint lists of distinct general variables, and $F$ is a formula over $\sigma _0$ such that each of its free variables belongs to $\textbf {X}$ or to $\textbf {V}$ . The number of arguments of each constant (4) is the length of $\textbf {V}$ plus 1; all arguments are of the sort general. If $n$ is a positive integer then the formula ${\mathit{Atleast}}^{\textbf {X},\textbf {V}}_F(\textbf {V},\overline n)$ is meant to express that $F$ holds for at least $n$ values of $\textbf {X}$ . The intuitive meaning of ${\mathit{Atmost}}^{\textbf {X},\textbf {V}}_F(\textbf {V},\overline n)$ is similar: $F$ holds for at most $n$ values of $\textbf {X}$ .

For an aggregate atom of the form ${{\mathit{count}}}\{\textbf {X}:\textbf {L}\}\geq t$ in the body of a rule, the corresponding formula over $\sigma _1$ is

\begin{equation*} \exists Z\left (\textit {val}_{t}(Z) \land {\mathit{Atleast}}^{\textbf {X};\textbf {V}}_{\exists \textbf {W}\tau ^B(\textbf {L})}(\textbf {V},Z)\right ), \end{equation*}

where

• • $\textbf {V}$ is the list of global variables that occur in $\textbf {L}$ , and

• • $\textbf {W}$ is the list of local variables that occur in $\textbf {L}$ and are different from the members of $\textbf {X}$ .

For example, the aggregate atom ${{\mathit{count}}}\{ X : p(X)\} \geq Y$ is represented by the formula

\begin{equation*} \exists Z\left (Z=Y\land {\mathit{Atleast}}^{X;}_{\exists Z(Z=X\land p(Z))}(Z)\right ) \end{equation*}

( $\textbf {V}$ and $\textbf {W}$ are empty).

The formula representing ${{\mathit{count}}}\{\textbf {X}:\textbf {L}\}\leq t$ is formed in a similar way, with $\mathit{Atmost}$ in place of $\mathit{Atleast}$ .

Now we are ready to define the translation $\tau ^*$ , which transforms pure rules into sentences over $\sigma _1$ . It converts a basic rule

\begin{equation*}p(t) \leftarrow B_1\land \cdots \land B_n\end{equation*}

into the universal closure of the formula

\begin{equation*}B^*_1\land \cdots \land B^*_n\land \textit {val}_{t}(Z)\to p(Z),\end{equation*}

where $B^*_i$ is

• • $\tau ^B(B_i)$ , if $B_i$ is a literal or comparison, and

• • the formula representation of $B_i$ formed as described above, if $B_i$ is an aggregate atom.

The definition of $\tau ^*$ for pure rules of other forms can be found in the previous paper on mini-gringo with counting (Lifschitz Reference Lifschitz2022, Sections 6 and 8). For any program $\Pi$ , $\tau ^*\Pi$ stands for the conjunction of the sentences $\tau ^*R$ for all rules $R$ of $\Pi$ .

2.5 Logic of here-and-there and standard interpretations

We are interested in deductive systems $S$ with the following property:

(5) \begin{equation} \begin{array} c {for\;any\;programs}\,\Pi _1\; {and}\, \Pi _2,\\[5pt] {if} \; \tau ^*\Pi _1\; {and} \, \tau ^*\Pi _2 \, {can\;be\;derived\;from\;each\;other\;in}\, S\\[5pt] {then} \,\Pi _1\, {is\;strongly\;equivalent\;to}\,\Pi _2. \end{array} \end{equation}

Systems with property (5) cannot possibly sanction unlimited use of classical propositional logic. Consider, for instance, the one-rule programs

\begin{equation*}p \leftarrow {\mathit{not}} q\quad \hbox{and}\quad q \leftarrow {\mathit{not}} p.\end{equation*}

They have different stable models, although the corresponding formulas

\begin{equation*}\neg q\to p\quad \hbox{and}\quad \neg p\to q\end{equation*}

have the same truth table.

This observation suggests that the study of subsystems of classical logic may be relevant. One such subsystem is first-order intuitionistic logic (with equality) adapted to the two-sorted signature $\sigma _1$ . Intuitionistic logic does have property (5). Furthermore, this property is preserved if we extend it by the axiom schema

(6) \begin{equation} F\lor (F\to G) \lor \neg G, \end{equation}

introduced by Hosoi (Reference Hosoi1966) as part of his formalization of the propositional logic known as the logic of here-and-there.

