dds/utils.rkt

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#lang racket
;;; dds/utils
;;; Various utilities.
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(require
graph
(for-syntax syntax/parse racket/list))
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(provide
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;; Functions
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(contract-out [eval-with (-> variable-mapping? any/c any)]
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[extract-symbols (-> any/c list?)]
[any->string (-> any/c string?)]
[stringify-variable-mapping (-> variable-mapping? string-variable-mapping?)]
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[string->any (-> string? any/c)]
[read-org-sexp (-> string? (listof any/c))]
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[map-sexp (-> procedure? any/c any/c)]
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[unorg (-> string? (listof any/c))]
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[unstringify-pairs (-> (listof (general-pair/c string? any/c))
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(listof (general-pair/c symbol? any/c)))]
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[read-org-variable-mapping (-> string? variable-mapping?)]
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[unorgv (-> string? variable-mapping?)]
[dotit (-> graph? void?)]
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[read-symbol-list (-> string? (listof symbol?))]
[drop-first-last (-> string? string?)]
[list-sets->list-strings (-> (listof (set/c any/c)) (listof string?))]
[pretty-print-set-sets (-> (set/c (set/c symbol?) #:kind 'dont-care) string?)]
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[update-vertices/unweighted (-> graph? (-> any/c any/c) graph?)]
[update-graph (->* (graph?)
(#:v-func (-> any/c any/c)
#:e-func (-> any/c any/c))
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graph?)]
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[pretty-print-set (-> generic-set? string?)]
[collect-by-key (-> (listof any/c) (listof any/c) (values (listof any/c) (listof (listof any/c))))]
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[collect-by-key/sets (-> (listof any/c) (listof any/c) (values (listof any/c) (listof (set/c any/c))))]
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[ht-values/list->set (-> (hash/c any/c (listof any/c)) (hash/c any/c (set/c any/c)))]
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[hash->list/ordered (-> hash? (listof (cons/c any/c any/c)))]
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[multi-split-at (-> (listof (listof any/c)) number?
(values (listof (listof any/c)) (listof (listof any/c))))]
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[lists-transpose (-> (listof (listof any/c)) (listof (listof any/c)))]
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[procedure-fixed-arity? (-> procedure? boolean?)]
[in-random (case-> (-> (stream/c (and/c real? inexact? (>/c 0) (</c 1))))
(-> (integer-in 1 4294967087) (stream/c exact-nonnegative-integer?))
(-> exact-integer? (integer-in 1 4294967087)
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(stream/c exact-nonnegative-integer?)))]
[cartesian-product/stream (->* () #:rest (listof stream?) stream?)])
;; Contracts
(contract-out [variable-mapping? contract?]
[string-variable-mapping? contract?]
[general-pair/c (-> contract? contract? contract?)])
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;; Syntax
auto-hash-ref/explicit auto-hash-ref/:)
(module+ test
(require rackunit))
;;; ===================
;;; HashTable Injection
;;; ===================
;;; This section of the file contains some utilities to streamline the
;;; usage of hash tables mapping symbols to values. The goal is
;;; essentially to avoid having to write explicit hash-ref calls.
;;; A variable mapping is a hash table mapping symbols to values.
(define (variable-mapping? dict) (hash/c symbol? any/c))
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;;; Given a (HashTable Symbol a) and a sequence of symbols, binds
;;; these symbols to the values they are associated to in the hash
;;; table, then puts the body in the context of these bindings.
;;;
;;; > (let ([ht #hash((a . 1) (b . 2))])
;;; (auto-hash-ref/explicit (ht a b) (+ a (* 2 b))))
;;; 5
;;;
;;; Note that only one expression can be supplied in the body.
(define-syntax (auto-hash-ref/explicit stx)
(syntax-parse stx
[(_ (ht:id xs:id ...) body:expr)
#`(let #,(for/list ([x (syntax->list #'(xs ...))])
#`[#,x (hash-ref ht '#,x)])
body)]))
(module+ test
(test-case "auto-hash-ref/explicit"
(define mytable #hash((a . 3) (b . 4)))
(check-equal? (auto-hash-ref/explicit (mytable b a)
(* a b))
12)
(define ht #hash((a . #t) (b . #f)))
(check-equal? (auto-hash-ref/explicit (ht a b)
(and (not a) b))
#f)))
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;;; Given an expression and a (HashTable Symbol a), looks up the
;;; symbols with a leading semicolon and binds them to the value they
;;; are associated to in the hash table.
