From 20c1d2ccc2990476aed22f74c484059c86fb88f7 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 5 Mar 2026 17:26:52 +0100 Subject: [PATCH] [FMFP] Typing --- ...methods-functional-programming-summary.pdf | Bin 402057 -> 433063 bytes ...methods-functional-programming-summary.tex | 6 +- .../02_typing/01_mini-haskell/00_syntax.tex | 6 +- .../01_mini-haskell/01_lambda-calculus.tex | 21 +++++++ .../01_mini-haskell/02_further-rules.tex | 58 ++++++++++++++++++ .../01_mini-haskell/03_type-inference.tex | 26 ++++++++ .../parts/02_typing/02_lambda-calculus.tex | 1 - 7 files changed, 114 insertions(+), 4 deletions(-) create mode 100644 semester4/fmfp/parts/02_typing/01_mini-haskell/01_lambda-calculus.tex create mode 100644 semester4/fmfp/parts/02_typing/01_mini-haskell/02_further-rules.tex create mode 100644 semester4/fmfp/parts/02_typing/01_mini-haskell/03_type-inference.tex delete mode 100644 semester4/fmfp/parts/02_typing/02_lambda-calculus.tex diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.pdf b/semester4/fmfp/formal-methods-functional-programming-summary.pdf index 4ae69a599b2dac5d76ec1943ffd7170fb62bfc05..c36dcf5d8aceb9d56f38e07e646ec8a684809065 100644 GIT binary patch delta 179076 zcmZs>Q*b6+(5@ZZwr!h}iEZ1q@x-?6iESGb6DR-seSg)ieX!4ZRiCU{)m`29 zweF}=IeYe%s*LpO^n6~R34 z2<%TpB9(j=p98vO=ELQJ4wtj6&z{k(4t?#8Z!*JZT;hTLgGV}sX$+Z&NSsL<8W)HR zhL9G_PSkp9hunA;6Oga3ZTXXFDz5Q-v`KqayUyRI*L3Q#1IJ^>5o^Z2*Vr-z5U{br zz3HvK?chKI(Mu1U@N?}x1U~vd@^@#;CpT+9B+`IUh4FzX7fzC3sk?Ynj&kP7heDvs z+*d!>{6!ZnF8a#Lq$)sPPx7@kUlg^1?|Y(a(!d+^nWREJJGaIcZ;89IF1fW%sr@6L zE``s^Hw9{6OfL1SJovABU=za|V73VDNHvzwSsf)8h#HZyryR3I$5ID>o<>&b`zN{) zdAMOKbuSIQlMTfVyWJuCV>_q#758uz12(3MFFo~+_F5-b!xpJ8(=yeuTWfqIvkMh` zUd2?m>NH#B{C$Yl5#*eRkmJnAH_;v(-wT261|fMZf+{ejzW{zD6*^Jpk%+v_Cx_2AWXMgHY~Dk6LMWaEPqcSEQ@O zUGqw#AtQOt_y_b{oz)l4+`-J%&Bff<{(mAz6B{^I4jvK~62~MA;$}*OuU%YLUY7qk z|CbREfMb?3cd&G`B4Op?<;eg=1EmG%=(?hFV+6d^ua(c<$Z4L{sTc4mftkx}2JX@z z$Mi~jG^|E3&p)lcoVBU=!GnTH-yszX%|$O$^>9B#Dx3xF8m9@^wpyR% zTg^NM%vqK{`taFztDcdiRXkw3uQF}5&uLEmM7IxPhm;Qwjs^yr@Q^E`221^EUU0iR zVVoVi+m1mXFRQ*PvF8m-M>C$-d!+Ccs$l_LM7a7k z4*lx>akDcF7MY|E#hr<)LMZ}FN>R)see^#)T4n1KUh_oLhUBuLx)DHii}Bf2G-;Ak zu@>h7Iwhh=%9WL(;ee+LZ|4*BB^!SQ^?CAm{KN;?%5KtNqggaC*TMj&k+9N%oTaQO z1Sqij92~IvJXF`nAgjTpgDj;GzEu(gq`yB zg4)L{QyEy5!5CR&X|lBw@DLL0HBfxM)SIf22Lq=t^Rte}WQ79l+3L@$z~ zjd3r6;e=;XXw8>rIJ8`kzuXY>{kyq+|Lt<op?!#IpTSUdm&<{j+V zqZu<3oaq5X)Nzc^O^xY5TTIUhUT9B94-tPxzZpa>^;8IeYBus5o+fpF^?2RlO?E*i`uvYi5me6f9MGiyFJh!b)jrfoN4{pm4O{pw5b|IMz*VfQnar_Q|bq-Dg$Z!dYL ztS*dVY{*j*>L^3%caOMepGlgCZ}5>yQsHH^#$MZP&L5n}*y@uB`i$R5 zPu@M<{881Z)%s{*Q>}u}t)O?~#{8W2`c=dnfi(rNfFpo;P#OYl7>T;;Iz zSg-qs`yKp)HriA|^L^xUcRGg~9G}ttD#`=Y7BU^}!9v?^*KnB?AOmjSnIEBR%eEtMyNECDE-af1?_M&wS<`SoGlOPnVxfrlV_~yC(wJMTK8RjWWWz;CwbD&?wu#=q!+H~b<=p=CdK^? 