The axiom schema

(7) \begin{equation} \exists X(F\to \forall X\,F) \end{equation}

(for a variable $X$ of either sort) can be included without losing property (5) as well. It was introduced to extend the logic of here-and-there to a language with variables and quantifiers (Lifschitz et al. Reference Lifschitz, Pearce and Valverde2007). Both (6) and (7) are provable classically, but not intuitionistically.

The axioms and inference rules discussed so far are abstract, in the sense that they are not related to any properties of the domains of variables (except that one is a subset of the other). To describe more specific axioms, we need the following definition. An interpretation of the signature $\sigma _0$ is standard if

• • its domain of the sort general is the set of precomputed terms;

• • its domain of the sort integer is the set of numerals;

• • every object constant represents itself;

• • the absolute value symbol and the binary function constants are interpreted as usual in arithmetic;

• • predicate constants (1) are interpreted in accordance with the total order on precomputed terms chosen in the definition of mgc (Section 2.1).

Two standard interpretations of $\sigma _0$ can differ only by how they interpret the predicate symbols $p/n$ . If a sentence over $\sigma _0$ does not contain these symbols then it is either satisfied by all standard interpretations or is not satisfied by any of them.

Let $\mathit{Std}$ be the set of sentences over $\sigma _0$ that do not contain predicate symbols of the form $p/n$ and are satisfied by standard interpretations. Property (5) will be preserved if we add any members of $\mathit{Std}$ to the set of axioms. The set $\mathit{Std}$ includes, for instance, the law of excluded middle $F\lor \neg F$ for every formula $F$ over $\sigma _0$ that does not contain symbols $p/n$ . Other examples of formulas from $\mathit{Std}$ are

\begin{equation*}\overline 2\times \overline 2 = \overline 4,\quad \forall N (N*N\geq \overline 0),\quad t_1\neq t_2, \end{equation*}

where $t_1$ , $t_2$ are distinct precomputed terms.

To reason about mgc programs, we need also axioms for $\mathit{Atleast}$ and $\mathit{Atmost}$ . A possible choice of such additional axioms is described in the next section.

3.1 Axioms of $\mathit{HT}_{\!\!\#}$

The axioms for $\mathit{Start}$ define these predicates recursively:

\begin{equation*} \begin{array} l \forall \textbf {X}\textbf {V} N(N\leq \overline 0 \to {\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N)),\\[5pt] \forall \textbf {X}\textbf {V} ({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},\overline 1)\leftrightarrow F),\\[5pt] \forall \textbf {X}\textbf {V} N(N\gt \overline 0 \to ({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N+\overline 1) \leftrightarrow \\[5pt] \hskip 4cm F\land \exists \textbf {U}(\textbf {X}\lt \textbf {U}\land {\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {U},\textbf {V},N)))). \end{array}\end{equation*}

Here $N$ is an integer variable, and $\textbf {U}$ is a list of distinct general variables of the same length as $\textbf {X}$ , which is disjoint from both $\textbf {X}$ and $\textbf {V}$ . The symbol $\lt$ in the last line denotes lexicographic order: $(X_1,\ldots ,X_m)\lt (U_1,\ldots ,U_m)$ stands for

\begin{equation*} \bigvee _{l=1}^m\left ((X_l\lt U_l) \land \bigwedge _{k=1}^{l-1}(X_k=U_k)\right ). \end{equation*}

This set of axioms for $\mathit{Start}$ will be denoted by $D_0$ .

The set of axioms for $\mathit{Atleast}$ and $\mathit{Atmost}$ , denoted by $D_1$ , defines these predicates in terms of $\mathit{Start}$ :

(8) \begin{align} \forall \textbf {V} Y({\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {V},Y) &\leftrightarrow \exists \textbf {X} N({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N)\land N\geq Y)), \end{align}

(9) \begin{align} \forall \textbf {V} Y({\mathit{Atmost}}^{\textbf {X};\textbf {V}}_F(\textbf {V},Y) &\leftrightarrow \forall \textbf {X} N({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N)\to N\leq Y)). \end{align}

In addition to the axioms listed above, we need the induction schemaFootnote ³

\begin{equation*} F^N_{\overline 0}\land \forall N\left (N\geq \overline 0 \land F \to F^N_{N+\overline 1}\right ) \to \forall N (N\geq \overline 0 \to F) \end{equation*}

for all formulas $F$ over $\sigma _2$ . The set of the universal closures of its instances will be denoted by $\mathit{Ind}$ .

