doc-src/Inductive/ind-defs.tex
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   217    \lfp(D,h)  & = & h(\lfp(D,h)) \\
   217    \lfp(D,h)  & = & h(\lfp(D,h)) \\
   218    \gfp(D,h)  & = & h(\gfp(D,h))
   218    \gfp(D,h)  & = & h(\gfp(D,h))
   219 \end{eqnarray*}
   219 \end{eqnarray*}
   220 These equations are instances of the Knaster-Tarski theorem, which states
   220 These equations are instances of the Knaster-Tarski theorem, which states
   221 that every monotonic function over a complete lattice has a
   221 that every monotonic function over a complete lattice has a
   222 fixedpoint~\cite{davey&priestley}.  It is obvious from their definitions
   222 fixedpoint~\cite{davey-priestley}.  It is obvious from their definitions
   223 that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.
   223 that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.
   224
   224
   225 This fixedpoint theory is simple.  The Knaster-Tarski theorem is easy to
   225 This fixedpoint theory is simple.  The Knaster-Tarski theorem is easy to
   226 prove.  Showing monotonicity of~$h$ is trivial, in typical cases.  We must
   226 prove.  Showing monotonicity of~$h$ is trivial, in typical cases.  We must
   227 also exhibit a bounding set~$D$ for~$h$.  Frequently this is trivial, as when
   227 also exhibit a bounding set~$D$ for~$h$.  Frequently this is trivial, as when