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zmgM?XOKT*#ealj}Bo9tINgkdGJgN&keja#c(iwPmDKOa-czHkY>P_HvF7R$%;8RZ^ zJ0AFUF7RU_F!dtv=V##Wq!SWFflMTG72RnMNdWw_J@J| diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.tex b/semester4/fmfp/formal-methods-functional-programming-summary.tex index 84b03dd..2700afe 100644 --- a/semester4/fmfp/formal-methods-functional-programming-summary.tex +++ b/semester4/fmfp/formal-methods-functional-programming-summary.tex @@ -68,13 +68,15 @@ % \input{parts/01_formal-reasoning/05_correctness/} % \input{parts/01_formal-reasoning/} + \newsection \section{Typing} \input{parts/02_typing/00_intro.tex} \input{parts/02_typing/01_mini-haskell/00_syntax.tex} +\input{parts/02_typing/01_mini-haskell/01_lambda-calculus.tex} +\input{parts/02_typing/01_mini-haskell/02_further-rules.tex} +\input{parts/02_typing/01_mini-haskell/03_type-inference.tex} % \input{parts/02_typing/01_mini-haskell/} -\input{parts/02_typing/02_lambda-calculus.tex} -% \input{parts/02_mini-haskell/} \end{document} diff --git a/semester4/fmfp/parts/02_typing/01_mini-haskell/00_syntax.tex b/semester4/fmfp/parts/02_typing/01_mini-haskell/00_syntax.tex index 74a71cd..f05c50d 100644 --- a/semester4/fmfp/parts/02_typing/01_mini-haskell/00_syntax.tex +++ b/semester4/fmfp/parts/02_typing/01_mini-haskell/00_syntax.tex @@ -4,8 +4,12 @@ This is a stripped down version of Haskell, used here to explore the type system \subsubsection{Syntax} Programs are terms, the core is the lambda-calculus, where $\cV$ is the set of variables and $\Z$ the set of integers: \[ - t :: = \cV \divider (\lambda x. t) \divider (t_1 t_2) \divider True \divider False \divider (\textbf{iszero } t) \divider \Z \divider (t_1 + t_2) \divider t_1 * t_2 + t :: = \cV \divider (\lambda x. t) \divider (t_1 t_2) \divider True \divider False \divider (\textbf{iszero } t) \divider \Z \divider (t_1 + t_2) \divider t_1 * t_2 \divider \textbf{if } t_0 \textbf{ then } t_1 \textbf{ else } t_2 \divider (t_1, t_2) \divider (\textbf{fst } t) \divider (\textbf{snd } t) \] It is easily possible to add additonal syntax and types and we employ syntactic sugar, such as omitting parenthesis. + +The types are given by $\tau ::= \cV_T \divider \texttt{Bool} \divider \texttt{Int} \divider (\tau, \tau) \divider (\tau \rightarrow \tau)$, where $\cV_T$ is a set of type variables. +The type system is based on typing judgement of form $\Gamma \vdash t :: r$, where $\Gamma$ is a set of bindings $x_i : \tau_i$ +that maps variables to types and can be understood as a typing symbol table. diff --git a/semester4/fmfp/parts/02_typing/01_mini-haskell/01_lambda-calculus.tex b/semester4/fmfp/parts/02_typing/01_mini-haskell/01_lambda-calculus.tex new file mode 100644 index 0000000..f46a3e8 --- /dev/null +++ b/semester4/fmfp/parts/02_typing/01_mini-haskell/01_lambda-calculus.tex @@ -0,0 +1,21 @@ +\subsubsection{Lambda calculus} +To prove that types are correct, the lambda calculus comes in handy. +It is based on the same concept as natural deduction trees + +\paragraph{Core rules for Lambda-Calculus} +\[ + \begin{prooftree} + \infer0[Var]{\Gamma, x : \tau \vdash x :: \tau} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma, x : \sigma \vdash t :: \tau} + \infer1[Abs]{\Gamma \vdash (\forall x. t) :: \sigma \rightarrow \tau} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t_1 :: \sigma \rightarrow \tau} + \hypo{\Gamma \vdash t_2 :: \sigma} + \infer2[App]{\Gamma \vdash (t_1 t_2) :: \tau} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/02_typing/01_mini-haskell/02_further-rules.tex b/semester4/fmfp/parts/02_typing/01_mini-haskell/02_further-rules.tex new