The deductive system $\mathit{HT}_{\!\!\#}$ is defined as first-order intuitionistic logic for the signature $\sigma _2$ extended by

• • axiom schemas (6) and (7) for all formulas $F$ , $G$ , $H$ over $\sigma _2$ , and

• • axioms $\mathit{Std}$ , $\mathit{Ind}$ , $D_0$ , and $D_1$ .

This deductive system has property (5):

The proof of this theorem can be found in Appendix B.8 (page 57). As discussed in Section 7, $\mathit{HT}_{\!\!\#}$ is an extension of the system with property (5) introduced by Lifschitz (Reference Lifschitz2022). Furthermore, in Section 4 we show that $\mathit{HT}_{\!\!\#}$ is sufficiently strong for proving the equivalence between $\tau ^*A$ and $\tau ^*B$ for the programs $A$ and $B$ from the introduction.

3.2 Some theorems of $\mathit{HT}_{\!\!\#}$

The characterization of $\mathit{Atleast}$ and $\mathit{Atmost}$ given by the axioms $D_1$ can be simplified, if we replace the variable $Y$ by an integer variable:

Proposition 1. The formulas

(10) \begin{equation} \forall \textbf {V} N({\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {V},N) \leftrightarrow \exists \textbf {X}\,{\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N)) \end{equation}

and

(11) \begin{equation} \forall \textbf {V} N({\mathit{Atmost}}^{\textbf {X};\textbf {V}}_F(\textbf {V},N) \leftrightarrow \neg {\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {V},N+\overline 1)) \end{equation}

are provable in $\mathit{HT}_{\!\!\#}$ .

A few other theorems of $\mathit{HT}_{\!\!\#}$ :

Proposition 2. The formulas

(12) \begin{align} &\forall \textbf {V} N(N\leq \overline 0\to {\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {V},N)), \end{align}

(13) \begin{align} &\forall \textbf {V} ({\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {V},\overline 1) \leftrightarrow \exists \textbf {X}\,F), \end{align}

(14) \begin{align} \forall \textbf {X}(F\to G)\to \forall \textbf {X}\textbf {V} N({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {X},\textbf {V},N) \to {\mathit{Start}}^{\textbf {X};\textbf {V}}_G(\textbf {X},\textbf {V},N)), \end{align}

(15) \begin{align} \forall \textbf {Z}\textbf {V} N ({\mathit{Start}}^{\textbf {X};\textbf {V}}_F(\textbf {Z},\textbf {V},N)\land N\gt \overline 0 \to F^{\textbf {X}}_{\textbf {Z}}) \end{align}

are provable in $\mathit{HT}_{\!\!\#}$ .

An expression of the form ${\mathit{Exactly}}^{\textbf {X};\textbf {V}}_F(\textbf {t},t)$ is shorthand for the conjunction

\begin{equation*}{\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {t},t)\land {\mathit{Atmost}}^{\textbf {X};\textbf {V}}_F(\textbf {t},t)\end{equation*}

( $\textbf {t}$ is a tuple of terms, and $t$ is a term). By (11), ${\mathit{Exactly}}^{\textbf {X};\textbf {V}}_F(\textbf {X},N)$ is equivalent in $\mathit{HT}_{\!\!\#}$ to

\begin{equation*}{\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {X},N)\land \neg {\mathit{Atleast}}^{\textbf {X};\textbf {V}}_F(\textbf {X},N+\overline 1).\end{equation*}

Proposition 3. The formulas

(16) \begin{equation} \forall \textbf {X} Y({\mathit{Exactly}}^{\textbf {X};\textbf {V}}_F(\textbf {X},Y) \to \exists N(Y=N\land N\geq \overline 0)) \end{equation}

and

(17) \begin{equation} \forall \textbf {X}(F\leftrightarrow G)\to \forall \textbf {X} Y({\mathit{Exactly}}^{\textbf {X};\textbf {V}}_F(\textbf {X},Y)\leftrightarrow {\mathit{Exactly}}^{\textbf {X};\textbf {V}}_G(\textbf {X},Y)) \end{equation}

are provable in $\mathit{HT}_{\!\!\#}$ .

