From 05989079b8877e912b7fe95371f24b4f67a890aa Mon Sep 17 00:00:00 2001 From: Karl Palmskog Date: Thu, 31 Aug 2023 14:07:06 +0200 Subject: [PATCH] add ott stuff --- index.html | 24 +++++++-- ott/functional.ott | 112 ++++++++++++++++++++++++++++++++++++++ ott/functional.pdf | Bin 0 -> 106355 bytes ott/lam.ott | 130 +++++++++++++++++++++++++++++++++++++++++++++ ott/lam.pdf | Bin 0 -> 109015 bytes 5 files changed, 262 insertions(+), 4 deletions(-) create mode 100644 ott/functional.ott create mode 100644 ott/functional.pdf create mode 100644 ott/lam.ott create mode 100644 ott/lam.pdf diff --git a/index.html b/index.html index fd2213f..bdf4563 100644 --- a/index.html +++ b/index.html @@ -26,7 +26,7 @@

Advanced Functional Programming Seminar Course

About the Course

-

This seminar course focuses on the principles and practice of purely functional programming, using the CakeML language as an example.

+

This seminar course focuses on the principles and practice of purely functional programming. The course features the CakeML language and the Ott tool for defining languages and inductive relations.

Seminars Fall 2023

During fall 2023 period 1 at KTH, seminars will occur twice per week:

@@ -47,13 +47,28 @@

Seminar Slides

1 -Functional languages, abstract syntax trees, variable binding, and inductive definitions -slides, PFPL chapters 1 and 2, introduction to inductive definitions by Aczel +Functional languages, abstract syntax trees, variable binding, and inductive definitions + + + 2 Untyped lambda calculus, operational semantics, combinators, encodings -slides, supplement by Harper + + + 3 @@ -125,6 +140,7 @@

