Tidied many proofs, using AddIffs to let equivalences take
the place of separate Intr and Elim rules. Also deleted most named clasets.
(* Title: HOL/ex/Simult.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Primitives for simultaneous recursive type definitions
includes worked example of trees & forests
This is essentially the same data structure that on ex/term.ML, which is
simpler because it uses list as a new type former. The approach in this
file may be superior for other simultaneous recursions.
*)
open Simult;
(*** Monotonicity and unfolding of the function ***)
goal Simult.thy "mono(%Z. A <*> Part Z In1 \
\ <+> ({Numb(0)} <+> Part Z In0 <*> Part Z In1))";
by (REPEAT (ares_tac [monoI, subset_refl, usum_mono, uprod_mono,
Part_mono] 1));
qed "TF_fun_mono";
val TF_unfold = TF_fun_mono RS (TF_def RS def_lfp_Tarski);
goalw Simult.thy [TF_def] "!!A B. A<=B ==> TF(A) <= TF(B)";
by (REPEAT (ares_tac [lfp_mono, subset_refl, usum_mono, uprod_mono] 1));
qed "TF_mono";
goalw Simult.thy [TF_def] "TF(sexp) <= sexp";
by (rtac lfp_lowerbound 1);
by (fast_tac (!claset addIs sexp.intrs@[sexp_In0I, sexp_In1I]
addSEs [PartE]) 1);
qed "TF_sexp";
(* A <= sexp ==> TF(A) <= sexp *)
val TF_subset_sexp = standard
(TF_mono RS (TF_sexp RSN (2,subset_trans)));
(** Elimination -- structural induction on the set TF **)
val TF_Rep_defs = [TCONS_def,FNIL_def,FCONS_def,NIL_def,CONS_def];
val major::prems = goalw Simult.thy TF_Rep_defs
"[| i: TF(A); \
\ !!M N. [| M: A; N: Part (TF A) In1; R(N) |] ==> R(TCONS M N); \
\ R(FNIL); \
\ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; R(M); R(N) \
\ |] ==> R(FCONS M N) \
\ |] ==> R(i)";
by (rtac ([TF_def, TF_fun_mono, major] MRS def_induct) 1);
by (fast_tac (!claset addIs (prems@[PartI])
addEs [usumE, uprodE, PartE]) 1);
qed "TF_induct";
(*This lemma replaces a use of subgoal_tac to prove tree_forest_induct*)
goalw Simult.thy [Part_def]
"!!A. ! M: TF(A). (M: Part (TF A) In0 --> P(M)) & \
\ (M: Part (TF A) In1 --> Q(M)) \
\ ==> (! M: Part (TF A) In0. P(M)) & (! M: Part (TF A) In1. Q(M))";
by (Fast_tac 1);
qed "TF_induct_lemma";
AddSIs [PartI];
AddSDs [In0_inject, In1_inject];
AddSEs [PartE];
(*Could prove ~ TCONS M N : Part (TF A) In1 etc. *)
(*Induction on TF with separate predicates P, Q*)
val prems = goalw Simult.thy TF_Rep_defs
"[| !!M N. [| M: A; N: Part (TF A) In1; Q(N) |] ==> P(TCONS M N); \
\ Q(FNIL); \
\ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; P(M); Q(N) \
\ |] ==> Q(FCONS M N) \
\ |] ==> (! M: Part (TF A) In0. P(M)) & (! N: Part (TF A) In1. Q(N))";
by (rtac (ballI RS TF_induct_lemma) 1);
by (etac TF_induct 1);
by (rewrite_goals_tac TF_Rep_defs);
by (ALLGOALS (fast_tac (!claset addIs prems)));
(*29 secs??*)
qed "Tree_Forest_induct";
(*Induction for the abstract types 'a tree, 'a forest*)
val prems = goalw Simult.thy [Tcons_def,Fnil_def,Fcons_def]
"[| !!x ts. Q(ts) ==> P(Tcons x ts); \
\ Q(Fnil); \
\ !!t ts. [| P(t); Q(ts) |] ==> Q(Fcons t ts) \
\ |] ==> (! t. P(t)) & (! ts. Q(ts))";
by (res_inst_tac [("P1","%z.P(Abs_Tree(z))"),
