From 6187ecc4a7b3df4e5766d7b19bf7b8a51454d11a Mon Sep 17 00:00:00 2001 From: shilangyu Date: Wed, 28 Feb 2024 10:51:24 +0100 Subject: [PATCH 1/6] Swap 'or' and 'and' in ligature column typo --- refman/quickguide.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/refman/quickguide.tex b/refman/quickguide.tex index b3c683d5..d6b4fc57 100644 --- a/refman/quickguide.tex +++ b/refman/quickguide.tex @@ -64,8 +64,8 @@ \subsection*{A Note on Special Characters} \begin{tabular}{c|c|c} Rendering & Input & Name \\ \hline \lstinline| === | & === & equality \\ \hline - \lstinline| \/ | & \textbackslash / & and \\ \hline - \lstinline| /\ | & /\textbackslash & or \\ \hline + \lstinline| \/ | & \textbackslash / & or \\ \hline + \lstinline| /\ | & /\textbackslash & and \\ \hline \lstinline| ==> | & ==> & implies \\ \hline \lstinline+ |- + & |- & vdash \\ \hline \lstinline| ∀ | & U+2200 & forall \\ \hline From f9af80f29c55ed9aa6bfad31e6ccb4edd5dbd815 Mon Sep 17 00:00:00 2001 From: shilangyu Date: Wed, 28 Feb 2024 10:54:22 +0100 Subject: [PATCH 2/6] Add missing tautology tactic reference --- refman/quickguide.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/refman/quickguide.tex b/refman/quickguide.tex index d6b4fc57..30eef1cf 100644 --- a/refman/quickguide.tex +++ b/refman/quickguide.tex @@ -225,7 +225,7 @@ \subsubsection*{Restate} \end{lstlisting} tries to justify \lstinline|statement| by showing it is equivalent to the previously proven \lstinline|premise|. -\subsubsection*{Tautology} +\subsubsection*{Tautology}\label{tact:Tautology} \lstinline|Tautology| is a propositional solver based upon restate, but complete. It is able to prove every formula inference that holds in classical propositional logic. However, in the worst case its complexity can be exponential in the size of the formula. Usage: \begin{lstlisting}[language=lisa] From b050e6ed2981ad5c2718245036437eec9eb02f52 Mon Sep 17 00:00:00 2001 From: shilangyu Date: Wed, 28 Feb 2024 11:18:12 +0100 Subject: [PATCH 3/6] Fix other small typos in manual --- refman/kernel.tex | 4 ++-- refman/prooflib.tex | 6 +++--- refman/theory.tex | 2 +- 3 files changed, 6 insertions(+), 6 deletions(-) diff --git a/refman/kernel.tex b/refman/kernel.tex index 340d278c..5be2eaa7 100644 --- a/refman/kernel.tex +++ b/refman/kernel.tex @@ -614,7 +614,7 @@ \subsection{Proofs} \end{figure} In the Lisa kernel, proof steps are organised linearly, in a list, to form actual proofs. Each proof step refers to its premises using numbers, which indicate the place of the premise in the proof. -a proof step can also be refered to by multiple subsequent proof steps, so that proofs are actually directed acyclic graphs (DAG) rather than trees. For the proof to be the linearization of a rooted DAG, the proof steps must only refer to numbers smaller than their own in the proof. Indeed, using topological sorting, it is always possible to order the nodes of a directed acyclic graph such that for any node, its predecessors appear earlier in the list. +a proof step can also be referred to by multiple subsequent proof steps, so that proofs are actually directed acyclic graphs (DAG) rather than trees. For the proof to be the linearization of a rooted DAG, the proof steps must only refer to numbers smaller than their own in the proof. Indeed, using topological sorting, it is always possible to order the nodes of a directed acyclic graph such that for any node, its predecessors appear earlier in the list. \autoref{fig:exampleProofLinear} shows the proof of Pierce's Law as linearized in Lisa's kernel. % \begin{figure}[ht] @@ -1106,7 +1106,7 @@ \subsection{How to write helpers} given Conversion[TermLabel, Term] = (t: Term) => Term(t, Seq()) \end{lstlisting} % -Now, every time a \lstinline|TermLabel| is writen in a place where a \lstinline|Term| is expected, it will be converted implicitly. +Now, every time a \lstinline|TermLabel| is written in a place where a \lstinline|Term| is expected, it will be converted implicitly. To learn more about Scala 3 and its capabilities, see its documentation at {\footnotesize\url{https://docs.scala-lang.org/scala3/book/introduction.html}}. diff --git a/refman/prooflib.tex b/refman/prooflib.tex index 08853d82..c6c20dad 100644 --- a/refman/prooflib.tex +++ b/refman/prooflib.tex @@ -53,7 +53,7 @@ \subsection{Instantiations} \subsection{Local Definitions} \label{sec:localDefinitions} -The following line of reasonning is standard in mathematical