hcju/proofwikips huggingface dataset:
Definition:Trivial Group Action
Let $G$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be the group action defined as: :$\forall \tuple {g, s} \in G \times S: g * s = s$ Then $*$ is the '''trivial group action''' of $G$ on $S$.
Definition:Group Action Axioms
{{begin-axiom}} {{axiom | n = \text {GA} 1         | q = \forall g, h \in G, x \in X         | m = g * \paren {h * x} = \paren {g \circ h} * x }} {{axiom | n = \text {GA} 2         | q = \forall x \in X         | m = e * x = x }} {{end-axiom}}
Definition:Algebra of Sets/Definition 1
Let $X$ be a set. Let $\powerset X$ be the power set of $X$. Let $\RR \subseteq \powerset X$ be a set of subsets of $X$. Then $\RR$ is an '''algebra of sets over $X$'''  the following conditions hold: {{begin-axiom}} {{axiom | n = \text {AS} 1         | lc= Unit:         | m = X \in \RR }} {{axiom | n = \text {AS} 2         | lc= Closure under Union:         | q = \forall A, B \in \RR         | m = A \cup B \in \RR }} {{axiom | n = \text {AS} 3         | lc= Closure under Complement Relative to $X$:         | q = \forall A \in \RR         | m = \relcomp X A \in \RR }} {{end-axiom}}
Definition:Algebra of Sets/Definition 2
An '''algebra of sets''' is a ring of sets with a unit.
Definition:Logical Argument/Finitary
A '''finitary argument''' is a logical argument which starts with a finite number of axioms, and can be translated into a finite number of statements.
Definition:Deductive Argument
A '''deductive argument''' is one whose conclusion follows unshakeable from the premises.
Definition:Argument Form
An '''argument form''' is a collation of symbols which contains statement variables such that: : when statements are used to replace statement variables (the same statement replacing the same statement variable throughout), the result is a logical argument. === Specific Form === {{:Definition:Argument Form/Specific Form}}
Definition:Argument Form/Specific Form
The '''specific form''' of a given logical argument is that argument form from which the logical argument results from replacing each distinct statement variable by a different simple statement.
Definition:Statement Form/Specific Form
The '''specific form''' of a given statement is that propositional formula from which the statement form results from replacing each distinct statement variable by a different simple statement.
Definition:Elementary Valid Argument
An '''elementary valid argument''' is a logical argument which is a substitution instance of an elementary valid argument form. The argument forms which are considered to be elementary valid argument forms will depend on the specific approach to propositional logic which is under discussion.
Definition:Operation/Arity/Ternary
A '''ternary operation''' (or '''three-place operation''') is an operation which takes three operands. That is, its arity is $3$.
Definition:Operation/Arity/Multiary
Certain types of operation have a variable number of operands. An operation which does not have a fixed arity is called '''multiary'''.
Definition:Operation/Arity/Finitary
A '''finitary operation''' is an operation which takes a finite number of operands.
Definition:Operation/Arity/Zero
It is possible to conceive of an operator which takes ''no'' operands. A constant can be considered as an operation which takes no operands. That is, it has an arity of zero.
Definition:Variable/Value
A variable $x$ may be (temporarily, conceptually) identified with a particular object. If so, then that object is called the '''value''' of $x$.
Definition:Propositional Function/Satisfaction
Let $\map P {x_1, x_2, \ldots, x_n}$ be an $n$-ary propositional function. If $a_1, a_2, \ldots, a_n$ have values which make $\map P {x_1, x_2, \ldots, x_n}$ true, then the ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ '''satisfies''' $\map P {x_1, x_2, \ldots, x_n}$.
Definition:Integer Division
Let $a, b \in \Z$ be integers such that $b \ne 0$.. From the Division Theorem: :$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ where $q$ is the quotient and $r$ is the remainder. The process of finding $q$ and $r$ is known as '''(integer) division'''.
Definition:Diagram (Graphical Technique)
A '''diagram''' is a graphical technique for illustrating a concept in picture form. It is generally considered that its use should be limited to that of an aid to understanding, and should not be used in order to prove something.
Definition:Quantified Statement
A '''quantified statement''' is a simple statement in predicate logic whose subject is qualified by either the universal quantifier or the existential quantifier. That is, it is either a universal statement or an existential statement.
