(* Title: HOL/HOLCF/Tools/Domain/domain_induction.ML
Author: David von Oheimb
Author: Brian Huffman
Proofs of high-level (co)induction rules for domain command.
*)
signature DOMAIN_INDUCTION =
sig
val comp_theorems :
binding list ->
Domain_Take_Proofs.take_induct_info ->
Domain_Constructors.constr_info list ->
theory -> thm list * theory
val quiet_mode: bool Unsynchronized.ref
val trace_domain: bool Unsynchronized.ref
end
structure Domain_Induction : DOMAIN_INDUCTION =
struct
val quiet_mode = Unsynchronized.ref false
val trace_domain = Unsynchronized.ref false
fun message s = if !quiet_mode then () else writeln s
fun trace s = if !trace_domain then tracing s else ()
open HOLCF_Library
(******************************************************************************)
(***************************** proofs about take ******************************)
(******************************************************************************)
fun take_theorems
(dbinds : binding list)
(take_info : Domain_Take_Proofs.take_induct_info)
(constr_infos : Domain_Constructors.constr_info list)
(thy : theory) : thm list list * theory =
let
val {take_consts, take_Suc_thms, deflation_take_thms, ...} = take_info
val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy
val n = Free ("n", @{typ nat})
val n' = @{const Suc} $ n
local
val newTs = map (#absT o #iso_info) constr_infos
val subs = newTs ~~ map (fn t => t $ n) take_consts
fun is_ID (Const (c, _)) = (c = @{const_name ID})
| is_ID _ = false
in
fun map_of_arg thy v T =
let val m = Domain_Take_Proofs.map_of_typ thy subs T
in if is_ID m then v else mk_capply (m, v) end
end
fun prove_take_apps
((dbind, take_const), constr_info) thy =
let
val {iso_info, con_specs, con_betas, ...} : Domain_Constructors.constr_info = constr_info
val {abs_inverse, ...} = iso_info
fun prove_take_app (con_const, args) =
let
val Ts = map snd args
val ns = Name.variant_list ["n"] (Old_Datatype_Prop.make_tnames Ts)
val vs = map Free (ns ~~ Ts)
val lhs = mk_capply (take_const $ n', list_ccomb (con_const, vs))
val rhs = list_ccomb (con_const, map2 (map_of_arg thy) vs Ts)
val goal = mk_trp (mk_eq (lhs, rhs))
val rules =
[abs_inverse] @ con_betas @ @{thms take_con_rules}
@ take_Suc_thms @ deflation_thms @ deflation_take_thms
fun tac ctxt = simp_tac (put_simpset HOL_basic_ss ctxt addsimps rules) 1
in
Goal.prove_global thy [] [] goal (tac o #context)
end
val take_apps = map prove_take_app con_specs
in
yield_singleton Global_Theory.add_thmss
((Binding.qualified true "take_rews" dbind, take_apps),
[Simplifier.simp_add]) thy
end
in
fold_map prove_take_apps
(dbinds ~~ take_consts ~~ constr_infos) thy
end
(******************************************************************************)
(****************************** induction rules *******************************)
(******************************************************************************)
val case_UU_allI =
@{lemma "(!!x. x ~= UU ==> P x) ==> P UU ==> ALL x. P x" by metis}
fun prove_induction
(comp_dbind : binding)
(constr_infos : Domain_Constructors.constr_info list)
(take_info : Domain_Take_Proofs.take_induct_info)
(take_rews : thm list)
(thy : theory) =
let
val comp_dname = Binding.name_of comp_dbind
val iso_infos = map #iso_info constr_infos