proofs. Suppose we have already proven the following fact: +The following line of reasoning is standard in mathematical proofs. Suppose we have already proven the following fact: $$\exists x. P(x)$$ And want to prove the property $\phi$. A proof of $\phi$ using the previous theorem would naturally be obtained the following way: @@ -104,7 +104,7 @@ \subsection{Instantiations with ``of''} \end{lstlisting} \lstinline|x := d| is called a substitution pair, and is equivalent to the tuple \lstinline|(x, d)|. Arbitrarily many substitution pairs can be given as argument to \lstinline|of|, and the instantiations are done simultaneously. \lstinline|ax of (x := c)| is called an \lstinline|InstantiatedFact|, whose statement is $P(c)$, and which can be used exactly like theorems, axioms and intermediate steps in the proof. Internally, Lisa produces a proof step corresponding to the instantiation using the \texttt{Inst} rule. -The \lstinline|of| keyword can also instantiate universaly quantified formulas of a fact, when it contains a single formula. For example, the following code is valid: +The \lstinline|of| keyword can also instantiate universally quantified formulas of a fact, when it contains a single formula. For example, the following code is valid: \begin{lstlisting}[language=lisa, frame=single] val ax = Axiom(∀(x, P(x))) val thm = Theorem(P(c) /\ P(d)) { @@ -113,7 +113,7 @@ \subsection{Instantiations with ``of''} \end{lstlisting} Here, \lstinline|ax of c| is a fact whose proven statement is again $P(c)$. It is possible to instantiate multiple $\forall$ quantifiers at once. For example if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, d)| is a fact whose proven statement is $P(c, d)$. It is also possible to combine instantiation of free symbols and quantified variables. For example, if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, y, P := ===)| is a fact whose proven statement is $(c = y)$. -Formally, the \lstinline|of| keyword takes as argument arbitrarily many terms and substitution pairs. If there is at least on term given as argument, the base fact must have a single universaly quantified formula on the right (an arbitrarily many formulas on the left). The number of given terms must be at most the number of leading universal quantifiers. Moreover, a substitution cannot instantiate any locked symbol (i.e. a symbol part of an assumption or definition). The ordering of substitution pairs does not matter, but the ordering of terms does. The resulting fact is obtained by firstreplacing the free symbols in the formula by the given substitution pairs, and then instantiating the quantified variables in the formula by the given terms +Formally, the \lstinline|of| keyword takes as argument arbitrarily many terms and substitution pairs. If there is at least on term given as argument, the base fact must have a single universally quantified formula on the right (an arbitrarily many formulas on the left). The number of given terms must be at most the number of leading universal quantifiers. Moreover, a substitution cannot instantiate any locked symbol (i.e. a symbol part of an assumption or definition). The ordering of substitution pairs does not matter, but the ordering of terms does. The resulting fact is obtained by first replacing the free symbols in the formula by the given substitution pairs, and then instantiating the quantified variables in the formula by the given terms In general, for the following proof \begin{lstlisting}[language=lisa, frame=single] diff --git a/refman/theory.tex b/refman/theory.tex index 7eaa52d7..76f5f340 100644 --- a/refman/theory.tex +++ b/refman/theory.tex @@ -15,7 +15,7 @@ \chapter{Library Development: Set Theory} In a more typical mathematical introduction to Set Theory, ZF would naturally only contain the set membership symbol $\in$. Axioms defining the other symbols would then only express the existence of functions or predicates with those properties, from which we could get the same symbols using extensions by definitions. In a very traditional sense, an axiomatization is any possibly infinite semi-recursive set of axioms. Hence, in its full generality, Axioms should be any function producing possibly infinitely many formulas. -This is however not a convenient definition. In practice, all infinite axiomatizations are schematic, meaning that they are expressable using schematic variables. Axioms \ref{axz:comprehension} (comprehension