;;;
;;; > (let ([ht #hash((a . 1) (b . 2))])
;;; (auto-hash-ref/: ht (+ :a (* 2 :b))))
;;; 5
;;;
;;; Note that the symbol :a is matched to the key 'a in the hash
;;; table.
;;;
;;; Note that only one expression can be supplied in the body.
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(define-syntax (auto-hash-ref/: stx)
(syntax-parse stx
[(_ ht:id body)
(let* ([names/: (collect-colons (syntax->datum #'body))])
#`(let #,(for/list ([x names/:])
;; put x in the same context as body
#`[#,(datum->syntax #'body x)
(hash-ref ht '#,(strip-colon x))])
body))]))
(module+ test
(test-case "auto-hash-ref/:"
(define ht1 #hash((x . #t) (y . #t) (t . #f)))
(define z #t)
(check-equal? (auto-hash-ref/: ht1
(and :x (not :y) z (or (and :t) :x)))
#f)
(define ht2 #hash((a . 1) (b . 2)))
(check-equal? (auto-hash-ref/: ht2 (+ :a (* 2 :b)))
5)))
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;;; The helper functions for auto-hash-ref/:.
(begin-for-syntax
;; Collect all the symbols starting with a colon in datum.
(define (collect-colons datum)
(remove-duplicates
(flatten
(for/list ([token datum])
(cond
[(symbol? token)
(let ([name (symbol->string token)])
(if (eq? #\: (string-ref name 0))
token
'()))]
[(list? token)
(collect-colons token)]
[else '()])))))
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;; Strip the leading colon off x.
(define (strip-colon x)
(let ([x-str (symbol->string x)])
(if (eq? #\: (string-ref x-str 0))
(string->symbol (substring x-str 1))
x))))
;;; Temporarily injects the mappings from the given hash table as
;;; bindings in a namespace including racket/base and then evaluates
;;; the expression.
;;;
;;; > (let ([ht #hash((a . 1) (b . 1))])
;;; (eval-with ht '(+ b a 1)))
;;; 3
;;;
;;; The local bindings from the current lexical scope are not
;;; conserved. Therefore, the following outputs an error about a
;;; missing identifier:
;;;
;;; > (let ([ht #hash((a . 1) (b . 1))]
;;; [z 1])
;;; (eval-with ht '(+ b z a 1)))
;;;
(define (eval-with ht expr)
(parameterize ([current-namespace (make-base-namespace)])
(for ([(x val) ht]) (namespace-set-variable-value! x val))
(eval expr)))
(module+ test
(test-case "eval-with"
(check-equal? (let ([ht #hash((a . 1) (b . 1))])
(eval-with ht '(+ b a 1)))
3)))
;;; Same as eval-with, but returns only the first value produced by
;;; the evaluated expression.
(define (eval-with1 ht expr)
(let ([vals (call-with-values (λ () (eval-with ht expr))
(λ vals vals))])
(car vals)))
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;;; ==============================
;;; Analysis of quoted expressions
;;; ==============================
;;; Produces a list of symbols appearing in the quoted expression
;;; passed in the first argument.
(define (extract-symbols form)
(match form
[(? symbol?) (list form)]
[(? list?) (flatten (for/list ([x form])
(extract-symbols x)))]
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[else '()]))
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(module+ test
(test-case "extract-symbols"
(check-equal? (extract-symbols '(1 (2 3) x (y z 3)))
'(x y z))))
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;;; =========================
;;; Interaction with Org-mode
;;; =========================
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;;; Org-mode supports laying out the output of code blocks as tables,
;;; which is very practical for various variable mappings (e.g.,
;;; states). However, when the hash table maps variables to lists,
;;; Org-mode will create a column per list element, which may or may
;;; not be the desired effect. In this section I define some
;;; utilities for nicer interoperation with Org-mode tables. I also
;;; define some shortcuts to reduce the number of words to type when
;;; using dds with Org-mode. See example.org for examples of usage.
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;;; Converts any value to string.
(define (any->string x)
(with-output-to-string (λ () (display x))))
(module+ test
(test-case "any->string"
(check-equal? (any->string 'a) "a")
(check-equal? (any->string '(a 1 (x y))) "(a 1 (x y))")
(check-equal? (any->string "hello") "hello")))
;;; A string variable mapping is a mapping from variables to strings.