val exhausts = map #exhaust constr_infos
val con_rews = maps #con_rews constr_infos
val {take_consts, take_induct_thms, ...} = take_info
val newTs = map #absT iso_infos
val P_names = Old_Datatype_Prop.indexify_names (map (K "P") newTs)
val x_names = Old_Datatype_Prop.indexify_names (map (K "x") newTs)
val P_types = map (fn T => T --> HOLogic.boolT) newTs
val Ps = map Free (P_names ~~ P_types)
val xs = map Free (x_names ~~ newTs)
val n = Free ("n", HOLogic.natT)
fun con_assm defined p (con, args) =
let
val Ts = map snd args
val ns = Name.variant_list P_names (Old_Datatype_Prop.make_tnames Ts)
val vs = map Free (ns ~~ Ts)
val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs))
fun ind_hyp (v, T) t =
case AList.lookup (op =) (newTs ~~ Ps) T of NONE => t
| SOME p' => Logic.mk_implies (mk_trp (p' $ v), t)
val t1 = mk_trp (p $ list_ccomb (con, vs))
val t2 = fold_rev ind_hyp (vs ~~ Ts) t1
val t3 = Logic.list_implies (map (mk_trp o mk_defined) nonlazy, t2)
in fold_rev Logic.all vs (if defined then t3 else t2) end
fun eq_assms ((p, T), cons) =
mk_trp (p $ HOLCF_Library.mk_bottom T) :: map (con_assm true p) cons
val assms = maps eq_assms (Ps ~~ newTs ~~ map #con_specs constr_infos)
val take_ss =
simpset_of (put_simpset HOL_ss @{context} addsimps (@{thm Rep_cfun_strict1} :: take_rews))
fun quant_tac ctxt i = EVERY
(map (fn name =>
Rule_Insts.res_inst_tac ctxt [((("x", 0), Position.none), name)] [] spec i) x_names)
(* FIXME: move this message to domain_take_proofs.ML *)
val is_finite = #is_finite take_info
val _ = if is_finite
then message ("Proving finiteness rule for domain "^comp_dname^" ...")
else ()
val _ = trace " Proving finite_ind..."
val finite_ind =
let
val concls =
map (fn ((P, t), x) => P $ mk_capply (t $ n, x))
(Ps ~~ take_consts ~~ xs)
val goal = mk_trp (foldr1 mk_conj concls)
fun tacf {prems, context = ctxt} =
let
(* Prove stronger prems, without definedness side conditions *)
fun con_thm p (con, args) =
let
val subgoal = con_assm false p (con, args)
val rules = prems @ con_rews @ @{thms simp_thms}
val simplify = asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps rules)
fun arg_tac (lazy, _) =
resolve_tac ctxt [if lazy then allI else case_UU_allI] 1
fun tacs ctxt =
rewrite_goals_tac ctxt @{thms atomize_all atomize_imp} ::
map arg_tac args @
[REPEAT (resolve_tac ctxt [impI] 1), ALLGOALS simplify]
in
Goal.prove ctxt [] [] subgoal (EVERY o tacs o #context)
end
fun eq_thms (p, cons) = map (con_thm p) cons
val conss = map #con_specs constr_infos
val prems' = maps eq_thms (Ps ~~ conss)
val tacs1 = [
quant_tac ctxt 1,
simp_tac (put_simpset HOL_ss ctxt) 1,
Induct_Tacs.induct_tac ctxt [[SOME "n"]] NONE 1,
simp_tac (put_simpset take_ss ctxt addsimps prems) 1,
TRY (safe_tac (put_claset HOL_cs ctxt))]
fun con_tac _ =
asm_simp_tac (put_simpset take_ss ctxt) 1 THEN
(resolve_tac ctxt prems' THEN_ALL_NEW eresolve_tac ctxt [spec]) 1
fun cases_tacs (cons, exhaust) =
Rule_Insts.res_inst_tac ctxt [((("y", 0), Position.none), "x")] [] exhaust 1 ::
asm_simp_tac (put_simpset take_ss ctxt addsimps prems) 1 ::
map con_tac cons
val tacs = tacs1 @ maps cases_tacs (conss ~~ exhausts)
in
EVERY (map DETERM tacs)
end
in Goal.prove_global thy [] assms goal tacf end
val _ = trace " Proving ind..."
