[FMFP] Mostly summarized formal reasoning

semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex

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+\newsection
+\subsection{First-Order Logic}
+\subsubsection{Syntax}
+\inlinedefinition[Signature] consists of a set of function symbols $\cF$ and a set of predicate symbols $\cP$, as well as their arities.
+
+We write $f^k$ (or $p^k$, for predicates) to indicate that it has \textit{arity} $k \in \N$.
+Constant functions have arity $0$, linear functions have arity $1$, thus, the arity of a given function (or predicate) is given by
+the number of parameters to uniquely describe it, minus one.
+
+\inlinedefinition[Term] is the terms of first-order logic is smallest set, where (with $\cV$ again a set of variables)
+\begin{enumerate}
+    \item $x \in \textit{Term}$ if $x \in \cV$
+    \item $f^n(t_1, \ldots, t_n) \in \textit{Term}$ if $f^n \in \cF$ and $t_i \in \textit{Term}$, $\forall 1 \leq i \leq n$ (description of form of formulas)
+\end{enumerate}
+
+\inlinedefinition[Form] is the formulas of first-order logic, is the smallest set where
+\begin{enumerate}
+    \item $\bot \in \textit{Form}$
+    \item $p^n(t_1, \ldots, t_n) \in \textit{Form}$ if $p^n \in \cP$  and $t_j \in \textit{Term}$, $\forall 1\leq j \leq n$ (description of form of predicates)
+    \item $A \circ B \in \textit{Form}$ if $A \in \textit{Form}$, $B \in \textit{Form}$ and $\circ \in \{ \land, \lor, \rightarrow \}$ (i.e. formulas with logic symbols)
+    \item $Qx.A \in \textit{Form}$ if $A \in \textit{Form}$, $x \in \cV$ and $Q \in \{ \forall, \exists \}$ (i.e. formulas with quantifiers)
+\end{enumerate}
+
+
+\paragraph{Binding}
+\inlinedefinition[Bound variable] A variable that occurs in a quantifier in scope ({\color{blue} blue in example below})
+
+\inlinedefinition[Free variable] A variable that is not bound by a quantifier in scope ({\color{red} red in example below})
+
+\inlineexample $(q({\color{red} x}) \lor \exists {\color{blue} x}.\forall {\color{blue} y}. p(f({\color{blue} x}), {\color{red} z}) \land q({\color{red} a}))
+\lor \forall {\color{blue} x}.r({\color{blue} x}, {\color{red} z}, g({\color{blue} x}))$
+
+\inlinedefinition[$\alpha$-conversion] A \bi{bound} variable can be renamed at any time, but must preserve binding structure.
+
+
+\paragraph{Precedences} 
+$\neg > \land > \lor > \rightarrow$, with $>$ a total order of precedences.
+Quantifiers extend as far right as possible, bounded by the end of line or a going out of scope by closing parenthesis.

semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex

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+\subsubsection{Semantics}
+\inlinedefinition[Structure] is a pair $\cS = \langle U_\cS, I_\cS \rangle$, where $U_\cS$ is the universe and it is a non-empty set and $I_\cS$ is a mapping with
+\begin{enumerate}
+    \item $I_\cS(p^n)$ is an $n$-ary relation on $U_\cS$ for $p^n \in \cP$ (short $p^\cS$)
+    \item $I_\cS(f^n)$ is an $n$-ary (total) function on $U_\cS$ for $f^n \in \cF$ short ($f^\cS$)
+\end{enumerate}
+\inlineintuition The $I_\cS$ is essentially assigning to each predicate and formula its definition in the universe of the structure, noted as a relation.
+
+\inlinedefinition[Interpretation] is a pair $\cI = \langle \cS, v \rangle$, with $v : \cV \rightarrow U_\cS$ a valuation\\
+\inlineintuition it assigns definitions to the formulas and predicates (through the structure), as well as values to the variables (through the valuation).
+
+\inlinedefinition[Value] of a term $t$ under $\cI$ is written as $\cI(t)$ and defined by $\cI(x) = v(x)$ for $x \in \cV$ and\\
+$\cI(f(t_1, \ldots, t_n)) = f^\cS(\cI(t_1), \ldots, \cI(t_n))$
+
+\inlinedefinition[Satisfiability] $\models \subseteq$ Interpretations $\times$ \textit{Form} is the smallest relation satisfying
+\begin{align*}
+     & \langle \cS, v \rangle \models p(t_1, \ldots, t_n) &  & \text{ if } (\cI(t_1), \ldots, \cI(t_n)) \in p^\cS \text{ where } \cI = \langle \cS, v \rangle \\
+     & \langle \cS, v \rangle \models \forall x. A        &  & \text{ if } \langle \cS, v[x \mapsto a] \rangle \models A, \text{ for all } a \in \U_\cS       \\
+     & \langle \cS, v \rangle \models \exists x. A        &  & \text{ if } \langle \cS, v[x \mapsto a] \rangle \models A, \text{ for some } a \in \U_\cS      \\
+\end{align*}
+
+\inlinedefinition[Model] When $\langle \cS, v \rangle \models A$, then $\langle \cS, v \rangle$ is a \bi{model} for $A$.
+If $A$ does not have free variables, the satisfaction does not depend on the valuation $v$ and we write $\cS \models A$
+
+\inlinedefinition[Validity] When every interpretation is a model, we write $\models A$, and we say that $A$ is \bi{valid}
+
+\inlinedefinition[Satisfiability] $A$ is \bi{satisfiable}, if there exists at least one model for $A$.
+
+\inlineexample Given $\forall x.p(x, s(x))$, we a model would be
+\begin{align*}
+    U_\cS & = \N                                                                   \\
+    p^\cS & = \{ (m, n) \divider m, n \in U_\cS \text{ and } m < n \}              \\
+    s^\cS & = \text{ successor function on } U_\cS, \text{ i.e. } s^\cS(x) = x + 1
+\end{align*}
+
+
+\paragraph{Substitution}
+\inlinedefinition[Substitution] Replace all occurrences of a free variable $x$ with some term $t$ in $A$.
+To denote a substitution, we write $A[x \mapsto t]$. \hl{Important} All free variables in $t$ must still be free in $A[x \mapsto t]$.
+If that would not be true anymore, do a $\alpha$-conversion first.

semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex

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+\subsubsection{Quantifiers}
+\paragraph{Universal quantification}
+Additional rules are needed for the universal quantifier. * side condition is that $x$ is not free in any assumption of $\Gamma$
+\begin{align*}
+    \begin{prooftree}
+        \hypo{\Gamma \vdash A}
+        \infer1[$\forall$-I*]{\Gamma \vdash \forall x.A}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\Gamma \vdash \forall x.A}
+        \infer1[$\forall$-E]{\Gamma \vdash A[x \rightarrow t]}
+    \end{prooftree}
+\end{align*}
+Again here be mindful not to capture free variables.
+
+\paragraph{Existential quantification}
+Additional rules are needed for the existential quantifier. ** side condition is that $x$ is neither free in B nor free in $\Gamma$
+\begin{align*}
+    \begin{prooftree}
+        \hypo{\Gamma \vdash A[x \mapsto t]}
+        \infer1[$\exists$-I]{\Gamma \vdash \exists x.A}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\Gamma \vdash \exists x.A}
+        \hypo{\Gamma, A \vdash B}
+        \infer2[$\exists$-E**]{\Gamma \vdash A[x \rightarrow t]}
+    \end{prooftree}
+\end{align*}
+Again here be mindful not to capture free variables.