(define (string-variable-mapping? dict) (hash/c symbol? string?))
;;; Converts all the values of a variable mapping to string.
(define (stringify-variable-mapping ht)
(for/hash ([(key val) ht]) (values key (any->string val))))
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(module+ test
(test-case "stringify-variable-mapping"
(define mp (stringify-variable-mapping #hash((a . (and a b)) (b . (not b)))))
(check-equal? (hash-ref mp 'a) "(and a b)")
(check-equal? (hash-ref mp 'b) "(not b)")))
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;;; Reads any value from string.
(define (string->any str)
(with-input-from-string str (λ () (read))))
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(module+ test
(test-case "string->any"
(check-equal? (string->any "(or b (not a))") '(or b (not a)))
(check-equal? (string->any "14") 14)))
;;; Given a sexp, converts all "#f" to #f and "#t" to #t.
;;;
;;; When I read Org-mode tables, I pump them through a call to the
;;; prin1 because the elisp sexp seems incompatible with Racket. On
;;; the other hand, Racket Booleans seem to upset elisp a little, so
;;; prin1 wraps them in additional double quotes. This function
;;; removes those quotes.
(define/match (handle-org-booleans datum)
[("#t") #t]
[("#f") #f]
[((? list?)) (map handle-org-booleans datum)]
[(_) datum])
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;;; Given a sexp, applies the given function to any object which is
;;; not a list.
;;;
;;; The contract of this function will not check whether func is
;;; indeed applicable to every non-list element of the sexp. If this
;;; is not the case, a contract violation for func will be generated.
(define (map-sexp func sexp)
(match sexp
[(? list?) (map ((curry map-sexp) func) sexp)]
[datum (func datum)]))
(module+ test
(test-case "map-sexp"
(check-equal? (map-sexp add1 '(1 2 (4 10) 3)) '(2 3 (5 11) 4))))
;;; Reads a sexp from a string produced by Org-mode for a named table.
;;; See example.org for examples.
(define read-org-sexp
(compose ((curry map-sexp) (match-lambda
[(and (? string?) str) (string->any str)]
[x x]))
string->any))
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;;; A shortcut for read-org-sexp.
(define unorg read-org-sexp)
(module+ test
(test-case "read-org-sexp"
(check-equal? (read-org-sexp "((\"a\" \"(and a b)\") (\"b\" \"(or b (not a))\"))")
'((a (and a b)) (b (or b (not a)))))
(check-equal? (read-org-sexp "(#t \"#t\" \"#t \" '(1 2 \"#f\"))")
'(#t #t #t '(1 2 #f)))))
;;; A contract allowing pairs constructed via cons or via list.
(define (general-pair/c key-contract val-contract)
(or/c (list/c key-contract val-contract)
(cons/c key-contract val-contract)))
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;;; Given a list of pairs of strings and some other values (possibly
;;; strings), converts the first element of each pair to a string, and
;;; reads the second element with string->any or keeps it as is if it
;;; is not a string.
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(define (unstringify-pairs pairs)
(for/list ([p pairs])
(match p
[(list key val)
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(cons (string->symbol key) (if (string? val)
(string->any val)
val))]
[(cons key val) ; also handle improper pairs
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(cons (string->symbol key) (if (string? val)
(string->any val)
val))])))
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(module+ test
(test-case "unstringify-pairs"
(check-equal? (unstringify-pairs '(("a" . "1") ("b" . "(and a (not b))")))
'((a . 1) (b . (and a (not b)))))
(check-equal? (unstringify-pairs '(("a" . 1) ("b" . "(and a (not b))")))
'((a . 1) (b . (and a (not b)))))))
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;;; Reads a variable mapping from a string, such as the one which
;;; Org-mode produces from tables.
(define read-org-variable-mapping
(compose make-immutable-hash unstringify-pairs string->any))
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(module+ test
(test-case "read-org-variable-mapping"
(define m1 (read-org-variable-mapping "((\"a\" \"(and a b)\") (\"b\" \"(or b (not a))\"))"))
(define m2 (read-org-variable-mapping "((\"a\" . \"(and a b)\") (\"b\" . \"(or b (not a))\"))"))
(define m3 (unorgv "((\"a\" . \"(and a b)\") (\"b\" . \"(or b (not a))\"))"))
(check-equal? (hash-ref m1 'a) '(and a b))
(check-equal? (hash-ref m2 'a) '(and a b))
(check-equal? (hash-ref m3 'a) '(and a b))
(check-equal? (hash-ref m1 'b) '(or b (not a)))
(check-equal? (hash-ref m2 'b) '(or b (not a)))
(check-equal? (hash-ref m3 'b) '(or b (not a)))))
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;;; A synonym for read-org-variable-mapping.