("Q1","%z.Q(Abs_Forest(z))")]
(Tree_Forest_induct RS conjE) 1);
(*Instantiates ?A1 to range(Leaf). *)
by (fast_tac (!claset addSEs [Rep_Tree_inverse RS subst,
Rep_Forest_inverse RS subst]
addSIs [Rep_Tree,Rep_Forest]) 4);
(*Cannot use simplifier: the rewrites work in the wrong direction!*)
by (ALLGOALS (fast_tac (!claset addSEs [Abs_Tree_inverse RS subst,
Abs_Forest_inverse RS subst]
addSIs prems)));
qed "tree_forest_induct";
(*** Isomorphisms ***)
goal Simult.thy "inj(Rep_Tree)";
by (rtac inj_inverseI 1);
by (rtac Rep_Tree_inverse 1);
qed "inj_Rep_Tree";
goal Simult.thy "inj_onto Abs_Tree (Part (TF(range Leaf)) In0)";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Tree_inverse 1);
qed "inj_onto_Abs_Tree";
goal Simult.thy "inj(Rep_Forest)";
by (rtac inj_inverseI 1);
by (rtac Rep_Forest_inverse 1);
qed "inj_Rep_Forest";
goal Simult.thy "inj_onto Abs_Forest (Part (TF(range Leaf)) In1)";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Forest_inverse 1);
qed "inj_onto_Abs_Forest";
(** Introduction rules for constructors **)
(* c : A <*> Part (TF A) In1
<+> {Numb(0)} <+> Part (TF A) In0 <*> Part (TF A) In1 ==> c : TF(A) *)
val TF_I = TF_unfold RS equalityD2 RS subsetD;
(*For reasoning about the representation*)
AddIs [TF_I, uprodI, usum_In0I, usum_In1I];
AddSEs [Scons_inject];
val prems = goalw Simult.thy TF_Rep_defs
"[| a: A; M: Part (TF A) In1 |] ==> TCONS a M : Part (TF A) In0";
by (fast_tac (!claset addIs prems) 1);
qed "TCONS_I";
(* FNIL is a TF(A) -- this also justifies the type definition*)
goalw Simult.thy TF_Rep_defs "FNIL: Part (TF A) In1";
by (Fast_tac 1);
qed "FNIL_I";
val prems = goalw Simult.thy TF_Rep_defs
"[| M: Part (TF A) In0; N: Part (TF A) In1 |] ==> \
\ FCONS M N : Part (TF A) In1";
by (fast_tac (!claset addIs prems) 1);
qed "FCONS_I";
(** Injectiveness of TCONS and FCONS **)
goalw Simult.thy TF_Rep_defs "(TCONS K M=TCONS L N) = (K=L & M=N)";
by (Fast_tac 1);
qed "TCONS_TCONS_eq";
bind_thm ("TCONS_inject", (TCONS_TCONS_eq RS iffD1 RS conjE));
goalw Simult.thy TF_Rep_defs "(FCONS K M=FCONS L N) = (K=L & M=N)";
by (Fast_tac 1);
qed "FCONS_FCONS_eq";
bind_thm ("FCONS_inject", (FCONS_FCONS_eq RS iffD1 RS conjE));
(** Distinctness of TCONS, FNIL and FCONS **)
goalw Simult.thy TF_Rep_defs "TCONS M N ~= FNIL";
by (Fast_tac 1);
qed "TCONS_not_FNIL";
bind_thm ("FNIL_not_TCONS", (TCONS_not_FNIL RS not_sym));
bind_thm ("TCONS_neq_FNIL", (TCONS_not_FNIL RS notE));
val FNIL_neq_TCONS = sym RS TCONS_neq_FNIL;
goalw Simult.thy TF_Rep_defs "FCONS M N ~= FNIL";
by (Fast_tac 1);
qed "FCONS_not_FNIL";
bind_thm ("FNIL_not_FCONS", (FCONS_not_FNIL RS not_sym));
bind_thm ("FCONS_neq_FNIL", (FCONS_not_FNIL RS notE));
val FNIL_neq_FCONS = sym RS FCONS_neq_FNIL;
goalw Simult.thy TF_Rep_defs "TCONS M N ~= FCONS K L";
by (Fast_tac 1);
qed "TCONS_not_FCONS";
bind_thm ("FCONS_not_TCONS", (TCONS_not_FCONS RS not_sym));
bind_thm ("TCONS_neq_FCONS", (TCONS_not_FCONS RS notE));
val FCONS_neq_TCONS = sym RS TCONS_neq_FCONS;
(*???? Too many derived rules ????