Definition:Square of Opposition
The '''square of opposition''' is a diagram whose purpose is to illustrate the relations between the various categories of quantified statement. ::$\begin{xy} <-10em,0em>*+{\forall x: \map \Phi x} = "A", <10em,0em>*+{\forall x: \neg \map \Phi x} = "E", <-10em,-20em>*+{\exists x: \map \Phi x} = "I", <10em,-20em>*+{\exists x: \neg \map \Phi x} = "O", "A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$ This therefore illustrates the relations: :''All $x$ have $\Phi$'' is contrary to ''All $x$ do not have $\Phi$'' :''All $x$ have $\Phi$'' is contradictory to ''Some $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is contradictory to ''All $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is subimplicant to ''All $x$ have $\Phi$'' :''Some $x$ do not have $\Phi$'' is subimplicant to ''All $x$ do not have $\Phi$'' :''Some $x$ have $\Phi$'' is subcontrary to ''Some $x$ do not have $\Phi$'' where $x$ is an object and $\Phi$ is a property. === Vacuous Universe === ;Beware: Note that if the universe of discourse is empty, then the square of opposition no longer holds. Although $\forall x: \map \Phi x$ is vacuously true for such an empty universe, $\exists x: \map \Phi x$ is not. Thus $\exists x: \map \Phi x$ is no longer subimplicant to $\forall x: \map \Phi x$. Similarly, as $\exists x: \neg \map \Phi x$ is also false, it follows that $\exists x: \map \Phi x$ and $\exists x: \neg \map \Phi x$ are no longer subcontrary.
Definition:Predicate/Is of Identity
Consider the sentence: :'''Socrates is the philosopher who taught Plato.''' This could be reworded as: :'''The object named Socrates has the property of being the philosopher who taught Plato.''' However, the meaning that is ''really'' being conveyed here is that of: :'''The object named Socrates ''is the same object as'' the object which is the philosopher who taught Plato.''' In this context, '''is''' is not being used in the same way as the ''is'' of predication. When being used to indicate that one object is the same object as another object, '''is''' is called '''the ''is'' of identity'''. In this context, ''is'' means the same as equals.
Definition:Predicate/Is of Predication
Consider the statement: :'''Socrates is a man.''' This means: :'''The object named Socrates has the property of being a man.''' Thus we see that '''is''' here means '''has the property of being'''. In this context, '''is''' here is called '''the ''is'' of predication'''.
Definition:Universal Affirmative
:''Every $S$ is $P$'' where $S$ and $P$ are predicates.
Definition:Universal Affirmative/Set Theory
The universal affirmative $\forall x: \map S x \implies \map P x$ can be expressed in set language as: :$\set {x: \map S x} \subseteq \set {x: \map P x}$ or, more compactly: :$S \subseteq P$
Definition:Universal Negative
:''No $S$ is $P$'' where $S$ and $P$ are predicates.
Definition:Universal Negative/Set Theory
The universal negative $\forall x: \map S x \implies \neg \map P x$ can be expressed in set language as: :$\set {x: \map S x} \implies \set {x: \map P x} = \O$ or, more compactly: :$S \subseteq \map \complement P$
Definition:Particular Affirmative
:''Some $S$ is $P$'' where $S$ and $P$ are predicates.
Definition:Particular Affirmative/Set Theory
The particular affirmative $\exists x: \map S x \land \map P x$ can be expressed in set language as: :$\set {x: \map S x} \cap \set {x: \map P x} \ne \O$ or, more compactly: :$S \cap P \ne \O$
Definition:Particular Negative/Set Theory
The particular negative $\exists x: \map S x \land \neg \map P x$ can be expressed in set language as: :$\set {x: \map S x} \cap \set {x: \neg \map P x} \ne \O$ or, more compactly: :$S \cap \map \complement P \ne \O$
Definition:Particular Negative
:''Some $S$ is not $P$'' where $S$ and $P$ are predicates.