Homework

Resources

diff --git a/ott/functional.ott b/ott/functional.ott new file mode 100644 index 0000000..7961f71 --- /dev/null +++ b/ott/functional.ott @@ -0,0 +1,112 @@ +embed +{{ coq +Require Export Ascii. +Require Export String. + +Hint Resolve ascii_dec : ott_coq_equality. +}} +metavar n ::= + {{ lex numeral }} + {{ coq nat }} + {{ coq-equality }} + {{ com number }} +metavar x ::= + {{ lex alphanum }} + {{ coq string }} + {{ coq-equality }} + {{ com variable }} +grammar +e :: e_ ::= + | x :: :: var {{ com variable }} + | n :: :: num {{ com number }} + | e + e' :: :: plus {{ com plus }} + | e * e' :: :: times {{ com times }} + | let x := e in e' :: :: def (+ bind x in e' +) + {{ com let }} + | e [ e' / x ] :: M :: subst + {{ com substitution }} + {{ coq (subst_e [[e']] [[x]] [[e]]) }} + | ( e ) :: S :: parentheses + {{ coq ([[e]]) }} + +terminals :: terminals_ ::= + | -> :: :: red + {{ tex \rightarrow }} + | \||/ :: :: down + {{ tex \Downarrow }} + +formula :: formula_ ::= + {{ com formulas }} + | judgement :: :: judgement + | n + n' = n'' :: M :: num_plus_eq + {{ coq ([[n]] + [[n']] = [[n'']]) }} + | n * n' = n'' :: M :: num_times_eq + {{ coq ([[n]] * [[n']] = [[n'']]) }} + +substitutions + single e x :: subst + +%freevars +% e x :: free + +defns + operational_semantics :: os_ ::= + +defn + e -> e' :: :: red :: red_ + {{ com reduction step }} by + + n1 + n2 = n + ------------ :: plus + n1 + n2 -> n + + e1 -> e'1 + ------------------- :: plus_l + e1 + e2 -> e'1 + e2 + + e -> e' + --------------- :: plus_r + n + e -> n + e' + + n1 * n2 = n + ------------ :: times + n1 * n2 -> n + + e1 -> e'1 + ------------------- :: times_l + e1 * e2 -> e'1 * e2 + + e -> e' + --------------- :: times_r + n * e -> n * e' + + e1 -> e'1 + --------------------------------------- :: let + let x := e1 in e2 -> let x := e'1 in e2 + + -------------------------------- :: bind + let x := n in e2 -> e2 [ n / x ] + +defn + e \||/ n :: :: eval :: eval_ + {{ com evaluates to }} by + + -------- :: num + n \||/ n + + e1 \||/ n1 + e2 \||/ n2 + n1 + n2 = n + -------------- :: plus + e1 + e2 \||/ n + + e1 \||/ n1 + e2 \||/ n2 + n1 * n2 = n + -------------- :: times + e1 * e2 \||/ n + + e1 \||/ n1 + e2 [ n1 / x ] \||/ n2 + ------------------------- :: let + let x := e1 in e2 \||/ n2 diff --git a/ott/functional.pdf b/ott/functional.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ff88c90c2adf9a4d9806aacffb678a61d6a30023 GIT binary patch literal 106355 zcma&MQ>-vt(5$&_+qP}*ciXmY+qP}nwr$(C?fK4R{z)chE>2%n->qaNRjJNHD*sQE zmXVGXigbQ?WF3l`fPuiy&=QJ=2Z~TI3u|W+M*@0LYXfJK zeqQJa2{V zKkDjpmv4z2%!;DAx2abn zAGTP#H>Z{jQno1ux)Ij3UZo$5N0*b95*3%sa3a>Qj8@h?h8+@;3DSw3BsS5Jm)Z=s z6p?2;j!t>fpc^dTR+5_bdT$tXbgVqF&CSsWm!$_|HeaczIvQ^34{W6W7=07`Bt=UZ z9vi$SteK>-b7Ul@P;lLDd@R>+Sess5B*o7|UfYek`nGOUyMAmx>N&X&Yiy#4eq*t7 z9~Ff&ae8SHh!ta@Y!x)b*mp_3%1Hu5Er)=KE8sAi+_mZ1pH^JAL%EPuFE5(v@#@I? 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b/ott/lam.ott @@ -0,0 +1,130 @@ +% all +metavar termvar, x, y ::= {{ com term variable }} +{{ isa string}} {{ coq nat}} {{ hol string}} {{ coq-equality }} +{{ ocaml int}} {{ lex alphanum}} {{ tex \mathit{[[termvar]]} }} + +grammar +t :: 't_' ::= {{ com term }} + | x :: :: Var {{ com variable}} + | \ x . t :: :: Lam (+ bind x in t +) {{ com lambda }} + | t t' :: :: App {{ com app }} + | ( t ) :: S:: Paren {{ icho [[t]] }} + | [ t / x ] t' :: M:: Tsub + {{ icho (tsubst_t [[t]] [[x]] [[t']])}} + +v :: 'v_' ::= {{ com value }} + | \ x . t :: :: Lam {{ com lambda }} + +terminals :: 'terminals_' ::= + | \ :: :: lambda {{ tex \lambda }} + | --> :: :: red {{ tex \longrightarrow }} + | in :: :: in {{ tex \in }} + | <> :: :: neq {{ tex \neq }} + | =a :: :: eqa {{ tex \equiv_\alpha }} + | =b :: :: eqb {{ tex \equiv_\beta }} + | FV :: :: FV {{ tex \mathrm{FV} }} + | notin :: :: notin {{ tex \notin }} + +formula :: formula_ ::= + {{ com formulas }} + | judgement :: :: judgement + | x <> x' :: M :: var_neq + | x notin FV(t) :: M :: notin_fv + +subrules + v <:: t + +substitutions + single t x :: tsubst + +defns +Jop :: '' ::= + + defn + t1 --> t2 :: ::reduce::'' {{ com $[[t1]]$ reduces to $[[t2]]$}} by + + + -------------------------- :: ax_app + (\x.t12) v2 --> [v2/x]t12 + + t1 --> t1' + -------------- :: ctx_app_fun + t1 t --> t1' t + + t1 --> t1' + -------------- :: ctx_app_arg + v t1 --> v t1' + + defn + x in FV ( t ) :: :: fv ::'' {{ com free variable }} by + + ---------- :: var + x in FV(x) + + x in FV(t1) + -------------- :: app_l + x in FV(t1 t2) + + x in FV(t2) + -------------- :: app_r + x in FV(t1 t2) + + x in FV(t) + x <> y + ------------- :: lam + x in FV(\y.t) + + defn + t =a t' :: :: aeq :: aeq_ {{ com alpha equivalence }} by + + ------ :: id + t =a t + + t =a t' + ------- :: sym + t' =a t + + t =a t' + t' =a t'' + --------- :: trans + t =a t'' + + t1 =a t1' + t2 =a t2' + ---------------- :: app + t1 t2 =a t1' t2' + + t =a t' + ------------- :: lam + \x.t =a \x.t' + + x' notin FV(t) + ------------------- :: subst + \x.t =a \x'.[x'/x]t + + defn + t =b t' :: :: beq :: beq_ {{ com beta equivalence }} by + + ------ :: id + t =b t + + t =b t' + ------- :: sym + t' =b t + + t =b t' + t' =b t'' + --------- :: trans + t =b t'' + + t1 =b t1' + t2 =b t2' + --------- :: app + t1 t2 =b t1' t2' + + t =b t' + ------------- :: lam + \x.t =b \x.t' + + --------------------- :: subst + (\x.t) t' =b [t'/x] t diff --git a/ott/lam.pdf b/ott/lam.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dbec272475d11a4444c8f15ace29312fe05f59c0 GIT binary patch literal 109015 zcma&NL$EN=lC61c+xU-d+qP}nwr$(CZQHhO>)m?M)zJ~vgUUfh=4g*HR<6izlgJB; z(lF4nK#|NZkE}y65zrIZ8CpVdb3@Tdo7kE;n-egwu&@&R-+-bMwXk+JaU`GT%8j5@JVH#?gwq?xtVMuzp+nY73n`Jg32uZ_h ze7c(U6bNPfM>C&({Y}}Ge)o8@vt-%zZvXqbk}-JecoKXdXWVkXW+`E;pl+3gs;4Mg zEnf_7Bj0pr)dJ+>2Fe~VPfPJnmpVo9sVs!yY30@R#UZDRnr;Hh#<~@RCfb%Y`*FW` zexBX^=X1so(!8X?TcwL}=Or}2*;=hZM~~-)0u@kFKvM)jBuj}n+S+Z*tMMRLa88Oo 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