(* Title: HOL/Tools/function_package/scnp_reconstruct.ML
Author: Armin Heller, TU Muenchen
Author: Alexander Krauss, TU Muenchen
Proof reconstruction for SCNP
*)
signature SCNP_RECONSTRUCT =
sig
val sizechange_tac : Proof.context -> tactic -> tactic
val decomp_scnp : ScnpSolve.label list -> Proof.context -> Proof.method
val setup : theory -> theory
datatype multiset_setup =
Multiset of
{
msetT : typ -> typ,
mk_mset : typ -> term list -> term,
mset_regroup_conv : int list -> conv,
mset_member_tac : int -> int -> tactic,
mset_nonempty_tac : int -> tactic,
mset_pwleq_tac : int -> tactic,
set_of_simps : thm list,
smsI' : thm,
wmsI2'' : thm,
wmsI1 : thm,
reduction_pair : thm
}
val multiset_setup : multiset_setup -> theory -> theory
end
structure ScnpReconstruct : SCNP_RECONSTRUCT =
struct
val PROFILE = FundefCommon.PROFILE
fun TRACE x = if ! FundefCommon.profile then Output.tracing x else ()
open ScnpSolve
val natT = HOLogic.natT
val nat_pairT = HOLogic.mk_prodT (natT, natT)
(* Theory dependencies *)
datatype multiset_setup =
Multiset of
{
msetT : typ -> typ,
mk_mset : typ -> term list -> term,
mset_regroup_conv : int list -> conv,
mset_member_tac : int -> int -> tactic,
mset_nonempty_tac : int -> tactic,
mset_pwleq_tac : int -> tactic,
set_of_simps : thm list,
smsI' : thm,
wmsI2'' : thm,
wmsI1 : thm,
reduction_pair : thm
}
structure MultisetSetup = TheoryDataFun
(
type T = multiset_setup option
val empty = NONE
val copy = I;
val extend = I;
fun merge _ (v1, v2) = if is_some v2 then v2 else v1
)
val multiset_setup = MultisetSetup.put o SOME
fun undef x = error "undef"
fun get_multiset_setup thy = MultisetSetup.get thy
|> the_default (Multiset
{ msetT = undef, mk_mset=undef,
mset_regroup_conv=undef, mset_member_tac = undef,
mset_nonempty_tac = undef, mset_pwleq_tac = undef,
set_of_simps = [],reduction_pair = refl,
smsI'=refl, wmsI2''=refl, wmsI1=refl })
fun order_rpair _ MAX = @{thm max_rpair_set}
| order_rpair msrp MS = msrp
| order_rpair _ MIN = @{thm min_rpair_set}
fun ord_intros_max true =
(@{thm smax_emptyI}, @{thm smax_insertI})
| ord_intros_max false =
(@{thm wmax_emptyI}, @{thm wmax_insertI})
fun ord_intros_min true =
(@{thm smin_emptyI}, @{thm smin_insertI})
| ord_intros_min false =
(@{thm wmin_emptyI}, @{thm wmin_insertI})
fun gen_probl D cs =
let
val n = Termination.get_num_points D
val arity = length o Termination.get_measures D
fun measure p i = nth (Termination.get_measures D p) i
fun mk_graph c =
let
val (_, p, _, q, _, _) = Termination.dest_call D c
fun add_edge i j =
case Termination.get_descent D c (measure p i) (measure q j)
of SOME (Termination.Less _) => cons (i, GTR, j)
| SOME (Termination.LessEq _) => cons (i, GEQ, j)
