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comparison src/Subset.v @ 82:15e2b3485dc4
Subset exercises
| author | Adam Chlipala <adamc@hcoop.net> |
|---|---|
| date | Sun, 05 Oct 2008 12:12:19 -0400 |
| parents | d07c77659c20 |
| children | 939add5a7db9 |
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| 81:f295a64bf9fd | 82:15e2b3485dc4 |
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| 756 | 756 |
| 757 = !! | 757 = !! |
| 758 : {t : type | hasType (Plus (Nat 1) (Bool false)) t} + | 758 : {t : type | hasType (Plus (Nat 1) (Bool false)) t} + |
| 759 {(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)} | 759 {(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)} |
| 760 ]] *) | 760 ]] *) |
| 761 | |
| 762 | |
| 763 (** * Exercises *) | |
| 764 | |
| 765 (** All of the notations defined in this chapter, plus some extras, are available for import from the module [MoreSpecif] of the book source. | |
| 766 | |
| 767 %\begin{enumerate}%#<ol># | |
| 768 %\item%#<li># Write a function of type [forall n m : nat, {n <= m} + {n > m}]. That is, this function decides whether one natural is less than another, and its dependent type guarantees that its results are accurate.#</li># | |
| 769 | |
| 770 %\item%#<li># %\begin{enumerate}%#<ol># | |
| 771 %\item%#<li># Define [var], a type of propositional variables, as a synonym for [nat].#</li># | |
| 772 %\item%#<li># Define an inductive type [prop] of propositional logic formulas, consisting of variables, negation, and binary conjunction and disjunction.#</li># | |
| 773 %\item%#<li># Define a function [propDenote] from variable truth assignments and [prop]s to [Prop], based on the usual meanings of the connectives. Represent truth assignments as functions from [var] to [bool].#</li># | |
| 774 %\item%#<li># Define a function [bool_true_dec] that checks whether a boolean is true, with a maximally expressive dependent type. That is, the function should have type [forall b, {b = true} + {b = true -> False}]. #</li># | |
| 775 %\item%#<li># Define a function [decide] that determines whether a particular [prop] is true under a particular truth assignment. That is, the function should have type [forall (truth : var -> bool) (p : prop), {propDenote truth p} + { ~propDenote truth p}]. This function is probably easiest to write in the usual tactical style, instead of programming with [refine]. [bool_true_dec] may come in handy as a hint.#</li># | |
| 776 %\item%#<li># Define a function [negate] that returns a simplified version of the negation of a [prop]. That is, the function should have type [forall p : prop, {p' : prop | forall truth, propDenote truth p <-> ~propDenote truth p'}]. To simplify a variable, just negate it. Simplify a negation by returning its argument. Simplify conjunctions and disjunctions using De Morgan's laws, negating the arguments recursively and switching the kind of connective. [decide] may be useful in some of the proof obligations, even if you do not use it in the computational part of [negate]'s definition. Lemmas like [decide] allow us to compensate for the lack of a general Law of the Excluded Middle in CIC.#</li># | |
| 777 #</ol>#%\end{enumerate}% #</li># | |
| 778 | |
| 779 %\item%#<li># Implement the DPLL satisfiability decision procedure for boolean formulas in conjunctive normal form, with a dependent type that guarantees its correctness. An example of a reasonable type for this function would be [forall f : formula, {truth : tvals | formulaTrue truth f} + {forall truth, ~formulaTrue truth f}]. Implement at least "the basic backtracking algorithm" as defined here: | |
| 780 %\begin{center}\url{http://en.wikipedia.org/wiki/DPLL_algorithm}\end{center}% | |
| 781 #<blockquote><a href="http://en.wikipedia.org/wiki/DPLL_algorithm">http://en.wikipedia.org/wiki/DPLL_algorithm</a></blockquote># | |
| 782 It might also be instructive to implement the unit propagation and pure literal elimination optimizations described there or some other optimizations that have been used in modern SAT solvers.#</li># | |
| 783 | |
| 784 #</ol>#%\end{enumerate}% *) |