"You know my methods. Apply them!"
Implementation
==============
- (#2 * x) = #2 * - x is not proved by arith
a simp command for terms
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primrec
"(foorec [] []) = []"
"(foorec [] (y # ys)) = (y # (foorec [] ys))"
*** Primrec definition error:
*** extra variables on rhs: "y", "ys"
*** in
*** "((foorec [] ((y::'a_1) # (ys::'a_1 list))) = (y # (foorec [] ys)))"
*** At command "primrec".
Bessere Fehlermeldung!
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Relation: comp -> composition
Add map_cong?? (upto 10% slower)
Recdef: Get rid of function name in header.
Support mutual recursion (Konrad?)
improve solver in simplifier: treat & and ! ...
better 1 point rules:
!x. !y. x = f y --> P x y should reduce to !y. P (f y) y.
use arith_tac in recdef to solve termination conditions?
-> new example in Recdef/termination
a tactic for replacing a specific occurrence:
apply(subst [2] thm)
it would be nice if @term could deal with ?-vars.
then a number of (unchecked!) @texts could be converted to @terms.
it would be nice if one could get id to the enclosing quotes in the [source] option.
More predefined functions for datatypes: map?
Induction rules for int: int_le/ge_induct?
Needed for ifak example. But is that example worth it?
Komischerweise geht das Splitten von _Annahmen_ auch mit simp_tac, was
ein generelles Feature ist, das man vielleicht mal abstellen sollte.
proper mutual simplification
defs with = and pattern matching??
Minor fixes in the tutorial
===========================
1/2 no longer abbrevs for Suc.
Warning: needs simplification to turn 1 into Suc 0. arith does so
automatically.
recdef: failed tcs no longer shown! (sec:Recursion over nested datatypes)
say something about how conditions are proved?
No, better show failed proof attempts.
Advanced recdef: explain recdef_tc? No. Adjust recdef!
Adjust FP textbook refs: new Bird, Hudak
Discrete math textbook: Rosen?
adjust type of ^ in tab:overloading
an example of induction: !y. A --> B --> C ??
Explain type_definition and mention pre-proved thms in subset.thy?
-> Types/typedef
Appendix: Lexical: long ids.
Warning: infixes automatically become reserved words!
Forward ref from blast proof of Puzzle (AdvancedInd) to Isar proof?
recdef with nested recursion: either an example or at least a pointer to the
literature. In Recdef/termination.thy, at the end.
%FIXME, with one exception: nested recursion.
Syntax section: syntax annotations not just for consts but also for constdefs and datatype.
Appendix with list functions.
All theory sources on the web?
Minor additions to the tutorial, unclear where
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unfold?
Possible exercises
==================
Exercises
For extensionality (in Sets chapter): prove
valif o norm = valif
in If-expression case study (Ifexpr)
Nested inductive datatypes: another example/exercise:
size(t) <= size(subst s t)?
insertion sort: primrec, later recdef
OTree:
first version only for non-empty trees:
Tip 'a | Node tree tree
Then real version?
First primrec, then recdef?
Ind. sets: define ABC inductively and prove
ABC = {rep A n @ rep B n @ rep C n. True}
Partial rekursive functions / Nontermination:
Exercise: ?! f. !i. f i = if i=0 then 1 else i*f(i-1)
(What about sum? Is there one, a unique one?)
Exercise
Better(?) sum i = fst(while (%(s,i). i=0) (%(s,i). (s+i,i-1)) (0,i))
Prove 0 <= i ==> sum i = i*(i+1) via while-rule
Possible examples/case studies
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Trie: Define functional version
datatype ('a,'b)trie = Trie ('b option) ('a => ('a,'b)trie option)
lookup t [] = value t
lookup t (a#as) = case tries t a of None => None | Some s => lookup s as
Maybe as an exercise?
Trie: function for partial matches (prefixes). Needs sets for spec/proof.
Sets via ordered list of intervals. (Isa/Interval(2))
propositional logic (soundness and completeness?),
predicate logic (soundness?),
Tautology checker. Based on Ifexpr or prop.logic?
Include forward reference in relevant section.
Sorting with comp-parameter and with type class (<)
Recdef:more example proofs:
if-normalization with measure function,
nested if-normalization,
quicksort
Trie?
New book by Bird?
Steps Towards Mechanizing Program Transformations Using PVS by N. Shankar,
Science of Computer Programming, 26(1-3):33-57, 1996.
You can get it from http://www.csl.sri.com/scp95.html
J Moore article Towards a ...
Mergesort, JVM
Additional topics
=================
Recdef with nested recursion?
Extensionality: applications in
- boolean expressions: valif o bool2if = value
- Advanced datatypes exercise subst (f o g) = subst f o subst g
A look at the library?
Map.
Prototyping?
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