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(* Title: ZF/ex/tf.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Trees & forests, a mutually recursive type definition.
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*)
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structure TF = Datatype_Fun
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(val thy = Univ.thy;
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val rec_specs =
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[("tree", "univ(A)",
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[(["Tcons"], "[i,i]=>i")]),
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("forest", "univ(A)",
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[(["Fnil"], "i"),
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(["Fcons"], "[i,i]=>i")])];
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val rec_styp = "i=>i";
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val ext = None
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val sintrs =
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["[| a:A; tf: forest(A) |] ==> Tcons(a,tf) : tree(A)",
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"Fnil : forest(A)",
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"[| t: tree(A); tf: forest(A) |] ==> Fcons(t,tf) : forest(A)"];
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val monos = [];
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val type_intrs = data_typechecks
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val type_elims = []);
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val [TconsI, FnilI, FconsI] = TF.intrs;
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(** tree_forest(A) as the union of tree(A) and forest(A) **)
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goalw TF.thy (tl TF.defs) "tree(A) <= tree_forest(A)";
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by (rtac Part_subset 1);
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val tree_subset_TF = result();
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goalw TF.thy (tl TF.defs) "forest(A) <= tree_forest(A)";
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by (rtac Part_subset 1);
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val forest_subset_TF = result();
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goalw TF.thy (tl TF.defs) "tree(A) Un forest(A) = tree_forest(A)";
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by (rtac (TF.dom_subset RS Part_sum_equality) 1);
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val TF_equals_Un = result();
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(** NOT useful, but interesting... **)
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goalw TF.thy (tl TF.defs) "tree(A) = {Inl(x). x: A*forest(A)}";
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by (res_inst_tac [("P", "%x.?t(x) = ?u::i")] (TF.unfold RS ssubst) 1);
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by (rewrite_goals_tac TF.con_defs);
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by (rtac equalityI 1);
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by (fast_tac (ZF_cs addIs [PartI] addSEs (PartE::TF.free_SEs)) 1);
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by (fast_tac (ZF_cs addIs ([PartI,InlI,InrI] @ [TF.dom_subset RS subsetD]
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@ data_typechecks)
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addSEs (PartE::TF.free_SEs)) 1);
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val tree_unfold = result();
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goalw TF.thy (tl TF.defs) "forest(A) = {Inr(x). x: {0} + tree(A)*forest(A)}";
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by (res_inst_tac [("P", "%x.?t(x) = ?u::i")] (TF.unfold RS ssubst) 1);
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by (rewrite_goals_tac TF.con_defs);
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by (rtac equalityI 1);
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by (fast_tac (ZF_cs addIs [PartI,InlI,InrI]
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addSEs (PartE::TF.free_SEs)) 1);
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by (fast_tac (ZF_cs addIs ([PartI,InlI,InrI] @ [TF.dom_subset RS subsetD]
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@ data_typechecks)
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addSEs ([PartE, sumE] @ TF.free_SEs)) 1);
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val forest_unfold = result();
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