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LambdaCalculus chapter: a nonterminating lambda term
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@ -81,6 +81,21 @@ Module Ulc.
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Notation "\ x , e" := (Abs x e) (at level 50).
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Notation "\ x , e" := (Abs x e) (at level 50).
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Infix "@" := App (at level 49, left associativity).
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Infix "@" := App (at level 49, left associativity).
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(* Believe it or not, this is a Turing-complete language! Here's an example
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* nonterminating program. *)
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Example omega := (\"x", "x" @ "x") @ (\"x", "x" @ "x").
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Theorem omega_no_eval : forall v, eval omega v -> False.
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Proof.
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induct 1.
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invert H.
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invert H0.
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simplify.
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apply IHeval3.
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trivial.
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Qed.
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(** * Church Numerals, everyone's favorite example of lambda terms in
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(** * Church Numerals, everyone's favorite example of lambda terms in
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* action *)
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* action *)
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@ -2095,6 +2095,17 @@ Substitute the argument value in the body of the abstraction, evaluate the resul
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Note that we only ever need to evaluate \emph{closed} terms\index{closed terms}, meaning terms that are not open, so we obey the restriction on substitution sketched above.
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Note that we only ever need to evaluate \emph{closed} terms\index{closed terms}, meaning terms that are not open, so we obey the restriction on substitution sketched above.
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It may be surprising that these two rules are enough to define the full semantics of a Turing-complete language!
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It may be surprising that these two rules are enough to define the full semantics of a Turing-complete language!
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Indeed, $\lambda$-calculus is Turing-complete, and we must be able to find nonterminating programs.
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Here is one example.
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\begin{eqnarray*}
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\Omega &=& (\lambda x. \; x \; x) \; (\lambda x. \; x \; x) \\
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\end{eqnarray*}
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\begin{theorem}
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$\Omega$ does not evaluate to anything. In other words, $\bigstep{\Omega}{v}$ implies a contradiction.
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\end{theorem}
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\begin{proof}
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By induction on the derivation of $\bigstep{\Omega}{v}$.
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\end{proof}
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\section{A Quick Case Study in Program Verification: Church Numerals}
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\section{A Quick Case Study in Program Verification: Church Numerals}
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