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(define unorgv read-org-variable-mapping)
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;;; Typeset the graph via graphviz and display it.
(define dotit (compose display graphviz))
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;;; Reads a list of symbols from a string.
(define (read-symbol-list str)
(string->any (string-append "(" str ")")))
(module+ test
(test-case "read-symbol-list"
(check-equal? (read-symbol-list "a b c") '(a b c))))
;;; Removes the first and the last symbol of a given string.
;;;
;;; Useful for removing the parentheses in string representations of
;;; lists.
(define (drop-first-last str)
(substring str 1 (- (string-length str) 1)))
(module+ test
(test-case "drop-first-last"
(check-equal? (drop-first-last "(a b)") "a b")))
;;; Converts a list of sets of symbols to a list of strings containing
;;; those symbols.
(define (list-sets->list-strings lst)
(map (compose drop-first-last any->string set->list) lst))
(module+ test
(test-case "list-sets->list-strings"
(check-equal? (list-sets->list-strings (list (set 'x 'y) (set 'z) (set) (set 't)))
'("y x" "z" "" "t"))))
;;; Pretty-prints a set of sets of symbols.
;;;
;;; Typically used for pretty-printing the annotations on the edges of
;;; the state graph.
(define (pretty-print-set-sets ms)
(string-join (for/list ([m ms]) (format "{~a}" (pretty-print-set m))) ""))
(module+ test
(test-case "pretty-print-set-sets"
(check-equal? (pretty-print-set-sets (set (set 'a 'b) (set 'c))) "{a b}{c}")))
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;;; ==========================
;;; Additional graph utilities
;;; ==========================
;;; Apply a transformation to every vertex in the unweighted graph,
;;; return the new graph. If the transformation function maps two
;;; vertices to the same values, these vertices will be merged in the
;;; resulting graph. The transformation function may be called
;;; multiple times for the same vertex.
;;;
;;; This function does not rely on rename-vertex!, so it can be used
;;; to permute vertex labels.
(define (update-vertices/unweighted gr func)
(unweighted-graph/directed
(for/list ([e (in-edges gr)])
(match-let ([(list u v) e])
(list (func u) (func v))))))
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(module+ test
(test-case "update-vertices/unweighted"
(define gr1 (directed-graph '((a b) (b c))))
(define gr2 (undirected-graph '((a b) (b c))))
(define dbl (λ (x) (let ([x-str (symbol->string x)])
(string->symbol (string-append x-str x-str)))))
(define new-gr1 (update-vertices/unweighted gr1 dbl))
(define new-gr2 (update-vertices/unweighted gr2 dbl))
(check-false (has-vertex? new-gr1 'a))
(check-true (has-vertex? new-gr1 'aa))
(check-false (has-vertex? new-gr1 'b))
(check-true (has-vertex? new-gr1 'bb))
(check-false (has-vertex? new-gr1 'c))
(check-true (has-vertex? new-gr1 'cc))
(check-true (has-edge? new-gr1 'aa 'bb))
(check-true (has-edge? new-gr1 'bb 'cc))
(check-true (has-edge? new-gr2 'aa 'bb))
(check-true (has-edge? new-gr2 'bb 'aa))
(check-true (has-edge? new-gr2 'bb 'cc))
(check-true (has-edge? new-gr2 'cc 'bb))))
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;;; Given a graph, apply a transformation v-func to every vertex label
;;; and, if the graph is a weighted graph, the transformation e-func
;;; to every edge label. Both transformations default to identity
;;; functions. If gr is an weighted graph, the result is a weighted
;;; graph. If gr is an unweighted graph, the result is an unweighted
;;; graph.
