(* Title: ZF/ex/Term.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Datatype definition of terms over an alphabet.
Illustrates the list functor (essentially the same type as in Trees & Forests)
*)
open Term;
goal Term.thy "term(A) = A * list(term(A))";
let open term; val rew = rewrite_rule con_defs in
by (fast_tac (sum_cs addSIs (equalityI :: map rew intrs)
addEs [rew elim]) 1)
end;
val term_unfold = result();
(*Induction on term(A) followed by induction on List *)
val major::prems = goal Term.thy
"[| t: term(A); \
\ !!x. [| x: A |] ==> P(Apply(x,Nil)); \
\ !!x z zs. [| x: A; z: term(A); zs: list(term(A)); P(Apply(x,zs)) \
\ |] ==> P(Apply(x, Cons(z,zs))) \
\ |] ==> P(t)";
by (rtac (major RS term.induct) 1);
by (etac list.induct 1);
by (etac CollectE 2);
by (REPEAT (ares_tac (prems@[list_CollectD]) 1));
val term_induct2 = result();
(*Induction on term(A) to prove an equation*)
val major::prems = goal Term.thy
"[| t: term(A); \
\ !!x zs. [| x: A; zs: list(term(A)); map(f,zs) = map(g,zs) |] ==> \
\ f(Apply(x,zs)) = g(Apply(x,zs)) \
\ |] ==> f(t)=g(t)";
by (rtac (major RS term.induct) 1);
by (resolve_tac prems 1);
by (REPEAT (eresolve_tac [asm_rl, map_list_Collect, list_CollectD] 1));
val term_induct_eqn = result();
(** Lemmas to justify using "term" in other recursive type definitions **)
goalw Term.thy term.defs "!!A B. A<=B ==> term(A) <= term(B)";
by (rtac lfp_mono 1);
by (REPEAT (rtac term.bnd_mono 1));
by (REPEAT (ares_tac (univ_mono::basic_monos) 1));
val term_mono = result();
(*Easily provable by induction also*)
goalw Term.thy (term.defs@term.con_defs) "term(univ(A)) <= univ(A)";
by (rtac lfp_lowerbound 1);
by (rtac (A_subset_univ RS univ_mono) 2);
by (safe_tac ZF_cs);
by (REPEAT (ares_tac [Pair_in_univ, list_univ RS subsetD] 1));
val term_univ = result();
val term_subset_univ =
term_mono RS (term_univ RSN (2,subset_trans)) |> standard;
goal Term.thy "!!t A B. [| t: term(A); A <= univ(B) |] ==> t: univ(B)";
by (REPEAT (ares_tac [term_subset_univ RS subsetD] 1));
val term_into_univ = result();
(*** term_rec -- by Vset recursion ***)
(*Lemma: map works correctly on the underlying list of terms*)
val [major,ordi] = goal list.thy
"[| l: list(A); Ord(i) |] ==> \
\ rank(l)<i --> map(%z. (lam x:Vset(i).h(x)) ` z, l) = map(h,l)";
by (rtac (major RS list.induct) 1);
by (simp_tac list_ss 1);
by (rtac impI 1);
by (forward_tac [rank_Cons1 RS lt_trans] 1);
by (dtac (rank_Cons2 RS lt_trans) 1);
by (asm_simp_tac (list_ss addsimps [ordi, VsetI]) 1);
val map_lemma = result();
(*Typing premise is necessary to invoke map_lemma*)
val [prem] = goal Term.thy
"ts: list(A) ==> \
\ term_rec(Apply(a,ts), d) = d(a, ts, map (%z. term_rec(z,d), ts))";
by (rtac (term_rec_def RS def_Vrec RS trans) 1);
by (rewrite_goals_tac term.con_defs);
val term_rec_ss = ZF_ss addsimps [Ord_rank, rank_pair2, prem RS map_lemma];
by (simp_tac term_rec_ss 1);
val term_rec = result();
(*Slightly odd typing condition on r in the second premise!*)
val major::prems = goal Term.thy
"[| t: term(A); \
\ !!x zs r. [| x: A; zs: list(term(A)); \
\ r: list(UN t:term(A). C(t)) |] \
\ ==> d(x, zs, r): C(Apply(x,zs)) \
\ |] ==> term_rec(t,d) : C(t)";
by (rtac (major RS term.induct) 1);
by (forward_tac [list_CollectD] 1);
by (rtac (term_rec RS ssubst) 1);
by (REPEAT (ares_tac prems 1));
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [term_rec])));
by (etac CollectE 1);
by (REPEAT (ares_tac [list.Cons_I, UN_I] 1));
val term_rec_type = result();
val [rew,tslist] = goal Term.thy
"[| !!t. j(t)==term_rec(t,d); ts: list(A) |] ==> \
\ j(Apply(a,ts)) = d(a, ts, map(%Z.j(Z), ts))";
by (rewtac rew);
by (rtac (tslist RS term_rec) 1);
val def_term_rec = result();