Automatically generate symmetric forms? Always expand TF_Rep_defs? *)
(** Injectiveness of Tcons and Fcons **)
(*For reasoning about abstract constructors*)
AddSIs [Rep_Tree, Rep_Forest, TCONS_I, FNIL_I, FCONS_I];
AddSEs [TCONS_inject, FCONS_inject,
TCONS_neq_FNIL, FNIL_neq_TCONS,
FCONS_neq_FNIL, FNIL_neq_FCONS,
TCONS_neq_FCONS, FCONS_neq_TCONS];
AddSDs [inj_onto_Abs_Tree RS inj_ontoD,
inj_onto_Abs_Forest RS inj_ontoD,
inj_Rep_Tree RS injD, inj_Rep_Forest RS injD,
Leaf_inject];
goalw Simult.thy [Tcons_def] "(Tcons x xs=Tcons y ys) = (x=y & xs=ys)";
by (Fast_tac 1);
qed "Tcons_Tcons_eq";
bind_thm ("Tcons_inject", (Tcons_Tcons_eq RS iffD1 RS conjE));
goalw Simult.thy [Fcons_def,Fnil_def] "Fcons x xs ~= Fnil";
by (Fast_tac 1);
qed "Fcons_not_Fnil";
bind_thm ("Fcons_neq_Fnil", Fcons_not_Fnil RS notE);
val Fnil_neq_Fcons = sym RS Fcons_neq_Fnil;
(** Injectiveness of Fcons **)
goalw Simult.thy [Fcons_def] "(Fcons x xs=Fcons y ys) = (x=y & xs=ys)";
by (Fast_tac 1);
qed "Fcons_Fcons_eq";
bind_thm ("Fcons_inject", Fcons_Fcons_eq RS iffD1 RS conjE);
(*** TF_rec -- by wf recursion on pred_sexp ***)
goal Simult.thy
"(%M. TF_rec M b c d) = wfrec (trancl pred_sexp) \
\ (%g. Case (Split(%x y. b x y (g y))) \
\ (List_case c (%x y. d x y (g x) (g y))))";
by (simp_tac (HOL_ss addsimps [TF_rec_def]) 1);
val TF_rec_unfold = (wf_pred_sexp RS wf_trancl) RS
((result() RS eq_reflection) RS def_wfrec);
(*---------------------------------------------------------------------------
* Old:
* val TF_rec_unfold =
* wf_pred_sexp RS wf_trancl RS (TF_rec_def RS def_wfrec);
*---------------------------------------------------------------------------*)
(** conversion rules for TF_rec **)
goalw Simult.thy [TCONS_def]
"!!M N. [| M: sexp; N: sexp |] ==> \
\ TF_rec (TCONS M N) b c d = b M N (TF_rec N b c d)";
by (rtac (TF_rec_unfold RS trans) 1);
by (simp_tac (!simpset addsimps [Case_In0, Split]) 1);
by (asm_simp_tac (!simpset addsimps [In0_def]) 1);
qed "TF_rec_TCONS";
goalw Simult.thy [FNIL_def] "TF_rec FNIL b c d = c";
by (rtac (TF_rec_unfold RS trans) 1);
by (simp_tac (HOL_ss addsimps [Case_In1, List_case_NIL]) 1);
qed "TF_rec_FNIL";
goalw Simult.thy [FCONS_def]
"!!M N. [| M: sexp; N: sexp |] ==> \
\ TF_rec (FCONS M N) b c d = d M N (TF_rec M b c d) (TF_rec N b c d)";
by (rtac (TF_rec_unfold RS trans) 1);
by (simp_tac (HOL_ss addsimps [Case_In1, List_case_CONS]) 1);
by (asm_simp_tac (!simpset addsimps [CONS_def,In1_def]) 1);
qed "TF_rec_FCONS";
(*** tree_rec, forest_rec -- by TF_rec ***)
val Rep_Tree_in_sexp =
[range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
Rep_Tree] MRS subsetD;
val Rep_Forest_in_sexp =
[range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
Rep_Forest] MRS subsetD;
val tf_rec_ss = HOL_ss addsimps
[TF_rec_TCONS, TF_rec_FNIL, TF_rec_FCONS,
TCONS_I, FNIL_I, FCONS_I, Rep_Tree, Rep_Forest,
Rep_Tree_inverse, Rep_Forest_inverse,
Abs_Tree_inverse, Abs_Forest_inverse,
inj_Leaf, Inv_f_f, sexp.LeafI, range_eqI,
Rep_Tree_in_sexp, Rep_Forest_in_sexp];
goalw Simult.thy [tree_rec_def, forest_rec_def, Tcons_def]
"tree_rec (Tcons a tf) b c d = b a tf (forest_rec tf b c d)";
by (simp_tac tf_rec_ss 1);
qed "tree_rec_Tcons";
goalw Simult.thy [forest_rec_def, Fnil_def] "forest_rec Fnil b c d = c";
by (simp_tac tf_rec_ss 1);
qed "forest_rec_Fnil";
goalw Simult.thy [tree_rec_def, forest_rec_def, Fcons_def]
"forest_rec (Fcons t tf) b c d = \
\ d t tf (tree_rec t b c d) (forest_rec tf b c d)";
by (simp_tac tf_rec_ss 1);
qed "forest_rec_Cons";