--- a/src/ZF/ex/TF.ML Fri Aug 12 11:13:23 1994 +0200
+++ b/src/ZF/ex/TF.ML Fri Aug 12 12:28:46 1994 +0200
@@ -4,39 +4,29 @@
Copyright 1993 University of Cambridge
Trees & forests, a mutually recursive type definition.
+
+Still needs
+
+"TF_reflect == (%z. TF_rec(z, %x ts r. Tcons(x,r), 0,
+ %t ts r1 r2. TF_of_list(list_of_TF(r2) @ <r1,0>)))"
*)
-structure TF = Datatype_Fun
- (val thy = Univ.thy
- val thy_name = "TF"
- val rec_specs = [("tree", "univ(A)",
- [(["Tcons"], "[i,i]=>i", NoSyn)]),
- ("forest", "univ(A)",
- [(["Fnil"], "i", NoSyn),
- (["Fcons"], "[i,i]=>i", NoSyn)])]
- val rec_styp = "i=>i"
- val sintrs =
- ["[| a:A; f: forest(A) |] ==> Tcons(a,f) : tree(A)",
- "Fnil : forest(A)",
- "[| t: tree(A); f: forest(A) |] ==> Fcons(t,f) : forest(A)"]
- val monos = []
- val type_intrs = datatype_intrs
- val type_elims = datatype_elims);
+open TF;
-val [TconsI, FnilI, FconsI] = TF.intrs;
+val [TconsI, FnilI, FconsI] = tree_forest.intrs;
(** tree_forest(A) as the union of tree(A) and forest(A) **)
-goalw TF.thy (tl TF.defs) "tree(A) <= tree_forest(A)";
+goalw TF.thy (tl tree_forest.defs) "tree(A) <= tree_forest(A)";
by (rtac Part_subset 1);
val tree_subset_TF = result();
-goalw TF.thy (tl TF.defs) "forest(A) <= tree_forest(A)";
+goalw TF.thy (tl tree_forest.defs) "forest(A) <= tree_forest(A)";
by (rtac Part_subset 1);
val forest_subset_TF = result();
-goalw TF.thy (tl TF.defs) "tree(A) Un forest(A) = tree_forest(A)";
-by (rtac (TF.dom_subset RS Part_sum_equality) 1);
+goalw TF.thy (tl tree_forest.defs) "tree(A) Un forest(A) = tree_forest(A)";
+by (rtac (tree_forest.dom_subset RS Part_sum_equality) 1);
val TF_equals_Un = result();
(** NOT useful, but interesting... **)
@@ -44,30 +34,240 @@
(*The (refl RS conjI RS exI RS exI) avoids considerable search!*)
val unfold_cs = sum_cs addSIs [PartI, refl RS conjI RS exI RS exI]
addIs datatype_intrs
- addDs [TF.dom_subset RS subsetD]
- addSEs ([PartE] @ datatype_elims @ TF.free_SEs);
+ addDs [tree_forest.dom_subset RS subsetD]
+ addSEs ([PartE] @ datatype_elims @ tree_forest.free_SEs);
-goalw TF.thy (tl TF.defs)
+goalw TF.thy (tl tree_forest.defs)
"tree_forest(A) = (A*forest(A)) + ({0} + tree(A)*forest(A))";
-by (rtac (TF.unfold RS trans) 1);
-by (rewrite_goals_tac TF.con_defs);
+by (rtac (tree_forest.unfold RS trans) 1);
+by (rewrite_goals_tac tree_forest.con_defs);
by (rtac equalityI 1);
by (fast_tac unfold_cs 1);
by (fast_tac unfold_cs 1);
val tree_forest_unfold = result();
-val tree_forest_unfold' = rewrite_rule (tl TF.defs) tree_forest_unfold;
+val tree_forest_unfold' = rewrite_rule (tl tree_forest.defs) tree_forest_unfold;
-goalw TF.thy (tl TF.defs) "tree(A) = {Inl(x). x: A*forest(A)}";
+goalw TF.thy (tl tree_forest.defs)
+ "tree(A) = {Inl(x). x: A*forest(A)}";
by (rtac (Part_Inl RS subst) 1);
by (rtac (tree_forest_unfold' RS subst_context) 1);
val tree_unfold = result();
-goalw TF.thy (tl TF.defs) "forest(A) = {Inr(x). x: {0} + tree(A)*forest(A)}";
+goalw TF.thy (tl tree_forest.defs)
+ "forest(A) = {Inr(x). x: {0} + tree(A)*forest(A)}";
by (rtac (Part_Inr RS subst) 1);
by (rtac (tree_forest_unfold' RS subst_context) 1);
val forest_unfold = result();
+(*** TF_rec -- by Vset recursion ***)
+(** conversion rules **)
+
