From 948bc1f5677cfaa5eb32f835d639a4329b6f1a2f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 2 Mar 2026 17:01:01 +0100 Subject: [PATCH] [FMFP] Correctness --- ...methods-functional-programming-summary.pdf | Bin 363069 -> 424004 bytes ...methods-functional-programming-summary.tex | 5 ++ .../05_correctness/00_intro.tex | 4 ++ .../05_correctness/01_termination.tex | 15 ++++++ .../05_correctness/02_behaviour.tex | 44 ++++++++++++++++++ .../05_correctness/03_induction.tex | 15 ++++++ 6 files changed, 83 insertions(+) create mode 100644 semester4/fmfp/parts/01_formal-reasoning/05_correctness/00_intro.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/05_correctness/01_termination.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/05_correctness/02_behaviour.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.pdf 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zI$*rz6G9juh4!C3h)-Z72Oy{W8#)SnGx{V0TA-{uKyar@4bUk@K07Q=WSt*)s|Q%p zM#`5BLzFT0@aTf3+etcs-1J0RC=yp2v(_>94@90E%y9g-)_N7F_Po{+>Vd7^vydd0 z*P>&T+_W@n6G6#|q@0o^s5O;;mlrETSD`497fLau7|=MX?3W+PYc`?qk$Xw)E`o>y zrWQ#|v$9bvYa3cLilGv<;=|*#e4t_yF>Q~lLR4V;M`>~E?)$H32^Ay!@5oc|sJIoJ z@(%^a0}?@XK?d`Kg`k5_Ajsi(i1ud-anL%R<|?EAJh4`Vitul?T4@CC<5!81t$tZA zm4n$|#L0}2E#Hk-I!5$(*e(SR2c4?{WTkuzmnx|v?0eg+wD#ROYcdmF)ZdL4HK7;e zt22}8FB_8Ti$ on a set $S$ is \bi{well-founded} if and only if there is no infinite decreasing chain $x_1 > x_2 > \ldots$ for $x_i \in S$ +An example of such an order is $>$ on $\N$, denoted $>_\N$. Counter example: $>_\Z$ (not bounded from below) + + +\inlinelemma Let $R \subseteq S \times S$ be a relation on $S$. Let $s_0, s_i \in S$ and $i \geq 1$. +Then $s_0 \; R^i \; s_i$ if and only if $s_1, \ldots s_{i - 1} \in S$ such that $s_0 \; R \; s_1 \R \ldots R \; s_{i - 1} \; R \; s_i$ + +\inlinedefinition $R^+ \equiv \bigcup_{n \geq 1} R^n$, where $R^n \equiv R \circ R^{n - 1}$ for $n \geq 2$ and $R^1 \equiv R$ + +\inlinetheorem If $>$ is a well-founded order on $S$, then $>^+$ is also well-founded on $S$ diff --git a/semester4/fmfp/parts/01_formal-reasoning/05_correctness/02_behaviour.tex b/semester4/fmfp/parts/01_formal-reasoning/05_correctness/02_behaviour.tex new file mode 100644 index 0000000..53561f7 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/05_correctness/02_behaviour.tex @@ -0,0 +1,44 @@ +\subsubsection{Correctness - Behaviour} +We denote that two functions \texttt{fac} and \texttt{fac2} compute the same function usually as +\[ + \forall n \in \N. \texttt{fac } n = \texttt{fac2 } (n, 1) +\] +The two functions are given as follows: +\begin{multicols}{2} + \begin{code}{haskell} + fac :: Int -> Int + fac 0 = 1 + fac n = n * fac (n - 1) + \end{code} + \begin{code}{haskell} + fac2 :: (Int, Int) -> Int + fac2 (0, a) = a + fac2 (n, a) = fac2 (n - 1, n * a) + \end{code} +\end{multicols} +An important fact to consider is that testing, while useful to find errors can't replace a formal proof to show that a function is correct! + +These proofs are based on a simple idea: \bi{functions are equations} and thus, we can reason about them through equational reasoning, or more generally, +proofs in first-order logic with equality. + +Often, especially in Haskell programs, we have cases depending on values. Logically, to prove such a function, we also use case distinction, also referred to as reasoning by cases. + +\inlineexample Consider the Haskell function below \rmvspace +\begin{code}{haskell} + maxi :: Int -> Int -> Int + maxi n m + | n >= m = n + | otherwise = m +\end{code} +To prove that it is correct, we can use reasoning by cases: + +We have $n \geq m \lor \neg(n \geq m)$. + +We then show that the function is correct for both cases (i.e. LHS and RHS of OR): +\begin{enumerate}[label=C\arabic*] + \item $n \geq m$: Then $\texttt{maxi } n \ m = n$ and $n \geq n$ + \item $\neg(n \geq m)$: Then $\texttt{maxi } n \ m = m$. But $m > n$, so we have $\texttt{maxi } n\ m \geq n$ +\end{enumerate} +In this proof we used the \textbf{TND} and \textbf{$\lor$-E} (here also called \bi{Case Split}) rules. + +So what we have to show, given $Q \lor R$ for any proposition $P$ with case split is that \bi{(1)} $P$ follows from $Q$ and \bi{(2)} $P$ follows from $R$ diff --git a/semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex b/semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex new file mode 100644 index 0000000..9a3ad43 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex @@ -0,0 +1,15 @@ +\subsubsection{Induction} +To prove recursive formulas, or more precisely formulated, a formula $P$ (with free variable $n$) for all $n \in \N$. + +A proof by cases is not possible, as there are infinitely many cases (one for each input). +Thus: We can use induction to prove recursive formulas or functions. + +\paragraph{The schema} +To prove $\forall n \in \N. P$ (with $n$ free in $P$), we do the following: + +\bi{Base case}: We show that $P[n \mapsto 0]$ is correct + +\bi{Step case}: For an arbitrary $m$ not free in $P$, we show that $P[n \mapsto m + 1]$ is correct under the assumption that $P[n \mapsto m]$ + +For \bi{well-founded} domains, we have to adjust the induction hypothesis slightly: We assume $\forall l \in \N. l < m \rightarrow P[n \mapsto l]$ +and then prove $P[n \mapsto m]$ under our assumption.