schema) and \ref{axzf:replacement} (replacement schema) are such examples of axiom schema, and motivates the use of schematic variables in Lisa. +This is however not a convenient definition. In practice, all infinite axiomatizations are schematic, meaning that they are expressible using schematic variables. Axioms \ref{axz:comprehension} (comprehension schema) and \ref{axzf:replacement} (replacement schema) are such examples of axiom schema, and motivates the use of schematic variables in Lisa. From 7fc1e3d40934e31203ed021c2601eb912de0e484 Mon Sep 17 00:00:00 2001 From: shilangyu Date: Sun, 3 Mar 2024 21:58:31 +0100 Subject: [PATCH 4/6] Fix alignment in kernel chapter --- refman/kernel.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/refman/kernel.tex b/refman/kernel.tex index 5be2eaa7..f93de1cb 100644 --- a/refman/kernel.tex +++ b/refman/kernel.tex @@ -682,7 +682,7 @@ \subsection{Proofs} \centering \begin{lstlisting}[language=scala, showspaces=false] val PierceLawProof = SCProof(IndexedSeq( - Hypothesis(φ |- φ, φ), + Hypothesis( φ |- φ, φ), Weakening( φ |- ( φ, ψ ), 0), RightImplies(() |- ( φ, φ ==> ψ ), 1, φ, ψ) LeftImplies(( φ ==> ψ ) ==> φ |- φ, 2, 0, φ ==> ψ, φ), @@ -811,7 +811,7 @@ \subsection{Definitions} Theorem( name: String, proposition: Sequent - ) + ) \end{lstlisting} \\ %\hline @@ -821,7 +821,7 @@ \subsection{Definitions} Axiom( name: String, ax: Formula - ) + ) \end{lstlisting} \\ %\hline @@ -831,7 +831,7 @@ \subsection{Definitions} PredicateDefinition( label: ConstantAtomicLabel, expression: LambdaTermFormula - ) + ) \end{lstlisting} \\ %\hline @@ -842,7 +842,7 @@ \subsection{Definitions} label: ConstantFunctionLabel, out: VariableLabel, expression: LambdaTermFormula - ) + ) \end{lstlisting} \\ %\hline @@ -867,7 +867,7 @@ \subsection{Definitions} statement: Sequent, proof: SCProof, justs: Seq[Justification] - ) + ) \end{lstlisting} \\ %\hline @@ -876,7 +876,7 @@ \subsection{Definitions} addAxiom( name: String, f: Formula - ) + ) \end{lstlisting} \\ %\hline @@ -885,7 +885,7 @@ \subsection{Definitions} makePredicateDefinition( label: ConstantAtomicLabel, expression: LambdaTermFormula - ) + ) \end{lstlisting} \\ %\hline @@ -897,7 +897,7 @@ \subsection{Definitions} label: ConstantFunctionLabel, out: VariableLabel, expression: LambdaTermFormula - ) + ) \end{lstlisting} \\ %\hline \end{tabular} From 3ea209ab3c743c09970fdc0a74e4cbe15f895409 Mon Sep 17 00:00:00 2001 From: shilangyu Date: Sun, 3 Mar 2024 22:23:58 +0100 Subject: [PATCH 5/6] Typos in prooflib and set theory --- refman/prooflib.tex | 8 ++++---- refman/theory.tex | 6 +++--- 2 files changed, 7 insertions(+), 7 deletions(-) diff --git a/refman/prooflib.tex b/refman/prooflib.tex index c6c20dad..7daebca4 100644 --- a/refman/prooflib.tex +++ b/refman/prooflib.tex @@ -80,10 +80,10 @@ \subsection{Local Definitions} \begin{figure} \begin{lstlisting}[language=lisa, frame=single] val existentialAxiom = Axiom(exists(x, in(x, emptySet))) - val falso = Theorem(⊥) { + val falso = Theorem( ⊥ ) { val c = witness(existentialAxiom) - have(⊥) by Tautology.from( - c.definition, emptySetAxiom of (x:= c)) + have( ⊥ ) by Tautology.from( + c.definition, emptySetAxiom of (x := c)) } \end{lstlisting} \caption{An example use of local definitions in Lisa} @@ -113,7 +113,7 @@ \subsection{Instantiations with ``of''} \end{lstlisting} Here, \lstinline|ax of c| is a fact whose proven statement is again $P(c)$. It is possible to instantiate multiple $\forall$ quantifiers at once. For example if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, d)| is a fact whose proven statement is $P(c, d)$. It is also possible to combine instantiation of free symbols and quantified variables. For example, if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, y, P := ===)| is a fact whose proven statement is $(c = y)$. -Formally, the \lstinline|of| keyword takes as argument arbitrarily many terms and substitution pairs. If there is at least on term given as argument, the base fact must have a single universally quantified formula on the right (an arbitrarily many formulas on the left). The number of given terms must be at most the number of leading universal quantifiers. Moreover, a substitution cannot instantiate any locked symbol (i.e. a symbol part of an assumption or definition). The ordering of substitution pairs does not matter, but the ordering of terms does. The resulting fact is obtained by first replacing the