+goal TF.thy "TF_rec(Tcons(a,f), b, c, d) = b(a, f, TF_rec(f,b,c,d))";
+by (rtac (TF_rec_def RS def_Vrec RS trans) 1);
+by (rewrite_goals_tac tree_forest.con_defs);
+by (simp_tac rank_ss 1);
+val TF_rec_Tcons = result();
+
+goal TF.thy "TF_rec(Fnil, b, c, d) = c";
+by (rtac (TF_rec_def RS def_Vrec RS trans) 1);
+by (rewrite_goals_tac tree_forest.con_defs);
+by (simp_tac rank_ss 1);
+val TF_rec_Fnil = result();
+
+goal TF.thy "TF_rec(Fcons(t,f), b, c, d) = \
+\ d(t, f, TF_rec(t, b, c, d), TF_rec(f, b, c, d))";
+by (rtac (TF_rec_def RS def_Vrec RS trans) 1);
+by (rewrite_goals_tac tree_forest.con_defs);
+by (simp_tac rank_ss 1);
+val TF_rec_Fcons = result();
+
+(*list_ss includes list operations as well as arith_ss*)
+val TF_rec_ss = list_ss addsimps
+ [TF_rec_Tcons, TF_rec_Fnil, TF_rec_Fcons, TconsI, FnilI, FconsI];
+
+(** Type checking **)
+
+val major::prems = goal TF.thy
+ "[| z: tree_forest(A); \
+\ !!x f r. [| x: A; f: forest(A); r: C(f) \
+\ |] ==> b(x,f,r): C(Tcons(x,f)); \
+\ c : C(Fnil); \
+\ !!t f r1 r2. [| t: tree(A); f: forest(A); r1: C(t); r2: C(f) \
+\ |] ==> d(t,f,r1,r2): C(Fcons(t,f)) \
+\ |] ==> TF_rec(z,b,c,d) : C(z)";
+by (rtac (major RS tree_forest.induct) 1);
+by (ALLGOALS (asm_simp_tac (TF_rec_ss addsimps prems)));
+val TF_rec_type = result();
+
+(*Mutually recursive version*)
+val prems = goal TF.thy
+ "[| !!x f r. [| x: A; f: forest(A); r: D(f) \
+\ |] ==> b(x,f,r): C(Tcons(x,f)); \
+\ c : D(Fnil); \
+\ !!t f r1 r2. [| t: tree(A); f: forest(A); r1: C(t); r2: D(f) \
+\ |] ==> d(t,f,r1,r2): D(Fcons(t,f)) \
+\ |] ==> (ALL t:tree(A). TF_rec(t,b,c,d) : C(t)) & \
+\ (ALL f: forest(A). TF_rec(f,b,c,d) : D(f))";
+by (rewtac Ball_def);
+by (rtac tree_forest.mutual_induct 1);
+by (ALLGOALS (asm_simp_tac (TF_rec_ss addsimps prems)));
+val tree_forest_rec_type = result();
+
+
+(** Versions for use with definitions **)
+
+val [rew] = goal TF.thy
+ "[| !!z. j(z)==TF_rec(z,b,c,d) |] ==> j(Tcons(a,f)) = b(a,f,j(f))";
+by (rewtac rew);
+by (rtac TF_rec_Tcons 1);
+val def_TF_rec_Tcons = result();
+
+val [rew] = goal TF.thy
+ "[| !!z. j(z)==TF_rec(z,b,c,d) |] ==> j(Fnil) = c";
+by (rewtac rew);
+by (rtac TF_rec_Fnil 1);
+val def_TF_rec_Fnil = result();
+
+val [rew] = goal TF.thy
+ "[| !!z. j(z)==TF_rec(z,b,c,d) |] ==> j(Fcons(t,f)) = d(t,f,j(t),j(f))";
+by (rewtac rew);
+by (rtac TF_rec_Fcons 1);
+val def_TF_rec_Fcons = result();
+
+fun TF_recs def = map standard
+ ([def] RL [def_TF_rec_Tcons, def_TF_rec_Fnil, def_TF_rec_Fcons]);
+
+
+(** list_of_TF and TF_of_list **)
+
+val [list_of_TF_Tcons, list_of_TF_Fnil, list_of_TF_Fcons] =
+ TF_recs list_of_TF_def;
+
+goalw TF.thy [list_of_TF_def]
+ "!!z A. z: tree_forest(A) ==> list_of_TF(z) : list(tree(A))";
+by (REPEAT (ares_tac ([TF_rec_type,TconsI] @ list_typechecks) 1));
+val list_of_TF_type = result();
+
+val [TF_of_list_Nil,TF_of_list_Cons] = list_recs TF_of_list_def;
+
+goalw TF.thy [TF_of_list_def]
+ "!!l A. l: list(tree(A)) ==> TF_of_list(l) : forest(A)";
+by (REPEAT (ares_tac [list_rec_type, FnilI, FconsI] 1));
+val TF_of_list_type = result();
+
+
+(** TF_map **)
+
+val [TF_map_Tcons, TF_map_Fnil, TF_map_Fcons] = TF_recs TF_map_def;
+
+val prems = goalw TF.thy [TF_map_def]
+ "[| !!x. x: A ==> h(x): B |] ==> \
+\ (ALL t:tree(A). TF_map(h,t) : tree(B)) & \