(define (update-graph gr
#:v-func [v-func identity]
#:e-func [e-func identity])
(define edges
(for/list ([e (in-edges gr)])
(match-let ([(list u v) e])
(cond
[(unweighted-graph? gr) (list (v-func u) (v-func v))]
[else (list (e-func (edge-weight gr u v))
(v-func u) (v-func v))]))))
(cond
[(unweighted-graph? gr) (unweighted-graph/directed edges)]
[else
(weighted-graph/directed edges)]))
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(module+ test
(test-case "update-graph"
(define gr1 (directed-graph '((a b) (b c))))
(define gr2 (undirected-graph '((a b) (b c))))
(define dbl (λ (x) (let ([x-str (symbol->string x)])
(string->symbol (string-append x-str x-str)))))
(define new-gr1-ug (update-graph gr1 #:v-func dbl))
(define new-gr2-ug (update-graph gr2 #:v-func dbl))
(define gr3 (weighted-graph/directed '((10 a b) (11 b c))))
(define new-gr3 (update-graph gr3 #:v-func dbl #:e-func (λ (x) (* 2 x))))
(check-false (has-vertex? new-gr1-ug 'a))
(check-true (has-vertex? new-gr1-ug 'aa))
(check-false (has-vertex? new-gr1-ug 'b))
(check-true (has-vertex? new-gr1-ug 'bb))
(check-false (has-vertex? new-gr1-ug 'c))
(check-true (has-vertex? new-gr1-ug 'cc))
(check-true (has-edge? new-gr1-ug 'aa 'bb))
(check-true (has-edge? new-gr1-ug 'bb 'cc))
(check-true (has-edge? new-gr2-ug 'aa 'bb))
(check-true (has-edge? new-gr2-ug 'bb 'aa))
(check-true (has-edge? new-gr2-ug 'bb 'cc))
(check-true (has-edge? new-gr2-ug 'cc 'bb))
(check-true (has-edge? new-gr3 'aa 'bb))
(check-false (has-edge? new-gr3 'bb 'aa))
(check-true (has-edge? new-gr3 'bb 'cc))
(check-false (has-edge? new-gr3 'cc 'bb))
(check-equal? (edge-weight new-gr3 'aa 'bb) 20)
(check-equal? (edge-weight new-gr3 'bb 'cc) 22)))
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;;; ===============
;;; Pretty printing
;;; ===============
;;; Pretty print a set by listing its elements in alphabetic order.
(define (pretty-print-set s)
(string-join (sort (set-map s any->string) string<?)))
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(module+ test
(test-case "pretty-print-set"
(check-equal? (pretty-print-set (set 'a 'b 1)) "1 a b")))
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;;; =========================
;;; Additional list utilities
;;; =========================
;;; Collects labels for duplicate edges into a sets of labels.
;;;
;;; More precisely, given a list of edges and weights, produces a new
;;; list of edges without duplicates, and a list of lists of weights
;;; in which each element corresponds to the edge (the input is
;;; suitable for graph constructors).
(define (collect-by-key edges labels)
(for/fold ([ht (make-immutable-hash)]
#:result (values (hash-keys ht) (hash-values ht)))
([e edges] [l labels])
(hash-update ht e (λ (ls) (cons l ls)) empty)))
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(module+ test
(test-case "collect-by-key"
(define-values (e1 l1) (collect-by-key '((1 2) (1 3)) '(a b)))
(define-values (e2 l2) (collect-by-key '((1 2) (1 2)) '(a b)))
(check-equal? e1 '((1 2) (1 3))) (check-equal? l1 '((a) (b)))
(check-equal? e2 '((1 2))) (check-equal? l2 '((b a)))))
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;;; Like collect-by-key, but returns a list of sets of weights.
(define (collect-by-key/sets edges labels)
(let-values ([(es ls) (collect-by-key edges labels)])
(values es (map list->set ls))))
(module+ test
(test-case "collect-by-key/sets"
(define-values (e3 l3) (collect-by-key/sets '(a b a) '(1 2 1)))
(check-equal? e3 '(a b)) (check-equal? l3 (list (set 1) (set 2)))))
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;;; Converts the values of a hash table from lists to sets.
(define (ht-values/list->set ht)
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(for/hash ([(k v) (in-hash ht)])
(values k (list->set v))))
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(module+ test
(test-case "ht-values/list->set"
(check-equal? (ht-values/list->set #hash((a . (1 1))))
(hash 'a (set 1)))))
;;; Returns the key-value pairs of a given hash table in the order in
;;; which the hash table orders them for hash-map and hash-for-each.
