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Revising for tomorrow's lecture
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2 changed files with 6 additions and 10 deletions
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@ -220,7 +220,7 @@ Proof.
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* an induction hypothesis. Let's ask Coq to search its library for lemmas
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* that would justify such a rewrite, giving a pattern with wildcards, to
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* specify the essential structure that the rewrite should match. *)
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SearchRewrite ((_ ++ _) ++ _).
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Search ((_ ++ _) ++ _).
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(* Ah, we see just the one! *)
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rewrite app_assoc_reverse.
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rewrite IHe1.
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@ -251,8 +251,8 @@ Proof.
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(* To match the form of our lemma, we need to replace [compile e] with
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* [compile e ++ nil], adding a "pointless" concatenation of the empty list.
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* [SearchRewrite] again helps us find a library lemma. *)
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SearchRewrite (_ ++ nil).
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* [Search] again helps us find a library lemma. *)
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Search (_ ++ nil).
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rewrite (app_nil_end (compile e)).
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(* Note that we can use [rewrite] with explicit values of the first few
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* quantified variables of a lemma. Otherwise, [rewrite] picks an
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@ -711,7 +711,6 @@ We apply the following notations throughout the book: \\
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As the name advertises, finite maps are functions with finite domains, where the domain may be expanded by each extension operation.
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Two axioms explain the essential interactions of the basic operators.
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$$\infer{\msel{\mupd{m}{k}{v}}{k} = v}{}
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\quad
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\infer{\msel{\mupd{m}{k_1}{v}}{k_2} = m(k_2)}{
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@ -723,7 +722,7 @@ $$\infer{\msel{\mupd{m}{k}{v}}{k} = v}{}
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With these operators in hand, we can write a semantics for arithmetic expressions.
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This is a recursive function that \emph{maps variable valuations to numbers}.
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We write $\denote{e}$ for the meaning of $e$; this notation is often referred to as \emph{Oxford brackets}\index{Oxford brackets}.
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Recall that we allow notations like this as syntactic sugar for arbitrary functions, even when giving the equations that define those functions.
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Recall that we allow this kind of notation as syntactic sugar for arbitrary functions, even when giving the equations that define those functions.
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We write $v$ for a valuation (finite map).
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\encoding
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\begin{eqnarray*}
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@ -758,7 +757,6 @@ We can prove a key compatibility property of these two recursive functions.
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That is, in some sense, the operations of interpretation and substitution \emph{commute} with each other.
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That intuition gives rise to the common notion of a \emph{commuting diagram}\index{commuting diagram}, like the one below for this particular example.
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\[
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\begin{tikzcd}
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(e, v) \arrow{r}{\subst{\ldots}{x}{e'}} \arrow{d}{\mupd{\ldots}{x}{\denote{e'}v}} & (\subst{e}{x}{e'}, v) \arrow{d}{\denote{\ldots}} \\
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@ -916,7 +914,7 @@ We'll write $\opt{\ldots}$ for this and other optimizations, which move neither
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Note that, when multiple defining equations apply to some function input, by convention we apply the \emph{earliest} equation that matches.
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Let's prove that this optimization preserves program behavior; that is, we prove that it is \emph{semantics preserving}\index{semantics preservation}.
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Let's prove that this optimization preserves program behavior; that is, we prove that it is \emph{semantics-preserving}\index{semantics preservation}.
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\begin{theorem}\label{unroll}
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$\denote{\opt{c}}v = \denote{c}v$.
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@ -925,7 +923,6 @@ Let's prove that this optimization preserves program behavior; that is, we prove
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It all looks so straightforward from that statement, doesn't it?
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Indeed, there actually isn't so much work to do to prove this theorem.
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We can also present it as a commuting diagram much like the prior one.
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\[
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\begin{tikzcd}
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c \arrow{r}{\opt{\ldots}} \arrow{dr}{\denote{\ldots}} & \opt{c} \arrow{d}{\denote{\ldots}} \\
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@ -933,14 +930,13 @@ c \arrow{r}{\opt{\ldots}} \arrow{dr}{\denote{\ldots}} & \opt{c} \arrow{d}{\denot
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\end{tikzcd}
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\]
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The statement of Theorem \ref{unroll} happens to already be in the right form to do induction directly, but we need a helper lemma, capturing the interaction of $^nc$ and the semantics.
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The statement of Theorem \ref{unroll} happens to be already in the right form to do induction directly, but we need a helper lemma, capturing the interaction of $^nc$ and the semantics.
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\begin{lemma}
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$\denote{^nc} = \denote{c}^n$.
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\end{lemma}
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Let us end the chapter with the commuting-diagram version of the lemma statement.
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\[
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\begin{tikzcd}
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c \arrow{r}{^n\ldots} \arrow{d}{\denote{\ldots}} & ^nc \arrow{d}{\denote{\ldots}} \\
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