(* 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)
*)
AddTCs [term.Apply_I];
Goal "term(A) = A * list(term(A))";
let open term; val rew = rewrite_rule con_defs in
by (fast_tac (claset() addSIs (map rew intrs) addEs [rew elim]) 1)
end;
qed "term_unfold";
(*Induction on term(A) followed by induction on List *)
val major::prems = Goal
"[| 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));
qed "term_induct2";
(*Induction on term(A) to prove an equation*)
val major::prems = Goal
"[| 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));
qed "term_induct_eqn";
(** Lemmas to justify using "term" in other recursive type definitions **)
Goalw term.defs "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));
qed "term_mono";
(*Easily provable by induction also*)
Goalw (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;
by (REPEAT (ares_tac [Pair_in_univ, list_univ RS subsetD] 1));
qed "term_univ";
val term_subset_univ =
term_mono RS (term_univ RSN (2,subset_trans)) |> standard;
Goal "[| t: term(A); A <= univ(B) |] ==> t: univ(B)";
by (REPEAT (ares_tac [term_subset_univ RS subsetD] 1));
qed "term_into_univ";
(*** term_rec -- by Vset recursion ***)
(*Lemma: map works correctly on the underlying list of terms*)
Goal "[| l: list(A); Ord(i) |] ==> \
\ rank(l)<i --> map(%z. (lam x:Vset(i).h(x)) ` z, l) = map(h,l)";
by (etac list.induct 1);
by (Simp_tac 1);
by (rtac impI 1);
by (subgoal_tac "rank(a)<i & rank(l) < i" 1);
by (asm_simp_tac (simpset() addsimps [VsetI]) 1);
by (full_simp_tac (simpset() addsimps list.con_defs) 1);
by (blast_tac (claset() addDs (rank_rls RL [lt_trans])) 1);
qed "map_lemma";
(*Typing premise is necessary to invoke map_lemma*)
Goal "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);
by (asm_simp_tac (simpset() addsimps [Ord_rank, rank_pair2, map_lemma]) 1);;
qed "term_rec";
(*Slightly odd typing condition on r in the second premise!*)
val major::prems = Goal
"[| 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 (stac term_rec 1);
by (REPEAT (ares_tac prems 1));
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [term_rec])));
by Auto_tac;
qed "term_rec_type";
val [rew,tslist] = Goal
"[| !!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);
qed "def_term_rec";
val prems = Goal
"[| 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);
qed "term_rec_simple_type";
AddTCs [term_rec_simple_type];
(** term_map **)
bind_thm ("term_map", term_map_def RS def_term_rec);
val prems = Goalw [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));
qed "term_map_type";
Goal "t: term(A) ==> term_map(f,t) : term({f(u). u:A})";
by (etac term_map_type 1);
by (etac RepFunI 1);
qed "term_map_type2";
(** term_size **)
bind_thm ("term_size", term_size_def RS def_term_rec);
Goalw [term_size_def] "t: term(A) ==> term_size(t) : nat";
by Auto_tac;
qed "term_size_type";
(** reflect **)
bind_thm ("reflect", reflect_def RS def_term_rec);
Goalw [reflect_def] "t: term(A) ==> reflect(t) : term(A)";
by Auto_tac;
qed "reflect_type";
(** preorder **)
bind_thm ("preorder", preorder_def RS def_term_rec);
Goalw [preorder_def] "t: term(A) ==> preorder(t) : list(A)";
by Auto_tac;
qed "preorder_type";
(** postorder **)
bind_thm ("postorder", postorder_def RS def_term_rec);
Goalw [postorder_def] "t: term(A) ==> postorder(t) : list(A)";
by Auto_tac;
qed "postorder_type";
(** Term simplification **)
AddTCs [term_map_type, term_map_type2, term_size_type,
reflect_type, preorder_type, postorder_type];
(*map_type2 and term_map_type2 instantiate variables*)
Addsimps [term_rec, term_map, term_size, reflect, preorder, postorder];
(** theorems about term_map **)
Addsimps [thm "List.map_compose"]; (*there is also TF.map_compose*)
Goal "t: term(A) ==> term_map(%u. u, t) = t";
by (etac term_induct_eqn 1);
by (Asm_simp_tac 1);
qed "term_map_ident";
Goal "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 1);
qed "term_map_compose";
Goal "t: term(A) ==> term_map(f, reflect(t)) = reflect(term_map(f,t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (simpset() addsimps [rev_map_distrib RS sym]) 1);
qed "term_map_reflect";
(** theorems about term_size **)
Goal "t: term(A) ==> term_size(term_map(f,t)) = term_size(t)";
by (etac term_induct_eqn 1);
by (Asm_simp_tac 1);
qed "term_size_term_map";
Goal "t: term(A) ==> term_size(reflect(t)) = term_size(t)";
by (etac term_induct_eqn 1);
by (asm_simp_tac(simpset() addsimps [rev_map_distrib RS sym, list_add_rev]) 1);
qed "term_size_reflect";
Goal "t: term(A) ==> term_size(t) = length(preorder(t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (simpset() addsimps [length_flat]) 1);
qed "term_size_length";
(** theorems about reflect **)
Goal "t: term(A) ==> reflect(reflect(t)) = t";
by (etac term_induct_eqn 1);
by (asm_simp_tac (simpset() addsimps [rev_map_distrib]) 1);
qed "reflect_reflect_ident";
(** theorems about preorder **)
Goal "t: term(A) ==> preorder(term_map(f,t)) = map(f, preorder(t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac (simpset() addsimps [map_flat]) 1);
qed "preorder_term_map";
Goal "t: term(A) ==> preorder(reflect(t)) = rev(postorder(t))";
by (etac term_induct_eqn 1);
by (asm_simp_tac(simpset() addsimps [rev_app_distrib, rev_flat,
rev_map_distrib RS sym]) 1);
qed "preorder_reflect_eq_rev_postorder";
writeln"Reached end of file.";