The proof of these results can be found in Appendix B.1 (page 28).

4.1 Part 1: Simplification

The translation $\tau ^*$ transforms program $A$ into the conjunction of the formulas

(18) \begin{equation} \forall Z(Z=a\to p(Z)) \end{equation}

and

(19) \begin{align} \!\!\!\forall Y\!Z\big (\exists Z_1(Z_1 = Y\! \land {\mathit{Atleast}}^{X;}_F(Z_1)) \land \exists Z_2(Z_2 = Y\! \land {\mathit{Atmost}}^{X;}_F(Z_2)) \land Z = Y \to q(Z)\big ), \end{align}

where $F$ stands for $\tau ^B(p(a)\land X\neq a)$ . Formula (18) is equivalent to $p(a)$ , and (19) is equivalent to

(20) \begin{equation} \forall Y({\mathit{Atleast}}^{X;}_F(Y)\land {\mathit{Atmost}}^{X;}_F(Y)\to q(Y)). \end{equation}

The antecedent of this implication can be written as ${\mathit{Exactly}}^{X;}_F(Y)$ . By (16), it follows that the variable $Y$ can be replaced by the integer variable $N$ . Furthermore, by (17), formula (20) can be further rewritten as

(21) \begin{equation} \forall N({\mathit{Exactly}}^{X;}_{p(a)\land X\neq a}(N)\to q(N)), \end{equation}

because $F$ is equivalent to $p(a)\land X\neq a$ .

The result of applying $\tau ^*$ to $B$ is the conjunction of (18) and

\begin{gather*} \begin{aligned} \forall YZ\big ( &\exists Z_1(Z_1=Y\land {\mathit{Atleast}}^{X;}_G(Z_1))\,\land \qquad \qquad \\ &\exists Z_2(Z_2=Y\land {\mathit{Atmost}}^{X;}_G(Z_2))\,\land \qquad \;\;\\ &\exists IJ(I=Y\land J=\overline 1\land Z=I+J) \to q(Z)\big ), \end{aligned} \end{gather*}

where $G$ stands for $\tau ^B(p(X))$ . This formula can be equivalently rewritten as:

\begin{equation*}\forall I({\mathit{Exactly}}^{X;}_G(I+\overline 1)\to q(I))\end{equation*}

and further as

(22) \begin{equation} \forall I({\mathit{Exactly}}^{X;}_{p(X)}(I+\overline 1)\to q(I)), \end{equation}

because $G$ is equivalent to $p(X)$ .

Thus the claim that $\tau ^*A$ is equivalent to $\tau ^*B$ will be proved if we prove the formula

\begin{equation*} p(a) \to \forall N({\mathit{Exactly}}^{X;}_{p(X)\land X\neq a}(N)\leftrightarrow {\mathit{Exactly}}^{X;}_{p(X)}(N+\overline 1)). \end{equation*}

It is clearly a consequence of the formula

(23) \begin{equation} p(a) \to \forall N({\mathit{Atleast}}^{X;}_{p(X)\land X\neq a}(N)\leftrightarrow {\mathit{Atleast}}^{X;}_{p(X)}(N+\overline 1)), \end{equation}

which is proven below.

4.2 Part 2: Three lemmas

Three lemmas will be proved in the next section:

(24) \begin{align} &\forall XN(N\gt \overline 0\land X \gt a\land {\mathit{Start}}^{X;}_{p(X)}(X,N) \to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)), \end{align}

(25) \begin{align} &\forall XN(N\gt \overline 0\land X\neq a\land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)\to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)), \end{align}

and

(26) \begin{equation} \begin{array} r \forall XN(N\gt \overline 0\land X\lt a\land p(a)\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)\to {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)). \end{array} \end{equation}

Using these lemmas, we will now prove (23). Assume $p(a)$ ; our goal is to show that

\begin{equation*} {\mathit{Atleast}}^{X;}_{p(X)\land X\neq a}(N)\leftrightarrow {\mathit{Atleast}}^{X;}_{p(X)}(N+\overline 1). \end{equation*}

We consider three cases, according to the axiom

\begin{equation*}\forall N(N\lt \overline 0 \lor N=\overline 0 \lor N\gt \overline 0)\end{equation*}

from $\mathit{Std}$ .

