diff options
| -rw-r--r-- | CHANGES | 11 | ||||
| -rw-r--r-- | doc/refman/RefMan-tac.tex | 85 |
2 files changed, 90 insertions, 6 deletions
@@ -6,6 +6,17 @@ Tactics - New tactic "extensionality in H" which applies (possibly dependent) functional extensionality in H supposed to be a quantified equality until giving a bare equality. +- New tactic "inversion_sigma" which turns equalities of dependent + pairs (e.g., "existT P x p = existT P y q", frequently left over by + "inversion" on a dependent type family) into pairs of equalities + (e.g., a hypothesis "H : x = y" and a hypothesis of type "rew H in p + = q"); these hypotheses can subsequently be simplified using + "subst", without ever invoking any kind of axiom asserting + uniqueness of identity proofs. If you want to explicitly specify the + hypothesis to be inverted, or name the generated hypotheses, you can + invoke "induction H as [H1 H2] using eq_sigT_rect". The tactic also + works for "sig", "sigT2", and "sig2", and there are similar + "eq_sig*_rect" induction lemmas. - Tactic "specialize with ..." now accepts any partial bindings. Missing bindings are either solved by unification or left quantified in the hypothesis. diff --git a/doc/refman/RefMan-tac.tex b/doc/refman/RefMan-tac.tex index 4931ca3b6..ecb89b5fb 100644 --- a/doc/refman/RefMan-tac.tex +++ b/doc/refman/RefMan-tac.tex @@ -352,7 +352,7 @@ Section~\ref{pattern} to transform the goal so that it gets the form The tactic {\tt eapply} behaves like {\tt apply} but it does not fail when no instantiations are deducible for some variables in the - premises. Rather, it turns these variables into + premises. Rather, it turns these variables into existential variables which are variables still to instantiate (see Section~\ref{evars}). The instantiation is intended to be found later in the proof. @@ -1411,7 +1411,7 @@ in the list of subgoals remaining to prove. quantifications or non-dependent implications) are instantiated by concrete terms coming either from arguments \term$_1$ $\ldots$ \term$_n$ or from a bindings list (see - Section~\ref{Binding-list} for more about bindings lists). + Section~\ref{Binding-list} for more about bindings lists). In the first form the application to \term$_1$ {\ldots} \term$_n$ can be partial. The first form is equivalent to {\tt assert ({\ident} := {\ident} {\term$_1$} \dots\ \term$_n$)}. @@ -2604,6 +2604,21 @@ Abort. This tactic behaves as generalizing \ident$_1$\dots\ \ident$_n$, then doing \texttt{inversion {\ident} using \ident$'$}. +\item \tacindex{inversion\_sigma} \texttt{inversion\_sigma} + + This tactic turns equalities of dependent pairs (e.g., + \texttt{existT P x p = existT P y q}, frequently left over by + \texttt{inversion} on a dependent type family) into pairs of + equalities (e.g., a hypothesis \texttt{H : x = y} and a hypothesis + of type \texttt{rew H in p = q}); these hypotheses can subsequently + be simplified using \texttt{subst}, without ever invoking any kind + of axiom asserting uniqueness of identity proofs. If you want to + explicitly specify the hypothesis to be inverted, or name the + generated hypotheses, you can invoke \texttt{induction H as [H1 H2] + using eq\_sigT\_rect}. This tactic also works for \texttt{sig}, + \texttt{sigT2}, and \texttt{sig2}, and there are similar + \texttt{eq\_sig\emph{*}\_rect} induction lemmas. + \end{Variants} \firstexample @@ -2698,6 +2713,64 @@ dependent inversion_clear H. Note that \texttt{H} has been substituted by \texttt{(LeS n m0 l)} and \texttt{m} by \texttt{(S m0)}. +\example{Using \texorpdfstring{\texttt{inversion\_sigma}}{inversion\_sigma}} + +Let us consider the following inductive type of length-indexed lists, +and a lemma about inverting equality of \texttt{cons}: + +\begin{coq_eval} +Reset Initial. +Set Printing Compact Contexts. +\end{coq_eval} + +\begin{coq_example*} +Require Coq.Logic.Eqdep_dec. + +Inductive vec A : nat -> Type := +| nil : vec A O +| cons {n} (x : A) (xs : vec A n) : vec A (S n). + +Lemma invert_cons : forall A n x xs y ys, + @cons A n x xs = @cons A n y ys + -> xs = ys. +Proof. +\end{coq_example*} + +\begin{coq_example} +intros A n x xs y ys H. +\end{coq_example} + +After performing \texttt{inversion}, we are left with an equality of +\texttt{existT}s: + +\begin{coq_example} +inversion H. +\end{coq_example} + +We can turn this equality into a usable form with +\texttt{inversion\_sigma}: + +\begin{coq_example} +inversion_sigma. +\end{coq_example} + +To finish cleaning up the proof, we will need to use the fact that +that all proofs of \texttt{n = n} for \texttt{n} a \texttt{nat} are +\texttt{eq\_refl}: + +\begin{coq_example} +let H := match goal with H : n = n |- _ => H end in +pose proof (Eqdep_dec.UIP_refl_nat _ H); subst H. +simpl in *. +\end{coq_example} + +Finally, we can finish the proof: + +\begin{coq_example} +assumption. +Qed. +\end{coq_example} + \subsection{\tt fix {\ident} {\num}} \tacindex{fix} \label{tactic:fix} @@ -2998,7 +3071,7 @@ activated, {\tt subst} also deals with the following corner cases: \item The presence of a recursive equation which without the option would be a cause of failure of {\tt subst}. - + \item A context with cyclic dependencies as with hypotheses {\tt \ident$_1$ = f~\ident$_2$} and {\tt \ident$_2$ = g~\ident$_1$} which without the option would be a cause of failure of {\tt subst}. @@ -3293,7 +3366,7 @@ a sort of strong normalization with two key differences: \begin{itemize} \item They unfold a constant if and only if it leads to a $\iota$-reduction, i.e. reducing a match or unfolding a fixpoint. -\item While reducing a constant unfolding to (co)fixpoints, +\item While reducing a constant unfolding to (co)fixpoints, the tactics use the name of the constant the (co)fixpoint comes from instead of the (co)fixpoint definition in recursive calls. @@ -4024,7 +4097,7 @@ Abort. & & e * & \text{ Kleene star } \\ & & \texttt{emp} & \text{ empty } \\ & & \texttt{eps} & \text{ epsilon } \\ - & & \texttt{(} e \texttt{)} & + & & \texttt{(} e \texttt{)} & \end{array}\] The \texttt{emp} regexp does not match any search path while @@ -5188,7 +5261,7 @@ Reset Initial. \subsection[\tt swap \num$_1$ \num$_2$]{\tt swap \num$_1$ \num$_2$\tacindex{swap}} -This tactic switches the position of the goals of indices $\num_1$ and $\num_2$. If either $\num_1$ or $\num_2$ is negative then goals are counted from the end of the focused goal list. Goals are indexed from $1$, there is no goal with position $0$. +This tactic switches the position of the goals of indices $\num_1$ and $\num_2$. If either $\num_1$ or $\num_2$ is negative then goals are counted from the end of the focused goal list. Goals are indexed from $1$, there is no goal with position $0$. \Example \begin{coq_example*} |