Definition:Categorical Syllogism
A '''categorical syllogism''' is a logical argument which is structured as follows: $(1): \quad$ It has exactly two premises and one conclusion. : The first premise is usually referred to as the major premise. : The second premise is usually referred to as the minor premise. $(2): \quad$ It concerns exactly three terms, which are usually denoted: {{begin-axiom}} {{axiom | q = P         | t = the primary term }} {{axiom | q = M         | t = the middle term }} {{axiom | q = S         | t = the secondary term }} {{end-axiom}} $(3): \quad$ Each of the premises and conclusion is a categorical statement.
Definition:Categorical Syllogism/Secondary Term
The '''secondary term''' of a categorical syllogism is the term that appears as the first predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the minor premise. It is usually denoted by $S$.
Definition:Categorical Syllogism/Middle Term
The '''middle term''' of a categorical syllogism is the term that does not appear in the conclusion of the syllogism. It appears once in each of the premises of the syllogism. It is usually denoted by $M$.
Definition:Categorical Syllogism/Primary Term
The '''primary term''' of a categorical syllogism is the term that appears as the second predicate of the conclusion of the syllogism. It also appears once in one of the premises of the syllogism, traditionally the major premise. It is usually denoted by $P$.
Definition:Categorical Syllogism/Premises
There are two premises in a categorical syllogism: === Major Premise === {{:Definition:Categorical Syllogism/Premises/Major Premise}} === Minor Premise === {{:Definition:Categorical Syllogism/Premises/Minor Premise}}
Definition:Categorical Syllogism/Terms
There are three terms in a categorical syllogism: === Primary Term of Syllogism === {{:Definition:Categorical Syllogism/Primary Term}} === Middle Term of Syllogism === {{:Definition:Categorical Syllogism/Middle Term}} === Secondary Term of Syllogism === {{:Definition:Categorical Syllogism/Secondary Term}}
Definition:Categorical Syllogism/Premises/Major Premise
The '''major premise''' of a categorical syllogism is conventionally stated first. It is a categorical statement which expresses the logical relationship between the primary term and the middle term of the syllogism.
Definition:Categorical Syllogism/Premises/Minor Premise
The '''minor premise''' of a categorical syllogism is conventionally stated second. It is a categorical statement which expresses the logical relationship between the secondary term and the middle term of the syllogism.
Definition:Categorical Syllogism/Conclusion
The '''conclusion''' of a categorical syllogism is stated last, conventionally prefaced with the word ''Therefore''. It is a categorical statement which expresses the logical relationship between the primary term and the secondary term of the syllogism. Furthermore, this categorical statement is ''specifically'' of the form in which the secondary term occurs ''first'' and the primary term occurs ''second''.
Definition:Figure of Categorical Syllogism
There are four patterns of categorical syllogism depending on the order of the terms in the major premise and minor premise. Let $\mathbf \Phi_1, \mathbf \Phi_2, \mathbf \Phi_3$ each be one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$. Let $P$ denote the primary term, $S$ denote the secondary term and $M$ denote the middle term. The four possible '''figures''' are as follows: === $\text I$ === {{:Definition:Figure of Categorical Syllogism/I}} === $\text {II}$ === {{:Definition:Figure of Categorical Syllogism/II}} === $\text {III}$ === {{:Definition:Figure of Categorical Syllogism/III}} === $\text {IV}$ === {{:Definition:Figure of Categorical Syllogism/IV}}
Definition:Figure of Categorical Syllogism/I
{| |- | &nbsp; Major Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_1} {M, P}$ |- | &nbsp; Minor Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_2} {S, M}$ |- | &nbsp; Conclusion: &nbsp; | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}
Definition:Figure of Categorical Syllogism/II
{| |- | &nbsp; Major Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_1} {P, M}$ |- | &nbsp; Minor Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_2 } {S, M}$ |- | &nbsp; Conclusion: &nbsp; | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}
Definition:Figure of Categorical Syllogism/III
{| |- | &nbsp; Major Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_1} {M, P}$ |- | &nbsp; Minor Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_2} {M, S}$ |- | &nbsp; Conclusion: &nbsp; | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}
Definition:Figure of Categorical Syllogism/IV
{| |- | &nbsp; Major Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_1} {P, M}$ |- | &nbsp; Minor Premise: &nbsp; | align="right" | $\map {\mathbf \Phi_2} {M, S}$ |- | &nbsp; Conclusion: &nbsp; | style="border-style: solid; border-width: 1px 0 0 0"  align="right" | $\map {\mathbf \Phi_3} {S, P}$ |}
Definition:Standard Instance of Categorical Syllogism
For each categorical syllogism, there are four figures: :$\begin{array}{r|rl} \text I & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {II} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$ :$\begin{array}{r|rl} \text {III} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {IV} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$ where $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ each denote one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$. A '''standard instance of a categorical syllogism''' is obtained by substituting $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$ for each of $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ in one of the above figures. Not all of these '''standard instances''' are valid.