| _ => I
val edges =
fold_product add_edge (0 upto arity p - 1) (0 upto arity q - 1) []
in
G (p, q, edges)
end
in
GP (map arity (0 upto n - 1), map mk_graph cs)
end
(* General reduction pair application *)
fun rem_inv_img ctxt =
let
val unfold_tac = LocalDefs.unfold_tac ctxt
in
rtac @{thm subsetI} 1
THEN etac @{thm CollectE} 1
THEN REPEAT (etac @{thm exE} 1)
THEN unfold_tac @{thms inv_image_def}
THEN rtac @{thm CollectI} 1
THEN etac @{thm conjE} 1
THEN etac @{thm ssubst} 1
THEN unfold_tac (@{thms split_conv} @ @{thms triv_forall_equality}
@ @{thms sum.cases})
end
(* Sets *)
val setT = HOLogic.mk_setT
fun set_member_tac m i =
if m = 0 then rtac @{thm insertI1} i
else rtac @{thm insertI2} i THEN set_member_tac (m - 1) i
val set_nonempty_tac = rtac @{thm insert_not_empty}
fun set_finite_tac i =
rtac @{thm finite.emptyI} i
ORELSE (rtac @{thm finite.insertI} i THEN (fn st => set_finite_tac i st))
(* Reconstruction *)
fun reconstruct_tac ctxt D cs (gp as GP (_, gs)) certificate =
let
val thy = ProofContext.theory_of ctxt
val Multiset
{ msetT, mk_mset,
mset_regroup_conv, mset_member_tac,
mset_nonempty_tac, mset_pwleq_tac, set_of_simps,
smsI', wmsI2'', wmsI1, reduction_pair=ms_rp }
= get_multiset_setup thy
fun measure_fn p = nth (Termination.get_measures D p)
fun get_desc_thm cidx m1 m2 bStrict =
case Termination.get_descent D (nth cs cidx) m1 m2
of SOME (Termination.Less thm) =>
if bStrict then thm
else (thm COMP (Thm.lift_rule (cprop_of thm) @{thm less_imp_le}))
| SOME (Termination.LessEq (thm, _)) =>
if not bStrict then thm
else sys_error "get_desc_thm"
| _ => sys_error "get_desc_thm"
val (label, lev, sl, covering) = certificate
fun prove_lev strict g =
let
val G (p, q, el) = nth gs g
fun less_proof strict (j, b) (i, a) =
let
val tag_flag = b < a orelse (not strict andalso b <= a)
val stored_thm =
get_desc_thm g (measure_fn p i) (measure_fn q j)
(not tag_flag)
|> Conv.fconv_rule (Thm.beta_conversion true)
val rule = if strict
then if b < a then @{thm pair_lessI2} else @{thm pair_lessI1}
else if b <= a then @{thm pair_leqI2} else @{thm pair_leqI1}
in
rtac rule 1 THEN PRIMITIVE (Thm.elim_implies stored_thm)
THEN (if tag_flag then Arith_Data.verbose_arith_tac ctxt 1 else all_tac)
end
fun steps_tac MAX strict lq lp =
let
val (empty, step) = ord_intros_max strict
in
if length lq = 0
then rtac empty 1 THEN set_finite_tac 1
THEN (if strict then set_nonempty_tac 1 else all_tac)
else
let
val (j, b) :: rest = lq
val (i, a) = the (covering g strict j)
fun choose xs = set_member_tac (Library.find_index (curry op = (i, a)) xs) 1
val solve_tac = choose lp THEN less_proof strict (j, b) (i, a)
in
rtac step 1 THEN solve_tac THEN steps_tac MAX strict rest lp