file mode 100644 index 0000000..7605e5b --- /dev/null +++ b/semester4/fmfp/parts/02_typing/01_mini-haskell/02_further-rules.tex @@ -0,0 +1,58 @@ +\subsubsection{Further rules for mini-Haskell} +\paragraph{Base types} +\[ + \begin{prooftree} + \infer0[Int]{\Gamma \vdash n :: \texttt{Int}} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[True]{\Gamma \vdash \texttt{True} :: \texttt{Bool}} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[False]{\Gamma \vdash \texttt{False} :: \texttt{Bool}} + \end{prooftree} +\] + +\paragraph{Operations} +Let $\textbf{op} \in \{ +, * \}$ +\[ + \begin{prooftree} + \hypo{\Gamma \vdash t :: \texttt{Int}} + \infer1[iszero]{\Gamma \vdash (\textbf{iszero } t) :: \texttt{Bool}} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t_1 :: \texttt{Int}} + \hypo{\Gamma \vdash t_2 :: \texttt{Int}} + \infer2[BinOp]{\Gamma \vdash (t_1 \textbf{ op } t_2) :: \texttt{Int}} + \end{prooftree} +\] +\[ + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t_0 :: \texttt{Bool}} + \hypo{\Gamma \vdash t_1 :: \tau} + \hypo{\Gamma \vdash t_2 :: \tau} + \infer3[if]{\Gamma \vdash (\textbf{if } t_0 \textbf{ then } t_1 \textbf{ else } t_2) :: \tau} + \end{prooftree} +\] + +\paragraph{Tuples} +\[ + \begin{prooftree} + \hypo{\Gamma \vdash t_1 :: \tau_1} + \hypo{\Gamma \vdash t_2 :: \tau_2} + \infer2[Tuple]{\Gamma \vdash (t_1, t_2) :: (\tau_1, \tau_2)} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t :: (\tau_1, \tau_2)} + \infer1[fst]{\Gamma \vdash (\textbf{fst } t) :: \tau_1} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t :: (\tau_1, \tau_2)} + \infer1[snd]{\Gamma \vdash (\textbf{snd } t) :: \tau_2} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/02_typing/01_mini-haskell/03_type-inference.tex b/semester4/fmfp/parts/02_typing/01_mini-haskell/03_type-inference.tex new file mode 100644 index 0000000..337b5e4 --- /dev/null +++ b/semester4/fmfp/parts/02_typing/01_mini-haskell/03_type-inference.tex @@ -0,0 +1,26 @@ +\subsubsection{Type inference} +Type inference in general fails, if two (or more) branches fail to resolve to unifiable types. + +We start a \bi{type judgement} with judgement $\vdash t :: \tau_0$, then build a derivation tree bottom-up. +Finally, apply constraints / unification to get possible types. + + +\paragraph{Self application} +This means that you apply a function to itself. +In Haskell, this is not typeable because there would need to be an infinite function type, but all \bi{Haskell types are finite} + + +\paragraph{Curry-Howard isomorphism} +We can also apply the implication introduction and implication elimination rules: +\[ + \begin{prooftree} + \hypo{\Gamma, \sigma \vdash \tau} + \infer1[$\rightarrow$-I]{\Gamma \vdash \sigma \rightarrow \tau} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash \sigma \rightarrow \tau} + \hypo{\Gamma \vdash \sigma} + \infer2[$\rightarrow$-E]{\Gamma \vdash \tau} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/02_typing/02_lambda-calculus.tex b/semester4/fmfp/parts/02_typing/02_lambda-calculus.tex deleted file mode 100644 index af7c720..0000000 --- a/semester4/fmfp/parts/02_typing/02_lambda-calculus.tex +++ /dev/null @@ -1 +0,0 @@ -\subsection{Lambda calculus}