Definition:Length of Curve
The '''length''' of a curve is defined as the limit of the length of a polygonal line inscribed within the curve as the maximum length of the chords which form that polygonal line tends to zero.
Definition:Excluded Point Topology/Finite
Let $S$ be finite. Then $\tau_{\bar p}$ is a '''finite excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is a '''finite excluded point space'''.
Definition:Excluded Point Topology/Countable
Let $S$ be countably infinite. Then $\tau_{\bar p}$ is a '''countable excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is a '''countable excluded point space'''.
Definition:Excluded Point Topology/Uncountable
Let $S$ be uncountable. Then $\tau_{\bar p}$ is an '''uncountable excluded point topology''', and $\struct {S, \tau_{\bar p} }$ is an '''uncountable excluded point space'''.
Definition:Excluded Point Topology/Infinite
Let $S$ be infinite. Then $\tau_{\bar p}$ is an '''infinite excluded point topology''', and $\left({S, \tau_{\bar p}}\right)$ is a '''infinite excluded point space'''.
Definition:Countable Complement Topology/Countable
It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space. This is why the countable complement topology is usually understood to apply to uncountable sets only.
Definition:Particular Point Topology/Finite
Let $S$ be finite. Then $\tau_p$ is a '''finite particular point topology''', and $\struct {S, \tau_p}$ is a '''finite particular point space'''.
Definition:Particular Point Topology/Infinite
Let $S$ be infinite. Then $\tau_p$ is an '''infinite particular point topology''', and $\left({S, \tau_p}\right)$ is an '''infinite particular point space'''.
Definition:Particular Point Topology/Countable
Let $S$ be countably infinite. Then $\tau_p$ is a '''countable particular point topology''', and $\struct {S, \tau_p}$ is a '''countable particular point space'''.
Definition:Particular Point Topology/Uncountable
Let $S$ be uncountable. Then $\tau_p$ is an '''uncountable particular point topology''', and $\struct {S, \tau_p}$ is an '''uncountable particular point space'''.
Definition:Finite Complement Topology/Finite
It is possible to define the finite complement topology on a finite set $S$, but as every subset of a finite set has a finite complement, it is clear that this is trivially equal to the discrete space. This is why the finite complement topology is usually understood to apply to infinite sets only.
Definition:Finite Complement Topology/Countable
Let $S$ be countably infinite. Then $\tau$ is a '''finite complement topology on a countable space''', and $\struct {S, \tau}$ is a '''countable finite complement space'''.
Definition:Finite Complement Topology/Uncountable
Let $S$ be uncountable. Then $\tau$ is a '''finite complement topology on an uncountable space''', and $\struct {S, \tau}$ is a '''uncountable finite complement space'''.
Definition:Fort Space/Countable
Let $S$ be countably infinite. Then $\tau_p$ is a '''countable Fort topology''', and $\struct {S, \tau_p}$ is a '''countable Fort space'''.
Definition:Fort Space/Uncountable
Let $S$ be uncountable. Then $\tau_p$ is an '''uncountable Fort topology''', and $\struct {S, \tau_p}$ is an '''uncountable Fort space'''.
Definition:Complete Metric Space/Definition 1
A metric space $M = \struct {A, d}$ is '''complete'''  every Cauchy sequence is convergent.
Definition:Complete Metric Space/Definition 2
A metric space $M = \struct {A, d}$ is '''complete'''  the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.
