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--- sav08:interval_analysis_and_widening [2008/05/20 20:33]
vkuncak
+++ sav08:interval_analysis_and_widening [2008/05/20 20:34]
vkuncak
@@ Line 54: / Line 54: @@
 Narrowing: after finding fixpoint of $(W \circ F)^{\#}$, apply $F^{\#}$ to improve precision.
-Observation: if $F^{\#}$ and $W$ are $\omega$-continuous functions and $x \sqsubseteq W(x)$ for all $x$, then narrowing will improve the result, that is, if $x_* = lfp (F^{\#})$ and $y_* = lfp (W \circ F^{\#})$, then
+Observation: if $F^{\#}$ and $W$ are $\omega$-continuous functions and $x \sqsubseteq W(x)$ for all $x$, then narrowing will improve the result, that is, if $x_* = lfp (F^{\#})$ and $y_* = lfp (W \circ F^{\#})$, then $x_* \sqsubseteq y_*$ and
 \[
     x_* = F(x_*) \sqsubseteq F^{\#}(y_*) \sqsubseteq (W \circ F^{\#})(y_*) \sqsubseteq y_*