free symbols in the formula by the given substitution pairs, and then instantiating the quantified variables in the formula by the given terms +Formally, the \lstinline|of| keyword takes as argument arbitrarily many terms and substitution pairs. If there is at least one term given as argument, the base fact must have a single universally quantified formula on the right (an arbitrarily many formulas on the left). The number of given terms must be at most the number of leading universal quantifiers. Moreover, a substitution cannot instantiate any locked symbol (i.e. a symbol part of an assumption or definition). The ordering of substitution pairs does not matter, but the ordering of terms does. The resulting fact is obtained by first replacing the free symbols in the formula by the given substitution pairs, and then instantiating the quantified variables in the formula by the given terms In general, for the following proof \begin{lstlisting}[language=lisa, frame=single] diff --git a/refman/theory.tex b/refman/theory.tex index 76f5f340..892d87b3 100644 --- a/refman/theory.tex +++ b/refman/theory.tex @@ -24,9 +24,9 @@ \chapter{Library Development: Set Theory} \begin{tabular}{l|c|l} {} & Math symbol & Lisa Kernel \\ \hline Set Membership predicate & $\in$ & \lstinline$in(s,t)$ \\ - Subset predicate & $\subset$ & \lstinline$subset(s,t)$ \\ + Subset predicate & $\subseteq$ & \lstinline$subset(s,t)$ \\ Empty Set constant & $\emptyset$ & \lstinline$emptyset()$ \\ - Unordered Pair constant & $(\cdot, \cdot )$ & \lstinline$pair(s,t)$ \\ + Ordered Pair constant & $(\cdot, \cdot )$ & \lstinline$pair(s,t)$ \\ Power Set function & $\mathcal P$ & \lstinline$powerSet(s)$ \\ Set Union/Flatten function & $\bigcup$ & \lstinline$union(x)$ \\ \end{tabular} @@ -94,7 +94,7 @@ \section{Using Comprehension and Replacement} \caption{Comprehensions in various programming languages} \label{tab:comprehensionsProgramming} \end{table} -Those are typicaly syntactic sugar for a more verbose expression. For example in scala, \lstinline|(0 to 9).filter(x => x % 2 == 0).map(x => -x)|. However this kind of expressions is not possible in first order logic: We can't built in any way a term that contains formulas as subexpressions, as in \lstinline|filter|. So if we want to use such constructions, we need to simulate it as we did for local definitions in \autoref{sec:localDefinitions}. +Those are typically syntactic sugar for a more verbose expression. For example in scala, \lstinline|(0 to 9).filter(x => x % 2 == 0).map(x => -x)|. However this kind of expressions is not possible in first order logic: We can't built in any way a term that contains formulas as subexpressions, as in \lstinline|filter|. So if we want to use such constructions, we need to simulate it as we did for local definitions in \autoref{sec:localDefinitions}. It turns out that the comprehension schema is a consequence of the replacement schema when the value plugged for $\psi(x, y)$ is $\phi(x) \land y = x$, i.e. when $\psi$ denotes a restriction of the diagonal relation. Hence, what follows is built only from replacement. Note that the replacement axiom \autoref{axzf:replacement} is conditional of the schematic symbol $\psi$ being a functional relation. It is more convenient to move this condition inside the axiom, to obtain a non-conditional equivalence. This is the approach adopted in Isabelle/ZF \cite{noelExperimentingIsabelleZF1993}. We instead can prove and use From 4017f7b4a3aaa0d22bc06140804f47f55aeb6095 Mon Sep 17 00:00:00 2001 From: shilangyu Date: Sun, 3 Mar 2024 22:25:01 +0100 Subject: [PATCH 6/6] Regenerate lisa manual --- refman/lisa.pdf | Bin 244809 -> 247380 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/refman/lisa.pdf b/refman/lisa.pdf index e9c6913a9f232484957e0db00e7e50c88028430b..cd1bb87ae8e25ca998ff5ee7995f293a13ab3729 100644 GIT binary patch delta 205770 zcmY&{pd_<+sQ>x8A1OT=JE)DUiRVqrHobEH}Xc;8XZE}jQ zUa8K0q9grCCOF5JEB!_$OpZA#{ZsiUAZVz^Hz=@v$tl9bbOe#L&J;1H+V30DHPs^x z(^y@lBTg-Gw!g|=_V^aMAsl5}>K}PIM@bE>Zt2&=H_;_*TOzYt@S0fOPLq7BhZ7K< z*8?OibBYYbC-YtCmKovSa_;IFYE*_Suby?ItMQ+r6`*dGS9bJYk-~J51^2;1%S}ph zSxiluyztYTW=8uH(rS(D?fwLBsrgV65WTKr2wedHi7o(vi_B@nXgW{7R zuAlD<^vvzD4A6NF19%Hc3Caq@0;3Tev@Sz|^yhYk6ayEO1D6vEOXPsmS0fh`Q-g<8 zLl=}omkUcww!#MoZ=~p4Lt%zoML_^JawEY8u@}Li1O|Zd0XVt;H|JM6lkpc^2-`38 zpWsEZ#kqYXF<@7a9k#5Ug)v@i=M^!}KiDba<6PCRBSfc#SFxxN&e>XpW90XV>MxpQ 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