author | blanchet |
Fri, 21 Sep 2012 16:34:40 +0200 | |
changeset 49509 | 163914705f8d |
parent 49508 | 1e205327f059 |
permissions | -rw-r--r-- |
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(* Title: HOL/BNF/Examples/HFset.thy |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 2012 |
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Hereditary sets. |
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*) |
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header {* Hereditary Sets *} |
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theory HFset |
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imports "../BNF" |
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begin |
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section {* Datatype definition *} |
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data_raw hfset: 'hfset = "'hfset fset" |
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section {* Customization of terms *} |
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subsection{* Constructors *} |
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definition "Fold hs \<equiv> hfset_ctor hs" |
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lemma hfset_simps[simp]: |
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"\<And>hs1 hs2. Fold hs1 = Fold hs2 \<longrightarrow> hs1 = hs2" |
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unfolding Fold_def hfset.ctor_inject by auto |
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theorem hfset_cases[elim, case_names Fold]: |
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assumes Fold: "\<And> hs. h = Fold hs \<Longrightarrow> phi" |
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shows phi |
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using Fold unfolding Fold_def |
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by (cases rule: hfset.ctor_exhaust[of h]) simp |
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lemma hfset_induct[case_names Fold, induct type: hfset]: |
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assumes Fold: "\<And> hs. (\<And> h. h |\<in>| hs \<Longrightarrow> phi h) \<Longrightarrow> phi (Fold hs)" |
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shows "phi t" |
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apply (induct rule: hfset.ctor_induct) |
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using Fold unfolding Fold_def fset_fset_member mem_Collect_eq .. |
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(* alternative induction principle, using fset: *) |
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lemma hfset_induct_fset[case_names Fold, induct type: hfset]: |
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assumes Fold: "\<And> hs. (\<And> h. h \<in> fset hs \<Longrightarrow> phi h) \<Longrightarrow> phi (Fold hs)" |
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shows "phi t" |
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apply (induct rule: hfset_induct) |
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using Fold by (metis notin_fset) |
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subsection{* Recursion and iteration (fold) *} |
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lemma hfset_ctor_rec: |
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"hfset_ctor_rec R (Fold hs) = R (map_fset <id, hfset_ctor_rec R> hs)" |
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using hfset.ctor_recs unfolding Fold_def . |
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(* The iterator has a simpler form: *) |
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lemma hfset_ctor_fold: |
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"hfset_ctor_fold R (Fold hs) = R (map_fset (hfset_ctor_fold R) hs)" |
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using hfset.ctor_folds unfolding Fold_def . |
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end |