Rertite a M (b | a) = b U a and a R (b | a) == b W a.

* src/ltlvisit/simplify.cc: Here.
* src/ltltest/reduccmp.test: Test it.
* doc/tl/tl.tex: Document it.

doc/tl/tl.tex

@@ -1322,7 +1322,8 @@ $b$ denotes a Boolean formula.
   f \M(\F f)          & \equiv \F f            & f \R(\F f)          & \equiv \F f            \\
   f \U (g \OR \G(f))  & \equiv f\W g           & f \W (g \OR \G(f))  & \equiv f\W g           \\
   f \M (g \AND \F(f)) & \equiv f\M g           & f \R (g \AND \F(f)) & \equiv f\M g           \\
-  f \U (g \AND f)     & \equiv g\M f           & f \W (g \AND f)     & \equiv g\R f
+  f \U (g \AND f)     & \equiv g\M f           & f \W (g \AND f)     & \equiv g\R f           \\
+  f \M (g \OR f)      & \equiv g\U f           & f \R (g \OR f)      & \equiv g\W f
 \end{align*}
 
 Here are the basic rewriting rules for $n$-ary operators ($\AND$ and
@@ -1618,23 +1619,19 @@ sometimes generalized to support operators such as $\M$ and $\W$.
 The operators \samp{F}, \samp{G}, \samp{U}, \samp{R}, \samp{M}, and
 \samp{W} can all be defined using only Boolean operators, \samp{X},
 and one of \samp{U}, \samp{W}, \samp{R}, or \samp{M}.  This property
-is usually used to simplify proofs.
-
-These equivalences can also help to understand the semantics of
-section~\ref{sec:opltl:sem} if you are only familiar with some of the
-operators.
-
-
+is usually used to simplify proofs.  These equivalences can also help
+to understand the semantics of section~\ref{sec:opltl:sem} if you are
+only familiar with some of the operators.
 {\allowdisplaybreaks
 \begin{align*}
 \intertext{Equivalences using $\U$:}
   \F f &\equiv \1\U f\\
   \G f &\equiv \NOT\F\NOT f\equiv \NOT(\1\U\NOT f)\\
-  f\W g &\equiv (f\U g)\OR(\G f)\equiv (f\U g)\OR\NOT(\1\U\NOT f)\\
+  f\W g &\equiv (f\U g)\OR\G f\equiv (f\U g)\OR\NOT(\1\U\NOT f)\\
         &\equiv f\U (g\OR \G f)\equiv f\U (g\OR\NOT(\1\U\NOT f))\\
   f\M g &\equiv g \U (f\AND g)\\
   f\R g &\equiv g \W (f\AND g) \equiv (g \U (f\AND g))\OR\NOT(\1\U\NOT g)\\
-        &\phantom{\equiv g \W (f\AND g)} \equiv g \U ((f\AND g)\OR\NOT(\1\U\NOT g))\\
+        &\phantom{{}\equiv g \W (f\AND g)} \equiv g \U ((f\AND g)\OR\NOT(\1\U\NOT g))\\
 \intertext{Equivalences using $\W$:}
   \F f &\equiv \NOT\G\NOT f\equiv \NOT((\NOT f)\W\0)\\
   \G f &\equiv \0\R f \equiv f\W \0\\
@@ -1645,13 +1642,15 @@ operators.
   \F f &\equiv \NOT\G\NOT f\equiv \NOT (\0\R\NOT f)\\
   \G f &\equiv \0 \R f \\
   f\U g&\equiv (((\X g) \R f)\AND\F g)\OR g \equiv (((\X g) \R f)\AND(\NOT (\0\R\NOT g)))\OR g \\
-  f \W g&\equiv ((\X g)\R f)\OR g\\
+  f \W g&\equiv g\R(f\OR g)\\
+        &\equiv ((\X g)\R f)\OR g\\
   f \M g&\equiv (f \R g)\AND\F f\equiv (f \R g)\AND\NOT (\0\R\NOT f)\\
         &\equiv f \R (g\AND\F g)\equiv f \R (g\AND\NOT (\0\R\NOT f))\\
 \intertext{Equivalences using $\M$:}
   \F f &\equiv f\M\1\\
   \G f &\equiv \NOT\F\NOT f \equiv \NOT((\NOT f)\M\1)\\
-  f\U g&\equiv ((\X g)\M f)\OR g \\
+  f\U g&\equiv g\M (f\OR g) \\
+       &\equiv ((\X g)\M f)\OR g \\
   f \W g&\equiv (f\U g)\OR \G f \equiv ((\X g)\M f)\OR g\OR \NOT((\NOT f)\M\1)\\
   f \R g&\equiv (f \M g)\OR\G g \equiv (f \M g)\OR \NOT((\NOT g)\M\1)
 \end{align*}}
@@ -1674,7 +1673,7 @@ to have trouble when the $\X$ operator is involved.  For instance $(f
 which looks like it should be reduced similarly to $f \U \X g$, will
 be rewritten instead to $(f \W \X g) \AND \X\F g$, because $\X\F g
 \equiv \F\X g$ is another rule that gets applied first during the
-bottom up rewriting.
+bottom-up rewriting.
 
 \chapter{Syntactic Implications}\label{ann:syntimpl}
 

src/ltltest/reduccmp.test

@@ -202,6 +202,10 @@ for x in ../reduccmp ../reductaustr; do
      run 0 $x 'a|(c&X((b&c) W a)&b)|d' '((b&c) W a)|d'
      run 0 $x 'a&(c|b|X((b|c) M a))&d' '((b|c) M a)&d'
      run 0 $x 'a&(c|X((b|c) R a)|b)&d' '((b|c) R a)&d'
+     run 0 $x 'g R (f|g|h)' '(f|h) W g'
+     run 0 $x 'g M (f|g|h)' '(f|h) U g'
+     run 0 $x 'g U (f&g&h)' '(f&h) M g'
+     run 0 $x 'g W (f&g&h)' '(f&h) R g'
 
      # Syntactic implication
      run 0 $x '(a & b) R (a R c)' '(a & b)R c'

src/ltlvisit/simplify.cc

@@ -2156,10 +2156,16 @@ namespace spot
 
 		    // a R (b & c & F(a)) = a M (b & c)
 		    // a M (b & c & F(a)) = a M (b & c)
+		    // a M (b | a | c) == (b | c) U a
+		    // a R (b | a | c) == (b | c) W a
 		    if (b->kind() == formula::MultOp)
 		      {
 			multop* fm2 = static_cast<multop*>(b);
-			if (fm2->op() == multop::And)
+			multop::type bt = fm2->op();
+
+			// a R (b & c & F(a)) = a M (b & c)
+			// a M (b & c & F(a)) = a M (b & c)
+			if (bt == multop::And)
 			  {
 			    int s = fm2->size();
 			    for (int i = 0; i < s; ++i)
@@ -2175,8 +2181,23 @@ namespace spot
 				return;
 			      }
 			  }
+			// a M (b | a | c) == (b | c) U a
+			// a R (b | a | c) == (b | c) W a
+			if (bt == multop::Or)
+			  {
+			    int s = fm2->size();
+			    for (int i = 0; i < s; ++i)
+			      {
+				if (fm2->nth(i) != a)
+				  continue;
+				result_ = recurse_destroy(binop::instance
+				  (op == binop::M ? binop::U : binop::W,
+				   fm2->all_but(i), a));
+				b->destroy();
+				return;
+			      }
+			  }
 		      }
-
 		    // If b is Boolean:
 		    // (Xc) R b = b & X(b W c)
 		    // (Xc) M b = b & X(b U c)