val prems = goal Term.thy
"[| t: term(A); \
\ !!x zs r. [| x: A; zs: list(term(A)); r: list(C) |] \
\ ==> d(x, zs, r): C \
\ |] ==> term_rec(t,d) : C";
by (REPEAT (ares_tac (term_rec_type::prems) 1));
by (etac (subset_refl RS UN_least RS list_mono RS subsetD) 1);
val term_rec_simple_type = result();
(** term_map **)
val term_map = standard (term_map_def RS def_term_rec);
val prems = goalw Term.thy [term_map_def]
"[| t: term(A); !!x. x: A ==> f(x): B |] ==> term_map(f,t) : term(B)";
by (REPEAT (ares_tac ([term_rec_simple_type, term.Apply_I] @ prems) 1));
val term_map_type = result();
val [major] = goal Term.thy
"t: term(A) ==> term_map(f,t) : term({f(u). u:A})";
by (rtac (major RS term_map_type) 1);
by (etac RepFunI 1);
val term_map_type2 = result();
(** term_size **)
val term_size = standard (term_size_def RS def_term_rec);
goalw Term.thy [term_size_def] "!!t A. t: term(A) ==> term_size(t) : nat";
by (REPEAT (ares_tac [term_rec_simple_type, list_add_type, nat_succI] 1));
val term_size_type = result();
(** reflect **)
val reflect = standard (reflect_def RS def_term_rec);
goalw Term.thy [reflect_def] "!!t A. t: term(A) ==> reflect(t) : term(A)";
by (REPEAT (ares_tac [term_rec_simple_type, rev_type, term.Apply_I] 1));
val reflect_type = result();
(** preorder **)
val preorder = standard (preorder_def RS def_term_rec);
goalw Term.thy [preorder_def]
"!!t A. t: term(A) ==> preorder(t) : list(A)";
by (REPEAT (ares_tac [term_rec_simple_type, list.Cons_I, flat_type] 1));
val preorder_type = result();
(** Term simplification **)
val term_typechecks =
[term.Apply_I, term_map_type, term_map_type2, term_size_type,
reflect_type, preorder_type];
(*map_type2 and term_map_type2 instantiate variables*)
val term_ss = list_ss
addsimps [term_rec, term_map, term_size, reflect, preorder]
setsolver type_auto_tac (list_typechecks@term_typechecks);
(** theorems about term_map **)
goal Term.thy "!!t A. t: term(A) ==> term_map(%u.u, t) = t";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [map_ident]) 1);
val term_map_ident = result();
goal Term.thy
"!!t A. t: term(A) ==> term_map(f, term_map(g,t)) = term_map(%u.f(g(u)), t)";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [map_compose]) 1);
val term_map_compose = result();
goal Term.thy
"!!t A. t: term(A) ==> term_map(f, reflect(t)) = reflect(term_map(f,t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [rev_map_distrib RS sym, map_compose]) 1);
val term_map_reflect = result();
(** theorems about term_size **)
goal Term.thy
"!!t A. t: term(A) ==> term_size(term_map(f,t)) = term_size(t)";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [map_compose]) 1);
val term_size_term_map = result();
goal Term.thy "!!t A. t: term(A) ==> term_size(reflect(t)) = term_size(t)";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [rev_map_distrib RS sym, map_compose,
list_add_rev]) 1);
val term_size_reflect = result();
goal Term.thy "!!t A. t: term(A) ==> term_size(t) = length(preorder(t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [length_flat, map_compose]) 1);
val term_size_length = result();
(** theorems about reflect **)
goal Term.thy "!!t A. t: term(A) ==> reflect(reflect(t)) = t";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [rev_map_distrib, map_compose,
map_ident, rev_rev_ident]) 1);
val reflect_reflect_ident = result();
(** theorems about preorder **)
goal Term.thy
"!!t A. t: term(A) ==> preorder(term_map(f,t)) = map(f, preorder(t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (term_ss addsimps [map_compose, map_flat]) 1);
val preorder_term_map = result();
(** preorder(reflect(t)) = rev(postorder(t)) **)
writeln"Reached end of file.";