(define (hash->list/ordered ht) (hash-map ht cons #t))
(module+ test
(test-case "hash->list/ordered"
(check-equal? (hash->list/ordered #hash((b . 1) (a . 1)))
'((a . 1) (b . 1)))))
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;;; Given a list of lists, splits every single list at the given
;;; position, and then returns two lists: one consisting of the first
;;; halves, and the one consisting of the second halves.
(define (multi-split-at lsts pos)
(define (split-1 lst res)
(define-values (l r) (split-at lst pos))
(match res [(cons left right)
(cons (cons l left) (cons r right))]))
(match (foldr split-1 (cons '() '()) lsts)
[(cons lefts rights) (values lefts rights)]))
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(module+ test
(test-case "multi-split-at"
(define-values (l1 l2) (multi-split-at '((1 2 3) (a b c)) 2))
(check-equal? l1 '((1 2) (a b))) (check-equal? l2 '((3) (c)))))
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;;; Given a list of lists of the same length, transposes them.
;;;
;;; > (lists-transpose '((1 2) (a b)))
;;; '((1 a) (2 b))
;;;
;;; This function is essentially in-parallel, wrapped in a couple
;;; conversions.
(define lists-transpose
(compose sequence->list
in-values-sequence
((curry apply) in-parallel)))
(module+ test
(test-case "lists-transpose"
(check-equal? (lists-transpose '((1 2) (a b))) '((1 a) (2 b)))))
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;;; =========
;;; Functions
;;; =========
;;; Returns #t if the function has fixed arity (i.e. if it does not
;;; take a variable number of arguments).
(define (procedure-fixed-arity? func)
(match (procedure-arity func)
[(arity-at-least _) #f] [arity #t]))
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(module+ test
(test-case "procedure-fixed-arity?"
(check-true (procedure-fixed-arity? not))
(check-false (procedure-fixed-arity? +))))
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;;; ==========
;;; Randomness
;;; ==========
;;; Generates a stream of inexact random numbers. The meaning of the
;;; arguments is the same as for the function random:
;;;
;;; (in-randoms k) — a sequence of random exact integers in the range
;;; 0 to k-1.
;;;
;;; (in-randoms min max) — a sequence of random exact integers the
;;; range min to max-1.
;;;
;;; (in-randoms) — a sequence of random inexact numbers between
;;; 0 and 1.
(define in-random
(case-lambda
[() (for/stream ([i (in-naturals)]) (random))]
[(k) (for/stream ([i (in-naturals)]) (random k))]
[(min max) (for/stream ([i (in-naturals)]) (random min max))]))
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(module+ test
(test-case "in-random"
(random-seed 0)
(check-equal? (stream->list (stream-take (in-random 100) 10))
'(85 65 20 40 89 45 54 38 26 62))
(check-equal? (stream->list (stream-take (in-random 50 100) 10))
'(75 59 82 85 61 85 59 64 75 53))
(check-equal? (stream->list (stream-take (in-random) 10))
'(0.1656109603231493
0.9680391127132195
0.051518813640790355
0.755901955353936
0.5923534604277275
0.5513340634474264
0.7022057040731392
0.48375400938578744
0.7538961707172924
0.01828428516237329))))
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;;; ===========================
;;; Additional stream utilities
;;; ===========================
;;; Returns the Cartesian product of the given streams. The result is
;;; a stream whose elements are the elements of the Cartesian product.
;;;
;;; The implementation is inspired from the implementation of
;;; cartesian-product in racket/list.
(define (cartesian-product/stream . ss)
;; Cartesian product of two streams, produces an improper pair.
(define (cp-2 ss1 ss2)
(for*/stream ([s1 (in-stream ss1)] [s2 (in-stream ss2)]) (cons s1 s2)))
;; Fold-right over the list of streams. The value for the fold is a
;; 1-value stream containing the empty list, which makes all the
;; lists proper.
(foldr cp-2 (sequence->stream (in-value (list))) ss))
(module+ test
(test-case "cartesian-product/stream"
(check-equal? (stream->list (cartesian-product/stream (in-range 3) (in-range 4 6) '(a b)))
'((0 4 a)
(0 4 b)
(0 5 a)
(0 5 b)
(1 4 a)
(1 4 b)
(1 5 a)
(1 5 b)
(2 4 a)
(2 4 b)
(2 5 a)
(2 5 b)))))