Mod because of new solver interface.
(* Title: ZF/inductive_package.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Fixedpoint definition module -- for Inductive/Coinductive Definitions
The functor will be instantiated for normal sums/products (inductive defs)
and non-standard sums/products (coinductive defs)
Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)
type inductive_result =
{defs : thm list, (*definitions made in thy*)
bnd_mono : thm, (*monotonicity for the lfp definition*)
dom_subset : thm, (*inclusion of recursive set in dom*)
intrs : thm list, (*introduction rules*)
elim : thm, (*case analysis theorem*)
mk_cases : string -> thm, (*generates case theorems*)
induct : thm, (*main induction rule*)
mutual_induct : thm}; (*mutual induction rule*)
(*Functor's result signature*)
signature INDUCTIVE_PACKAGE =
sig
(*Insert definitions for the recursive sets, which
must *already* be declared as constants in parent theory!*)
val add_inductive_i :
bool ->
term list * term * term list * thm list * thm list * thm list * thm list
-> theory -> theory * inductive_result
val add_inductive :
string list * string * string list *
thm list * thm list * thm list * thm list
-> theory -> theory * inductive_result
end;
(*Declares functions to add fixedpoint/constructor defs to a theory.
Recursive sets must *already* be declared as constants.*)
functor Add_inductive_def_Fun
(structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU)
: INDUCTIVE_PACKAGE =
struct
open Logic Ind_Syntax;
(*internal version, accepting terms*)
fun add_inductive_i verbose (rec_tms, dom_sum, intr_tms,
monos, con_defs, type_intrs, type_elims) thy =
let
val dummy = (*has essential ancestors?*)
Theory.requires thy "Inductive" "(co)inductive definitions"
val sign = sign_of thy;
(*recT and rec_params should agree for all mutually recursive components*)
val rec_hds = map head_of rec_tms;
val dummy = assert_all is_Const rec_hds
(fn t => "Recursive set not previously declared as constant: " ^
Sign.string_of_term sign t);
(*Now we know they are all Consts, so get their names, type and params*)
val rec_names = map (#1 o dest_Const) rec_hds
and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
val rec_base_names = map Sign.base_name rec_names;
val dummy = assert_all Syntax.is_identifier rec_base_names
(fn a => "Base name of recursive set not an identifier: " ^ a);
local (*Checking the introduction rules*)
val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
fun intr_ok set =
case head_of set of Const(a,recT) => a mem rec_names | _ => false;
in
val dummy = assert_all intr_ok intr_sets
(fn t => "Conclusion of rule does not name a recursive set: " ^
Sign.string_of_term sign t);
end;
val dummy = assert_all is_Free rec_params
(fn t => "Param in recursion term not a free variable: " ^
Sign.string_of_term sign t);
(*** Construct the fixedpoint definition ***)
val mk_variant = variant (foldr add_term_names (intr_tms,[]));
val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";
fun dest_tprop (Const("Trueprop",_) $ P) = P
| dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
Sign.string_of_term sign Q);
(*Makes a disjunct from an introduction rule*)
fun fp_part intr = (*quantify over rule's free vars except parameters*)
let val prems = map dest_tprop (strip_imp_prems intr)
val dummy = seq (fn rec_hd => seq (chk_prem rec_hd) prems) rec_hds
val exfrees = term_frees intr \\ rec_params
val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))
in foldr FOLogic.mk_exists
(exfrees, fold_bal (app FOLogic.conj) (zeq::prems))
end;
(*The Part(A,h) terms -- compose injections to make h*)
fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
| mk_Part h = Part_const $ Free(X',iT) $ Abs(w',iT,h);
(*Access to balanced disjoint sums via injections*)
val parts =
map mk_Part (accesses_bal (ap Su.inl, ap Su.inr, Bound 0)
(length rec_tms));
(*replace each set by the corresponding Part(A,h)*)
val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;
val fp_abs = absfree(X', iT,
mk_Collect(z', dom_sum,
fold_bal (app FOLogic.disj) part_intrs));
val fp_rhs = Fp.oper $ dom_sum $ fp_abs
val dummy = seq (fn rec_hd => deny (rec_hd occs fp_rhs)
"Illegal occurrence of recursion operator")
rec_hds;
(*** Make the new theory ***)
(*A key definition:
If no mutual recursion then it equals the one recursive set.