If $N\lt \overline 0$ then both sides of the equivalence are true by (12). If $N=\overline 0$ then the left-hand side is true by (12), and the right-hand side follows from $p(a)$ by (13). Hence, assume that $N\gt \overline 0$ .

Right-to-left: assume ${\mathit{Atleast}}^{X;}_{p(X)}(N+\overline 1)$ . By (10), there exists $X$ such that

(27) \begin{equation} {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1). \end{equation}

Case 1: $X=a$ , so that ${\mathit{Start}}^{X;}_{p(X)}(a,N+\overline 1)$ . By $D_0$ ,

\begin{equation*}p(a)\land \exists U(a\lt U\land {\mathit{Start}}^{X;}_{p(X)}(U,N)).\end{equation*}

Take $U$ such that $a\lt U$ and ${\mathit{Start}}^{X;}_{p(X)}(U,N)$ . By (24), it follows that

\begin{equation*}{\mathit{Start}}^{X;}_{p(X)\land X\neq a}(U,N).\end{equation*}

Then ${\mathit{Atleast}}^{X;}_{p(X)\land X\neq a}(N)$ by (10).

Case 2: $X\neq a$ . By (27) and (25),

\begin{equation*}{\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N).\end{equation*}

By (10), it follows that ${\mathit{Atleast}}^{X;}_{p(X)\land X\neq a}(N)$ .

Left-to-right: assume ${\mathit{Atleast}}^{X;}_{p(X)\land X\neq a}(N)$ . Then, for some $X$ ,

(28) \begin{equation} {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N) \end{equation}

by (10), and consequently ${\mathit{Start}}^{X;}_{p(X)}(X,N)$ by (14).

Case 1: $X\gt a$ . Then

\begin{equation*}p(a)\land \exists U(a\lt U\land {\mathit{Start}}^{X;}_{p(X)}(U,N))\end{equation*}

(take $U$ to be $X$ ). By $D_0$ , we can conclude that ${\mathit{Start}}^{X;}_{p(X)}(a,N+\overline 1)$ . Then ${\mathit{Atleast}}^{X;}_{p(X)}(N+\overline 1)$ follows by (10).

Case 2: $X\leq a$ . From (28) and (15), $X\neq a$ , so that $X\lt a$ . From (28) and (26), ${\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)$ ; ${\mathit{Atleast}}^{X;}_{p(X)}(N+\overline 1)$ follows by (10).

4.3 Part 3: Proofs of the lemmas

Proofs of all three lemmas use induction in the form

(29) \begin{equation} F^N_{\,\overline 1}\land \forall N\left (N\geq \overline 1 \land F \to F^N_{N+\overline 1}\right ) \to \forall N (N\geq \overline 1 \to F), \end{equation}

which follows from $\mathit{Ind}$ and $\mathit{Std}$ .

Proof of (24). We need to show that for all positive $N$ ,

(30) \begin{equation} \forall X(X\gt a \land {\mathit{Start}}^{X;}_{p(X)}(X,N)\to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)). \end{equation}

If $N$ is $\overline 1$ then (30) is equivalent to

\begin{equation*}\forall X(X\gt a \land p(X) \to p(X)\land X\neq a)\end{equation*}

by $D_0$ ; this formula follows from $\mathit{Std}$ . Assume (30) for a positive $N$ ; we need to prove

\begin{equation*}\forall X(X\gt a \land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)\to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N+\overline 1)).\end{equation*}

Assume $X\gt a \land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)$ . From the second conjunctive term,

\begin{equation*} p(X)\land \exists U(X\lt U\land {\mathit{Start}}^{X;}_{p(X)}(U,N)) \end{equation*}

by $D_0$ . Take $U$ such that $X\lt U$ and ${\mathit{Start}}^{X;}_{p(X)}(U,N)$ . Then $U\gt a$ , so that by the induction hypothesis, ${\mathit{Start}}^{X;}_{p(X)\land X\neq a}(U,N)$ . Since $p(X)$ , $X\neq a$ , and $X\lt U$ ,

\begin{equation*}{\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N+\overline 1))\end{equation*}

follows by $D_0$ .