+\ (ALL f: forest(A). TF_map(h,f) : forest(B))";
+by (REPEAT
+ (ares_tac ([tree_forest_rec_type, TconsI, FnilI, FconsI] @ prems) 1));
+val TF_map_type = result();
+
+
+(** TF_size **)
+
+val [TF_size_Tcons, TF_size_Fnil, TF_size_Fcons] = TF_recs TF_size_def;
+
+goalw TF.thy [TF_size_def]
+ "!!z A. z: tree_forest(A) ==> TF_size(z) : nat";
+by (REPEAT (ares_tac [TF_rec_type, add_type, nat_0I, nat_succI] 1));
+val TF_size_type = result();
+
+
+(** TF_preorder **)
+
+val [TF_preorder_Tcons, TF_preorder_Fnil, TF_preorder_Fcons] =
+ TF_recs TF_preorder_def;
+
+goalw TF.thy [TF_preorder_def]
+ "!!z A. z: tree_forest(A) ==> TF_preorder(z) : list(A)";
+by (REPEAT (ares_tac ([TF_rec_type, app_type] @ list.intrs) 1));
+val TF_preorder_type = result();
+
+
+(** Term simplification **)
+
+val treeI = tree_subset_TF RS subsetD
+and forestI = forest_subset_TF RS subsetD;
+
+val TF_typechecks =
+ [TconsI, FnilI, FconsI, treeI, forestI,
+ list_of_TF_type, TF_map_type, TF_size_type, TF_preorder_type];
+
+val TF_rewrites =
+ [TF_rec_Tcons, TF_rec_Fnil, TF_rec_Fcons,
+ list_of_TF_Tcons, list_of_TF_Fnil, list_of_TF_Fcons,
+ TF_of_list_Nil,TF_of_list_Cons,
+ TF_map_Tcons, TF_map_Fnil, TF_map_Fcons,
+ TF_size_Tcons, TF_size_Fnil, TF_size_Fcons,
+ TF_preorder_Tcons, TF_preorder_Fnil, TF_preorder_Fcons];
+
+val TF_ss = list_ss addsimps TF_rewrites
+ setsolver type_auto_tac (list_typechecks@TF_typechecks);
+
+(** theorems about list_of_TF and TF_of_list **)
+
+(*essentially the same as list induction*)
+val major::prems = goal TF.thy
+ "[| f: forest(A); \
+\ R(Fnil); \
+\ !!t f. [| t: tree(A); f: forest(A); R(f) |] ==> R(Fcons(t,f)) \
+\ |] ==> R(f)";
+by (rtac (major RS (tree_forest.mutual_induct RS conjunct2 RS spec RSN (2,rev_mp))) 1);
+by (REPEAT (ares_tac (TrueI::prems) 1));
+val forest_induct = result();
+
+goal TF.thy "!!f A. f: forest(A) ==> TF_of_list(list_of_TF(f)) = f";
+by (etac forest_induct 1);
+by (ALLGOALS (asm_simp_tac TF_ss));
+val forest_iso = result();
+
+goal TF.thy
+ "!!ts. ts: list(tree(A)) ==> list_of_TF(TF_of_list(ts)) = ts";
+by (etac list.induct 1);
+by (ALLGOALS (asm_simp_tac TF_ss));
+val tree_list_iso = result();
+
+(** theorems about TF_map **)
+
+goal TF.thy "!!z A. z: tree_forest(A) ==> TF_map(%u.u, z) = z";
+by (etac tree_forest.induct 1);
+by (ALLGOALS (asm_simp_tac TF_ss));
+val TF_map_ident = result();
+
+goal TF.thy
+ "!!z A. z: tree_forest(A) ==> TF_map(h, TF_map(j,z)) = TF_map(%u.h(j(u)), z)";
+by (etac tree_forest.induct 1);
+by (ALLGOALS (asm_simp_tac TF_ss));
+val TF_map_compose = result();
+
+(** theorems about TF_size **)
+
+goal TF.thy
+ "!!z A. z: tree_forest(A) ==> TF_size(TF_map(h,z)) = TF_size(z)";
+by (etac tree_forest.induct 1);
+by (ALLGOALS (asm_simp_tac TF_ss));
+val TF_size_TF_map = result();
+
+goal TF.thy
+ "!!z A. z: tree_forest(A) ==> TF_size(z) = length(TF_preorder(z))";
+by (etac tree_forest.induct 1);
+by (ALLGOALS (asm_simp_tac (TF_ss addsimps [length_app])));
+val TF_size_length = result();
+
+(** theorems about TF_preorder **)
+
+goal TF.thy "!!z A. z: tree_forest(A) ==> \
+\ TF_preorder(TF_map(h,z)) = map(h, TF_preorder(z))";
+by (etac tree_forest.induct 1);
+by (ALLGOALS (asm_simp_tac (TF_ss addsimps [map_app_distrib])));
+val TF_preorder_TF_map = result();