If mutual recursion then it differs from all the recursive sets. *)
val big_rec_base_name = space_implode "_" rec_base_names;
val big_rec_name = Sign.intern_const sign big_rec_base_name;
val dummy =
if verbose then
writeln ((if #1 (dest_Const Fp.oper) = "Fixedpt.lfp" then "Inductive"
else "Coinductive") ^ " definition " ^ big_rec_name)
else ();
(*Forbid the inductive definition structure from clashing with a theory
name. This restriction may become obsolete as ML is de-emphasized.*)
val dummy = deny (big_rec_base_name mem (Sign.stamp_names_of sign))
("Definition " ^ big_rec_base_name ^
" would clash with the theory of the same name!");
(*Big_rec... is the union of the mutually recursive sets*)
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
(*The individual sets must already be declared*)
val axpairs = map Logic.mk_defpair
((big_rec_tm, fp_rhs) ::
(case parts of
[_] => [] (*no mutual recursion*)
| _ => rec_tms ~~ (*define the sets as Parts*)
map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));
(*tracing: print the fixedpoint definition*)
val dummy = if !Ind_Syntax.trace then
seq (writeln o Sign.string_of_term sign o #2) axpairs
else ()
(*add definitions of the inductive sets*)
val thy1 = thy |> Theory.add_path big_rec_base_name
|> PureThy.add_defs_i (map Thm.no_attributes axpairs)
(*fetch fp definitions from the theory*)
val big_rec_def::part_rec_defs =
map (get_def thy1)
(case rec_names of [_] => rec_names
| _ => big_rec_base_name::rec_names);
val sign1 = sign_of thy1;
(********)
val dummy = writeln " Proving monotonicity...";
val bnd_mono =
prove_goalw_cterm []
(cterm_of sign1
(FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs)))
(fn _ =>
[rtac (Collect_subset RS bnd_monoI) 1,
REPEAT (ares_tac (basic_monos @ monos) 1)]);
val dom_subset = standard (big_rec_def RS Fp.subs);
val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);
(********)
val dummy = writeln " Proving the introduction rules...";
(*Mutual recursion? Helps to derive subset rules for the
individual sets.*)
val Part_trans =
case rec_names of
[_] => asm_rl
| _ => standard (Part_subset RS subset_trans);
(*To type-check recursive occurrences of the inductive sets, possibly
enclosed in some monotonic operator M.*)
val rec_typechecks =
[dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
RL [subsetD];
(*Type-checking is hardest aspect of proof;
disjIn selects the correct disjunct after unfolding*)
fun intro_tacsf disjIn prems =
[(*insert prems and underlying sets*)
cut_facts_tac prems 1,
DETERM (stac unfold 1),
REPEAT (resolve_tac [Part_eqI,CollectI] 1),
(*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
rtac disjIn 2,
(*Not ares_tac, since refl must be tried before equality assumptions;
backtracking may occur if the premises have extra variables!*)
DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 2 APPEND assume_tac 2),
(*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
rewrite_goals_tac con_defs,
REPEAT (rtac refl 2),
(*Typechecking; this can fail*)
if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
else all_tac,
REPEAT (FIRSTGOAL ( dresolve_tac rec_typechecks
ORELSE' eresolve_tac (asm_rl::PartE::SigmaE2::
type_elims)
ORELSE' hyp_subst_tac)),
if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
else all_tac,
DEPTH_SOLVE (swap_res_tac (SigmaI::subsetI::type_intrs) 1)];
(*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
val mk_disj_rls =
let fun f rl = rl RS disjI1
and g rl = rl RS disjI2
in accesses_bal(f, g, asm_rl) end;
fun prove_intr (ct, tacsf) = prove_goalw_cterm part_rec_defs ct tacsf;
val intrs = ListPair.map prove_intr
(map (cterm_of sign1) intr_tms,
map intro_tacsf (mk_disj_rls(length intr_tms)))
handle SIMPLIFIER (msg,thm) => (print_thm thm; error msg);
(********)
val dummy = writeln " Proving the elimination rule...";
(*Breaks down logical connectives in the monotonic function*)
val basic_elim_tac =
REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
ORELSE' bound_hyp_subst_tac))
THEN prune_params_tac
(*Mutual recursion: collapse references to Part(D,h)*)
THEN fold_tac part_rec_defs;
(*Elimination*)
val elim = rule_by_tactic basic_elim_tac
(unfold RS Ind_Syntax.equals_CollectD)
(*Applies freeness of the given constructors, which *must* be unfolded by
the given defs. Cannot simply use the local con_defs because
con_defs=[] for inference systems.
