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(* Title: ZF/ex/term.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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Datatype definition of terms over an alphabet.
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Illustrates the list functor (essentially the same type as in Trees & Forests)
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*)
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structure Term = Datatype_Fun
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(val thy = List.thy;
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val rec_specs =
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[("term", "univ(A)",
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[(["Apply"], "[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 = ["[| a: A; l: list(term(A)) |] ==> Apply(a,l) : term(A)"];
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val monos = [list_mono];
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val type_intrs = [SigmaI,Pair_in_univ, list_univ RS subsetD, A_into_univ];
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val type_elims = []);
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val [ApplyI] = Term.intrs;
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(*Induction on term(A) followed by induction on List *)
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val major::prems = goal Term.thy
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"[| t: term(A); \
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\ !!x. [| x: A |] ==> P(Apply(x,Nil)); \
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\ !!x z zs. [| x: A; z: term(A); zs: list(term(A)); P(Apply(x,zs)) \
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\ |] ==> P(Apply(x, Cons(z,zs))) \
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\ |] ==> P(t)";
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by (rtac (major RS Term.induct) 1);
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by (etac List.induct 1);
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by (etac CollectE 2);
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by (REPEAT (ares_tac (prems@[list_CollectD]) 1));
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val term_induct2 = result();
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(*Induction on term(A) to prove an equation*)
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val major::prems = goal (merge_theories(Term.thy,ListFn.thy))
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"[| t: term(A); \
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\ !!x zs. [| x: A; zs: list(term(A)); map(f,zs) = map(g,zs) |] ==> \
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\ f(Apply(x,zs)) = g(Apply(x,zs)) \
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\ |] ==> f(t)=g(t)";
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by (rtac (major RS Term.induct) 1);
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by (resolve_tac prems 1);
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by (REPEAT (eresolve_tac [asm_rl, map_list_Collect, list_CollectD] 1));
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val term_induct_eqn = result();
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(** Lemmas to justify using "term" in other recursive type definitions **)
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goalw Term.thy Term.defs "!!A B. A<=B ==> term(A) <= term(B)";
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by (rtac lfp_mono 1);
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by (REPEAT (rtac Term.bnd_mono 1));
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by (REPEAT (ares_tac (univ_mono::basic_monos) 1));
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val term_mono = result();
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(*Easily provable by induction also*)
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goalw Term.thy (Term.defs@Term.con_defs) "term(univ(A)) <= univ(A)";
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by (rtac lfp_lowerbound 1);
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by (rtac (A_subset_univ RS univ_mono) 2);
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by (safe_tac ZF_cs);
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by (REPEAT (ares_tac [Pair_in_univ, list_univ RS subsetD] 1));
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val term_univ = result();
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val term_subset_univ = standard
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(term_mono RS (term_univ RSN (2,subset_trans)));
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