end
end
| steps_tac MIN strict lq lp =
let
val (empty, step) = ord_intros_min strict
in
if length lp = 0
then rtac empty 1
THEN (if strict then set_nonempty_tac 1 else all_tac)
else
let
val (i, a) :: rest = lp
val (j, b) = the (covering g strict i)
fun choose xs = set_member_tac (Library.find_index (curry op = (j, b)) xs) 1
val solve_tac = choose lq THEN less_proof strict (j, b) (i, a)
in
rtac step 1 THEN solve_tac THEN steps_tac MIN strict lq rest
end
end
| steps_tac MS strict lq lp =
let
fun get_str_cover (j, b) =
if is_some (covering g true j) then SOME (j, b) else NONE
fun get_wk_cover (j, b) = the (covering g false j)
val qs = lq \\ map_filter get_str_cover lq
val ps = map get_wk_cover qs
fun indices xs ys = map (fn y => Library.find_index (curry op = y) xs) ys
val iqs = indices lq qs
val ips = indices lp ps
local open Conv in
fun t_conv a C =
params_conv ~1 (K ((concl_conv ~1 o arg_conv o arg1_conv o a) C)) ctxt
val goal_rewrite =
t_conv arg1_conv (mset_regroup_conv iqs)
then_conv t_conv arg_conv (mset_regroup_conv ips)
end
in
CONVERSION goal_rewrite 1
THEN (if strict then rtac smsI' 1
else if qs = lq then rtac wmsI2'' 1
else rtac wmsI1 1)
THEN mset_pwleq_tac 1
THEN EVERY (map2 (less_proof false) qs ps)
THEN (if strict orelse qs <> lq
then LocalDefs.unfold_tac ctxt set_of_simps
THEN steps_tac MAX true (lq \\ qs) (lp \\ ps)
else all_tac)
end
in
rem_inv_img ctxt
THEN steps_tac label strict (nth lev q) (nth lev p)
end
val (mk_set, setT) = if label = MS then (mk_mset, msetT) else (HOLogic.mk_set, setT)
fun tag_pair p (i, tag) =
HOLogic.pair_const natT natT $
(measure_fn p i $ Bound 0) $ HOLogic.mk_number natT tag
fun pt_lev (p, lm) = Abs ("x", Termination.get_types D p,
mk_set nat_pairT (map (tag_pair p) lm))
val level_mapping =
map_index pt_lev lev
|> Termination.mk_sumcases D (setT nat_pairT)
|> cterm_of thy
in
PROFILE "Proof Reconstruction"
(CONVERSION (Conv.arg_conv (Conv.arg_conv (FundefLib.regroup_union_conv sl))) 1
THEN (rtac @{thm reduction_pair_lemma} 1)
THEN (rtac @{thm rp_inv_image_rp} 1)
THEN (rtac (order_rpair ms_rp label) 1)
THEN PRIMITIVE (instantiate' [] [SOME level_mapping])
THEN unfold_tac @{thms rp_inv_image_def} (local_simpset_of ctxt)
THEN LocalDefs.unfold_tac ctxt
(@{thms split_conv} @ @{thms fst_conv} @ @{thms snd_conv})
THEN REPEAT (SOMEGOAL (resolve_tac [@{thm Un_least}, @{thm empty_subsetI}]))
THEN EVERY (map (prove_lev true) sl)
THEN EVERY (map (prove_lev false) ((0 upto length cs - 1) \\ sl)))
end
local open Termination in
fun print_cell (SOME (Less _)) = "<"
| print_cell (SOME (LessEq _)) = "≤"
| print_cell (SOME (None _)) = "-"
| print_cell (SOME (False _)) = "-"
| print_cell (NONE) = "?"