Definition:Affirmative Categorical Statement
An '''affirmative categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Particular Affirmative === {{:Definition:Particular Affirmative}}
Definition:Negative Categorical Statement
A '''negative categorical statement''' is either one of: === Universal Negative === {{:Definition:Universal Negative}} === Particular Negative === {{:Definition:Particular Negative}}
Definition:Reduction to First Figure
'''Reduction to the first figure''' is a method for determining the validity of a categorical syllogism, as follows: :$(1): \quad$ Certain "self-evident" patterns are identified in the first figure of the categorical syllogism. :$(2): \quad$ Using various rules of categorical statements, the valid patterns of other figures of the categorical syllogism are deduced. There are two forms this procedure takes: === Direct Reduction === {{:Definition:Reduction to First Figure/Direct}} === Indirect Reduction === {{:Definition:Reduction to First Figure/Indirect}}
Definition:Reduction to First Figure/Indirect
'''Indirect reduction to the first figure''' is a form of the reduction to the first figure method for determining the validity of a categorical syllogism. This form of the method works backward by supposing that a pattern is not valid, and then deriving a contradiction against a known valid pattern in the first figure. Hence by Reductio ad Absurdum the pattern is deduced to be valid.
Definition:Square of Opposition/Categorical Statements/Vacuous Terms
The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous. However, from the point of view of the full predicate logic, this assumption may not be valid. Note that if $S$ is empty, then the square of opposition no longer holds. Although ''All $S$ are $P$'' is vacuously true for such an empty universe, ''Some $S$ are $P$'' is not. Thus ''Some $S$ are $P$'' is no longer subimplicant to ''All $S$ are $P$''. Similarly, as ''Some $S$ are not $P$'' is also false, it follows that ''All $S$ are $P$'' and ''Some $S$ are not $P$'' are no longer subcontrary.
Definition:Categorical Syllogism/Shorthand
In order to specify the pattern of a categorical syllogism completely, it is necessary and sufficient to specify: :$(1) \quad$ The figure ($\text I$ to $\text {IV}$) of the syllogism and :$(2) \quad$ The types of the three categorical statements that compose the syllogism. Hence, for example, the following categorical syllogism, which is of the first figure: :$\begin{array}{r|rl} \text I & & \\ \hline \\ \text{Major Premise}: & \mathbf E & \left({M, P}\right) \\ \text{Minor Premise}: & \mathbf I & \left({S, M}\right) \\ \hline \\ \text{Conclusion}: & \mathbf O & \left({S, P}\right) \\ \end{array}$ is specified completely by: :$\text I: EIO$
Definition:Categorical Statement/Abbreviation
A '''categorical statement''' connecting $S$ and $P$ can be abbreviated as: :$\mathbf{\Phi} \left({S, P}\right)$ where $\Phi$ is one of either $\mathbf{A}$, $\mathbf{E}$, $\mathbf{I}$ or $\mathbf{O}$, signifying Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative respectively. Thus: :$\mathbf{A} \left({S, P}\right)$ denotes ''All $S$ are $P$'' :$\mathbf{E} \left({S, P}\right)$ denotes ''No $S$ are $P$'' :$\mathbf{I} \left({S, P}\right)$ denotes ''Some $S$ are $P$'' :$\mathbf{O} \left({S, P}\right)$ denotes ''Some $S$ are not $P$''.
Definition:Categorical Statement/Subject
The symbol $S$ can be referred to as the '''subject''' of $\map {\mathbf {\Phi} } {S, P}$.
Definition:Categorical Statement/Predicate
The symbol $P$ can be referred to as the '''predicate''' of $\map {\mathbf {\Phi} } {S, P}$.
Definition:Universal Categorical Statement
A '''universal categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Universal Negative === {{:Definition:Universal Negative}}
Definition:Particular Categorical Statement
A '''particular categorical statement''' is either one of: === Particular Affirmative === {{:Definition:Particular Affirmative}} === Particular Negative === {{:Definition:Particular Negative}}
Definition:Distributed Term of Categorical Syllogism/Subject
Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a universal categorical statement. Then $S$ is described as being '''distributed'''.
Definition:Distributed Term of Categorical Syllogism
A term in a syllogism is '''distributed''' if the categorical statement in which it occurs refers to the whole of the class designated by the term. === Distributed Subject === {{:Definition:Distributed Term of Categorical Syllogism/Subject}} === Distributed Predicate === {{:Definition:Distributed Term of Categorical Syllogism/Predicate}}