String s should have the form t:Si where Si is an inductive set*)
fun mk_cases s =
rule_by_tactic (basic_elim_tac THEN
ALLGOALS Asm_full_simp_tac THEN
basic_elim_tac)
(assume_read (theory_of_thm elim) s
(*Don't use thy1: it will be stale*)
RS elim)
|> standard;
fun induction_rules raw_induct thy =
let
val dummy = writeln " Proving the induction rule...";
(*** Prove the main induction rule ***)
val pred_name = "P"; (*name for predicate variables*)
(*Used to make induction rules;
ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
prem is a premise of an intr rule*)
fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $
(Const("op :",_)$t$X), iprems) =
(case gen_assoc (op aconv) (ind_alist, X) of
Some pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
| None => (*possibly membership in M(rec_tm), for M monotone*)
let fun mk_sb (rec_tm,pred) =
(rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
in subst_free (map mk_sb ind_alist) prem :: iprems end)
| add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
(*Make a premise of the induction rule.*)
fun induct_prem ind_alist intr =
let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
val iprems = foldr (add_induct_prem ind_alist)
(Logic.strip_imp_prems intr,[])
val (t,X) = Ind_Syntax.rule_concl intr
val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
val concl = FOLogic.mk_Trueprop (pred $ t)
in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
handle Bind => error"Recursion term not found in conclusion";
(*Minimizes backtracking by delivering the correct premise to each goal.
Intro rules with extra Vars in premises still cause some backtracking *)
fun ind_tac [] 0 = all_tac
| ind_tac(prem::prems) i =
DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN
ind_tac prems (i-1);
val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);
val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
intr_tms;
val dummy = if !Ind_Syntax.trace then
(writeln "ind_prems = ";
seq (writeln o Sign.string_of_term sign1) ind_prems;
writeln "raw_induct = "; print_thm raw_induct)
else ();
(*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
If the premises get simplified, then the proofs could fail.*)
val min_ss = empty_ss
setmksimps (map mk_eq o ZF_atomize o gen_all)
setSolver (mk_solver "minimal"
(fn prems => resolve_tac (triv_rls@prems)
ORELSE' assume_tac
ORELSE' etac FalseE));
val quant_induct =
prove_goalw_cterm part_rec_defs
(cterm_of sign1
(Logic.list_implies
(ind_prems,
FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp(big_rec_tm,pred)))))
(fn prems =>
[rtac (impI RS allI) 1,
DETERM (etac raw_induct 1),
(*Push Part inside Collect*)
full_simp_tac (min_ss addsimps [Part_Collect]) 1,
(*This CollectE and disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE]
ORELSE' hyp_subst_tac)),
ind_tac (rev prems) (length prems) ]);
val dummy = if !Ind_Syntax.trace then
(writeln "quant_induct = "; print_thm quant_induct)
else ();
(*** Prove the simultaneous induction rule ***)
(*Make distinct predicates for each inductive set*)
(*The components of the element type, several if it is a product*)
val elem_type = CP.pseudo_type dom_sum;
val elem_factors = CP.factors elem_type;
val elem_frees = mk_frees "za" elem_factors;
val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;
(*Given a recursive set and its domain, return the "fsplit" predicate
and a conclusion for the simultaneous induction rule.
NOTE. This will not work for mutually recursive predicates. Previously
a summand 'domt' was also an argument, but this required the domain of
mutual recursion to invariably be a disjoint sum.*)
fun mk_predpair rec_tm =
let val rec_name = (#1 o dest_Const o head_of) rec_tm
val pfree = Free(pred_name ^ "_" ^ Sign.base_name rec_name,
elem_factors ---> FOLogic.oT)
val qconcl =
foldr FOLogic.mk_all
(elem_frees,
FOLogic.imp $
(Ind_Syntax.mem_const $ elem_tuple $ rec_tm)
$ (list_comb (pfree, elem_frees)))
in (CP.ap_split elem_type FOLogic.oT pfree,
qconcl)
end;
val (preds,qconcls) = split_list (map mk_predpair rec_tms);
(*Used to form simultaneous induction lemma*)
fun mk_rec_imp (rec_tm,pred) =
FOLogic.imp $ (Ind_Syntax.mem_const $ Bound 0 $ rec_tm) $
(pred $ Bound 0);
(*To instantiate the main induction rule*)
val induct_concl =
FOLogic.mk_Trueprop
(Ind_Syntax.mk_all_imp
(big_rec_tm,
Abs("z", Ind_Syntax.iT,
fold_bal (app FOLogic.conj)
(ListPair.map mk_rec_imp (rec_tms, preds)))))
and mutual_induct_concl =
FOLogic.mk_Trueprop(fold_bal (app FOLogic.conj) qconcls);
val dummy = if !Ind_Syntax.trace then
(writeln ("induct_concl = " ^
Sign.string_of_term sign1 induct_concl);
writeln ("mutual_induct_concl = " ^
Sign.string_of_term sign1 mutual_induct_concl))
else ();
val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
resolve_tac [allI, impI, conjI, Part_eqI],
dresolve_tac [spec, mp, Pr.fsplitD]];
val need_mutual = length rec_names > 1;
val lemma = (*makes the link between the two induction rules*)
if need_mutual then
(writeln " Proving the mutual induction rule...";
prove_goalw_cterm part_rec_defs
(cterm_of sign1 (Logic.mk_implies (induct_concl,
mutual_induct_concl)))
(fn prems =>
[cut_facts_tac prems 1,
REPEAT (rewrite_goals_tac [Pr.split_eq] THEN
lemma_tac 1)]))
else (writeln " [ No mutual induction rule needed ]";
TrueI);
val dummy = if !Ind_Syntax.trace then
(writeln "lemma = "; print_thm lemma)
else ();
(*Mutual induction follows by freeness of Inl/Inr.*)
(*Simplification largely reduces the mutual induction rule to the
standard rule*)
val mut_ss =
min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
val all_defs = con_defs @ part_rec_defs;
(*Removes Collects caused by M-operators in the intro rules. It is very
hard to simplify
list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
where t==Part(tf,Inl) and f==Part(tf,Inr) to list({v: tf. P_t(v)}).