Proof of (25). We need to show that for all positive $N$ ,

(31) \begin{equation} \forall X(X\neq a\land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)\to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)). \end{equation}

To prove this formula for $N$ equal to $\overline 1$ , assume that $X\neq a\land {\mathit{Start}}^{X;}_{p(X)}(X,\overline 2)$ . By (15), the second conjunctive term implies $p(X)$ ; ${\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,\overline 1)$ follows by $D_0$ . Now assume (31) for a positive $N$ ; we need to prove

(32) \begin{equation} \forall X(X\neq a\land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 2)\to {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N+\overline 1)). \end{equation}

Assume $X\neq a\land {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 2)$ . From the second conjunctive term we conclude by $D_0$ that $p(X)$ and, for some $U$ ,

(33) \begin{equation} U\gt X\land {\mathit{Start}}^{X;}_{p(X)}(U,N+\overline 1). \end{equation}

We proceed by cases.

Case 1: $U=a$ , so that ${\mathit{Start}}^{X;}_{p(X)}(a,N+\overline 1)$ . By $D_0$ , it follows that for some $V$ , $V\gt a\land {\mathit{Start}}^{X;}_{p(X)}(V,N)$ . Then, by (24), ${\mathit{Start}}^{X;}_{p(X)\land X\neq a}(V,N)$ . On the other hand, $p(X)\land X\neq a$ and $V\gt a=U\gt X$ ; the consequent of (32) follows by $D_0$ .

Case 2: $U\neq a$ . By the induction hypothesis, from the second conjunctive term of (33) we conclude that ${\mathit{Start}}^{X;}_{p(X)\land X\neq a}(U,N)$ . Since $U\gt X$ and $p(X)\land X\neq a$ , the consequent of (32) follows by $D_0$ .

Proof of (26). We need to show that for all positive $N$ ,

(34) \begin{equation} \forall X(X\lt a \land p(a)\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N)\to {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 1)). \end{equation}

To prove this formula for $N$ equal to $\overline 1$ , assume that

\begin{equation*}X\lt a\land p(a)\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,\overline 1).\end{equation*}

By $D_0$ , the second conjunctive term implies ${\mathit{Start}}^{X;}_{p(X)}(a,\overline 1)$ , and the third term implies $p(X)$ . Hence,

\begin{equation*}p(X)\land \exists U(X\lt U \land {\mathit{Start}}^{X;}_{p(X)}(U,\overline 1))\end{equation*}

(take $U$ to be $a$ ). By $D_0$ , it follows that ${\mathit{Start}}^{X;}_{p(X)}(X,\overline 2)$ . Now assume (34) for a positive $N$ ; we need to prove

\begin{equation*}\forall X(X\lt a\land p(a)\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N+\overline 1)\to {\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 2)).\end{equation*}

Assume $X\lt a\land p(a)\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(X,N+\overline 1)$ . From the last conjunctive term, we conclude by $D_0$ that $p(X)$ , and there exists $U$ such that

(35) \begin{equation} X\lt U\land {\mathit{Start}}^{X;}_{p(X)\land X\neq a}(U,N). \end{equation}

From the second conjunctive term of (35), by (15), $p(U)$ and $U\neq a$ . We proceed by cases.

Case 1: $U\lt a$ . By the induction hypothesis, ${\mathit{Start}}^{X;}_{p(X)}(U,N+\overline 1)$ . Since $p(X)$ and $X\lt U$ , we can conclude by $D_0$ that ${\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 2)$ .

Case 2: $U\gt a$ . By (14), the second conjunctive term of (35) implies ${\mathit{Start}}^{X;}_{p(X)}(U,N)$ . Since $p(a)$ and $a\lt U$ , ${\mathit{Start}}^{X;}_{p(X)}(a,N+\overline 1)$ follows by $D_0$ . Then, since $p(X)$ and $X\lt a$ , ${\mathit{Start}}^{X;}_{p(X)}(X,N+\overline 2)$ follows in a similar way.