fun print_error ctxt D = CALLS (fn (cs, i) =>
let
val np = get_num_points D
val ms = map (get_measures D) (0 upto np - 1)
val tys = map (get_types D) (0 upto np - 1)
fun index xs = (1 upto length xs) ~~ xs
fun outp s t f xs = map (fn (x, y) => s ^ Int.toString x ^ t ^ f y ^ "\n") xs
val ims = index (map index ms)
val _ = Output.tracing (concat (outp "fn #" ":\n" (concat o outp "\tmeasure #" ": " (Syntax.string_of_term ctxt)) ims))
fun print_call (k, c) =
let
val (_, p, _, q, _, _) = dest_call D c
val _ = Output.tracing ("call table for call #" ^ Int.toString k ^ ": fn " ^
Int.toString (p + 1) ^ " ~> fn " ^ Int.toString (q + 1))
val caller_ms = nth ms p
val callee_ms = nth ms q
val entries = map (fn x => map (pair x) (callee_ms)) (caller_ms)
fun print_ln (i : int, l) = concat (Int.toString i :: " " :: map (enclose " " " " o print_cell o (uncurry (get_descent D c))) l)
val _ = Output.tracing (concat (Int.toString (p + 1) ^ "|" ^ Int.toString (q + 1) ^
" " :: map (enclose " " " " o Int.toString) (1 upto length callee_ms)) ^ "\n"
^ cat_lines (map print_ln ((1 upto (length entries)) ~~ entries)))
in
true
end
fun list_call (k, c) =
let
val (_, p, _, q, _, _) = dest_call D c
val _ = Output.tracing ("call #" ^ (Int.toString k) ^ ": fn " ^
Int.toString (p + 1) ^ " ~> fn " ^ Int.toString (q + 1) ^ "\n" ^
(Syntax.string_of_term ctxt c))
in true end
val _ = forall list_call ((1 upto length cs) ~~ cs)
val _ = forall print_call ((1 upto length cs) ~~ cs)
in
all_tac
end)
end
fun single_scnp_tac use_tags orders ctxt cont err_cont D = Termination.CALLS (fn (cs, i) =>
let
val ms_configured = is_some (MultisetSetup.get (ProofContext.theory_of ctxt))
val orders' = if ms_configured then orders
else filter_out (curry op = MS) orders
val gp = gen_probl D cs
(* val _ = TRACE ("SCNP instance: " ^ makestring gp)*)
val certificate = generate_certificate use_tags orders' gp
(* val _ = TRACE ("Certificate: " ^ makestring certificate)*)
in
case certificate
of NONE => err_cont D i
| SOME cert =>
SELECT_GOAL (reconstruct_tac ctxt D cs gp cert) i
THEN (rtac @{thm wf_empty} i ORELSE cont D i)
end)
fun gen_decomp_scnp_tac orders autom_tac ctxt err_cont =
let
open Termination
val derive_diag = Descent.derive_diag ctxt autom_tac
val derive_all = Descent.derive_all ctxt autom_tac
val decompose = Decompose.decompose_tac ctxt autom_tac
val scnp_no_tags = single_scnp_tac false orders ctxt
val scnp_full = single_scnp_tac true orders ctxt
fun first_round c e =
derive_diag (REPEAT scnp_no_tags c e)
val second_round =
REPEAT (fn c => fn e => decompose (scnp_no_tags c c) e)
val third_round =
derive_all oo
REPEAT (fn c => fn e =>
scnp_full (decompose c c) e)
fun Then s1 s2 c e = s1 (s2 c c) (s2 c e)
val strategy = Then (Then first_round second_round) third_round
in
TERMINATION ctxt (strategy err_cont err_cont)
end
fun gen_sizechange_tac orders autom_tac ctxt err_cont =
TRY (FundefCommon.apply_termination_rule ctxt 1)
THEN TRY (Termination.wf_union_tac ctxt)
THEN
(rtac @{thm wf_empty} 1
ORELSE gen_decomp_scnp_tac orders autom_tac ctxt err_cont 1)
fun sizechange_tac ctxt autom_tac =
gen_sizechange_tac [MAX, MS, MIN] autom_tac ctxt (K (K all_tac))
fun decomp_scnp orders ctxt =
let
val extra_simps = FundefCommon.TerminationSimps.get ctxt
val autom_tac = auto_tac (local_clasimpset_of ctxt addsimps2 extra_simps)
in
SIMPLE_METHOD
(gen_sizechange_tac orders autom_tac ctxt (print_error ctxt))
end
(* Method setup *)
val orders =
(Scan.repeat1
((Args.$$$ "max" >> K MAX) ||
(Args.$$$ "min" >> K MIN) ||
(Args.$$$ "ms" >> K MS))
|| Scan.succeed [MAX, MS, MIN])
val setup = Method.add_method
("sizechange", Method.sectioned_args (Scan.lift orders) clasimp_modifiers decomp_scnp,
"termination prover with graph decomposition and the NP subset of size change termination")
end