From 28d0696091b1bb1cf39254bc699d1a492a64f0b5 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 16 Mar 2026 17:43:49 +0100 Subject: [PATCH] [FMFP] algebraic data types --- ...methods-functional-programming-summary.pdf | Bin 433063 -> 443426 bytes ...methods-functional-programming-summary.tex | 2 + .../05_correctness/03_induction.tex | 8 +- .../00_correctness.tex | 88 ++++++++++++++++++ 4 files changed, 94 insertions(+), 4 deletions(-) create mode 100644 semester4/fmfp/parts/02_typing/02_algebraic-data-types/00_correctness.tex diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.pdf b/semester4/fmfp/formal-methods-functional-programming-summary.pdf index c36dcf5d8aceb9d56f38e07e646ec8a684809065..6864a5b39c70235892bf9c594e1ae8a2472480b2 100644 GIT binary patch delta 122818 zcmYJaV~{R9ur)fiXZF~(ZQC~X*fyTAvB$P;+qP}np1JQi=c~FuQkA5-Q`M=YI%{Qh zX)AU?I8L1ebSlv_I4US1J5wsM1Ozo8Syjer3~$*QG(Gp?KRqY`g!8+#@8(=zIa5W1SSjP3REDNEHqVlbfF!Iv(br!B1<`3 z%idTZ3lq^mMSPF_AR5UrCG?ft)y#ga~AUqyOdMlo@lf*7Ity#b}Aa<8Cf| zbySb1i9+@JPVyY{iy|bvy!} zFR>dq>i26&3ygk4I#)AW!o4LIfJ2|H@YrPMZKUeAB%7dh*@oRR zHa|*L_|-sl+GPKh-+=BH&j+*Tj1TW+#tRH?5d0jDDBMCN+N}GPz=0{$n*k;|ZGISI z)d_-*Y#^cky~$&h=W74X!8!wTQPQ#UtjeE#7G|~X)CF46Z1{&joKrWfQN$hjS*Zru ztSL_uKs1%SESZOg0C<15P}jx1u2WGF=QAxiV?=NZvsq(2R*vEfC^X|}rBpehixgKf z`sC&2_^vpphn|@A6j^jYdMLVoH-jK#8qmnj?L=CM_%& zC3<63qQy|CP9q@>A@&4d1c#Q}B*~`8I%+2ZIN`XbL&jTJBOCeB50HWp@@u<;f9f^D z+}-gqg=-TF+?R+fTZp8P!W<=I%}|cR$%*RCRnG&7%gM|UfXgFRuqu=qLaoqvz%c)2 zWWas{gHlnRtW&065m!iM6=MI-%F+!2>8NB_9ZFyw1)-FYqpt4=O^dw!*p`-k9x(vU zV~sQljv^W}!e~szF0>QC>_lFXtagpGYAe@=26R9_U7;r_7>J52

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P$ (with $n$ free in $P$), we do the following: -\bi{Base case}: We show that $P[n \mapsto 0]$ is correct +\shade{blue}{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 $P[n \mapsto m]$ +\shade{green}{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. @@ -18,8 +18,8 @@ and then prove $P[n \mapsto m]$ under our assumption. \paragraph{Induction over Lists} To prove $P$ for all $xs$ in \texttt{[T]}, we do the following: -\bi{Base case}: We prove that $P[xs \mapsto []]$ is correct +\shade{blue}{Base case} We prove that $P[xs \mapsto []]$ is correct -\bi{Step case}: We prove that $\forall y :: T, ys :: [T]. P[xs \mapsto ys] \rightarrow P[xs \mapsto y : ys]$, or in other words: +\shade{green}{Step case} We prove that $\forall y :: T, ys :: [T]. P[xs \mapsto ys] \rightarrow P[xs \mapsto y : ys]$, or in other words: We fix arbitrary $y :: T$ and $ys :: [T]$, which both are not free in $P$. We then apply our induction hypothesis $P[xs \mapsto ys]$ to prove $P[xs \mapsto y : ys]$ diff --git a/semester4/fmfp/parts/02_typing/02_algebraic-data-types/00_correctness.tex b/semester4/fmfp/parts/02_typing/02_algebraic-data-types/00_correctness.tex new file mode 100644 index 0000000..00d9b11 --- /dev/null +++ b/semester4/fmfp/parts/02_typing/02_algebraic-data-types/00_correctness.tex @@ -0,0 +1,88 @@ +\subsection{Natural Number Proofs} +To prove $\forall n \in \N. P$, we of course again use induction: + +\shade{blue}{Base Case} Show $P[n \mapsto 0]$ + +\shade{green}{Step Case} Let $m \in \N$ be arbitrary and not free in $P$. We then assume that $P[n \mapsto m]$ and show that $P[n \mapsto m + 1]$ + +Or the same as a natural deduction rule: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash P[n \mapsto 0]} + \hypo{\Gamma, P[n \mapsto m] \vdash P[n \mapsto m + 1]} + \infer2[$m$ not free in $\Gamma, P$]{\Gamma \vdash \forall n \in \N. P} + \end{prooftree} +\] + + +\subsubsection{Induction over the natural numbers} +In Haskell, we can also define all the natural numbers using +\mint{haskell}+data Nat = Zero | Succ Nat deriving (Eq, Ord, Show)+ + +Thus the natural numbers are (isomorphic to) the set +\[ + \texttt{Nat} = \{ \texttt{Zero}, \texttt{Succ Zero}, \texttt{Succ (Succ Zero)}, \ldots \} +\] + +The data type provides two crucial rules for constructing members of \texttt{Nat}: +\begin{itemize} + \item $\texttt{Zero} \in \texttt{Nat}$ + \item If $x \in \texttt{Nat}$, then $\texttt{Succ} x \in \texttt{Nat}$ +\end{itemize} + +The induction stated as a natural deduction rule: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash P[n \mapsto \texttt{Zero}]} + \hypo{\Gamma, P[n \mapsto m] \vdash P[n \mapsto \texttt{Succ}\ m]} + \infer2[$m$ not free in $\Gamma, P$]{\Gamma \vdash \forall n \in \texttt{Nat}. P} + \end{prooftree} +\] + + +\subsubsection{Lists} +A possible data type for lists in Haskell is: +\mint{haskell}+data L t = Nil | Cons t (L t)+ + +A natural deduction rule for induction over lists is: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash P[xs \mapsto \texttt{Nil}]} + \hypo{\Gamma, P[xs \mapsto ys] \vdash P[xs \mapsto \texttt{Cons}\ y\ ys]} + \infer2[$y$, $ys$ not free in $\Gamma, P$]{\Gamma \vdash \forall xs \in \texttt{L}\ t. P} + \end{prooftree} +\] + + +\subsubsection{Trees} +A possible data type for trees in Haskell is: +\mint{haskell}+data Tree t = Leaf | Node t (Tree t) (Tree t)+ + +A natural deduction rule for induction over trees is: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash P[x \mapsto \texttt{Leaf}]} + \hypo{\Gamma, P[x \mapsto l] \vdash P[xs \mapsto \texttt{Node}\ a\ l\ r]} + \infer2[$a$, $l$, $r$ not free in $\Gamma, P$]{\Gamma \vdash \forall x \in \texttt{Tree}\ t. P} + \end{prooftree} +\] + + +\subsubsection{Structural Induction} +Induction is based on the structure of terms +\mint{haskell}+data T t = Leaf t | Node1 (T t) | Node2 t (T t) (T t)+ + +\shade{blue}{Base Case} $T_0 = \{ \texttt{Leaf}\ a \divider a \in t \}$ + +\shade{green}{Step Case} $T_i = T_{i - 1} \cup \{ \texttt{Node1}\ s \divider s \in T_{i - 1} \} \cup \{ \texttt{Node2}\ a\ l\ r \divider a \in t \text{ and } l, r \in T_{i - 1} \}$ + +A natural deduction rule structural induction is: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash P[x \mapsto \texttt{Leaf}\ a]} + \hypo{\Gamma, P[x \mapsto s] \vdash P[xs \mapsto \texttt{Node1}\ s]} + \hypo{\Gamma, P[x \mapsto l], P[x \mapsto r] \vdash P[ \mapsto \texttt{Node2}\ a\ l\ r]} + \infer3[$(*)$]{\Gamma \vdash \forall x \in \texttt{T}\ t. P} + \end{prooftree} +\] +(*) $a$, $l$, $r$, $s$ not free in $\Gamma, P$