Instead the following rules extract the relevant conjunct.
*)
val cmonos = [subset_refl RS Collect_mono] RL monos
RLN (2,[rev_subsetD]);
(*Minimizes backtracking by delivering the correct premise to each goal*)
fun mutual_ind_tac [] 0 = all_tac
| mutual_ind_tac(prem::prems) i =
DETERM
(SELECT_GOAL
(
(*Simplify the assumptions and goal by unfolding Part and
using freeness of the Sum constructors; proves all but one
conjunct by contradiction*)
rewrite_goals_tac all_defs THEN
simp_tac (mut_ss addsimps [Part_iff]) 1 THEN
IF_UNSOLVED (*simp_tac may have finished it off!*)
((*simplify assumptions*)
(*some risk of excessive simplification here -- might have
to identify the bare minimum set of rewrites*)
full_simp_tac
(mut_ss addsimps conj_simps @ imp_simps @ quant_simps) 1
THEN
(*unpackage and use "prem" in the corresponding place*)
REPEAT (rtac impI 1) THEN
rtac (rewrite_rule all_defs prem) 1 THEN
(*prem must not be REPEATed below: could loop!*)
DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
eresolve_tac (conjE::mp::cmonos))))
) i)
THEN mutual_ind_tac prems (i-1);
val mutual_induct_fsplit =
if need_mutual then
prove_goalw_cterm []
(cterm_of sign1
(Logic.list_implies
(map (induct_prem (rec_tms~~preds)) intr_tms,
mutual_induct_concl)))
(fn prems =>
[rtac (quant_induct RS lemma) 1,
mutual_ind_tac (rev prems) (length prems)])
else TrueI;
(** Uncurrying the predicate in the ordinary induction rule **)
(*instantiate the variable to a tuple, if it is non-trivial, in order to
allow the predicate to be "opened up".
The name "x.1" comes from the "RS spec" !*)
val inst =
case elem_frees of [_] => I
| _ => instantiate ([], [(cterm_of sign1 (Var(("x",1), Ind_Syntax.iT)),
cterm_of sign1 elem_tuple)]);
(*strip quantifier and the implication*)
val induct0 = inst (quant_induct RS spec RSN (2,rev_mp));
val Const ("Trueprop", _) $ (pred_var $ _) = concl_of induct0
val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
|> standard
and mutual_induct = CP.remove_split mutual_induct_fsplit
in
(thy
|> PureThy.add_thms
[(("induct", induct), []),
(("mutual_induct", mutual_induct), [])],
induct, mutual_induct)
end; (*of induction_rules*)
val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)
val (thy2, induct, mutual_induct) =
if #1 (dest_Const Fp.oper) = "Fixedpt.lfp" then induction_rules raw_induct thy1
else (thy1, raw_induct, TrueI)
and defs = big_rec_def :: part_rec_defs
in
(thy2
|> (PureThy.add_thms o map Thm.no_attributes)
[("bnd_mono", bnd_mono),
("dom_subset", dom_subset),
("elim", elim)]
|> (PureThy.add_thmss o map Thm.no_attributes)
[("defs", defs),
("intrs", intrs)]
|> Theory.parent_path,
{defs = defs,
bnd_mono = bnd_mono,
dom_subset = dom_subset,
intrs = intrs,
elim = elim,
mk_cases = mk_cases,
induct = induct,
mutual_induct = mutual_induct})
end;
(*external version, accepting strings*)
fun add_inductive (srec_tms, sdom_sum, sintrs, monos,
con_defs, type_intrs, type_elims) thy =
let val rec_tms = map (readtm (sign_of thy) Ind_Syntax.iT) srec_tms
and dom_sum = readtm (sign_of thy) Ind_Syntax.iT sdom_sum
and intr_tms = map (readtm (sign_of thy) propT) sintrs
in
add_inductive_i true (rec_tms, dom_sum, intr_tms,
monos, con_defs, type_intrs, type_elims) thy
end
end;