val ind =
let
val concls = map (op $) (Ps ~~ xs)
val goal = mk_trp (foldr1 mk_conj concls)
val adms = if is_finite then [] else map (mk_trp o mk_adm) Ps
fun tacf {prems, context = ctxt} =
let
fun finite_tac (take_induct, fin_ind) =
resolve_tac ctxt [take_induct] 1 THEN
(if is_finite then all_tac else resolve_tac ctxt prems 1) THEN
(resolve_tac ctxt [fin_ind] THEN_ALL_NEW solve_tac ctxt prems) 1
val fin_inds = Project_Rule.projections ctxt finite_ind
in
TRY (safe_tac (put_claset HOL_cs ctxt)) THEN
EVERY (map finite_tac (take_induct_thms ~~ fin_inds))
end
in Goal.prove_global thy [] (adms @ assms) goal tacf end
(* case names for induction rules *)
val dnames = map (fst o dest_Type) newTs
val case_ns =
let
val adms =
if is_finite then [] else
if length dnames = 1 then ["adm"] else
map (fn s => "adm_" ^ Long_Name.base_name s) dnames
val bottoms =
if length dnames = 1 then ["bottom"] else
map (fn s => "bottom_" ^ Long_Name.base_name s) dnames
fun one_eq bot (constr_info : Domain_Constructors.constr_info) =
let fun name_of (c, _) = Long_Name.base_name (fst (dest_Const c))
in bot :: map name_of (#con_specs constr_info) end
in adms @ flat (map2 one_eq bottoms constr_infos) end
val inducts = Project_Rule.projections (Proof_Context.init_global thy) ind
fun ind_rule (dname, rule) =
((Binding.empty, rule),
[Rule_Cases.case_names case_ns, Induct.induct_type dname])
in
thy
|> snd o Global_Theory.add_thms [
((Binding.qualified true "finite_induct" comp_dbind, finite_ind), []),
((Binding.qualified true "induct" comp_dbind, ind ), [])]
|> (snd o Global_Theory.add_thms (map ind_rule (dnames ~~ inducts)))
end (* prove_induction *)
(******************************************************************************)
(************************ bisimulation and coinduction ************************)
(******************************************************************************)
fun prove_coinduction
(comp_dbind : binding, dbinds : binding list)
(constr_infos : Domain_Constructors.constr_info list)
(take_info : Domain_Take_Proofs.take_induct_info)
(take_rews : thm list list)
(thy : theory) : theory =
let
val iso_infos = map #iso_info constr_infos
val newTs = map #absT iso_infos
val {take_consts, take_0_thms, take_lemma_thms, ...} = take_info
val R_names = Old_Datatype_Prop.indexify_names (map (K "R") newTs)
val R_types = map (fn T => T --> T --> boolT) newTs
val Rs = map Free (R_names ~~ R_types)
val n = Free ("n", natT)
val reserved = "x" :: "y" :: R_names
(* declare bisimulation predicate *)
val bisim_bind = Binding.suffix_name "_bisim" comp_dbind
val bisim_type = R_types ---> boolT
val (bisim_const, thy) =
Sign.declare_const_global ((bisim_bind, bisim_type), NoSyn) thy
(* define bisimulation predicate *)
local
fun one_con T (con, args) =
let
val Ts = map snd args
val ns1 = Name.variant_list reserved (Old_Datatype_Prop.make_tnames Ts)
val ns2 = map (fn n => n^"'") ns1
val vs1 = map Free (ns1 ~~ Ts)
val vs2 = map Free (ns2 ~~ Ts)
val eq1 = mk_eq (Free ("x", T), list_ccomb (con, vs1))
val eq2 = mk_eq (Free ("y", T), list_ccomb (con, vs2))
fun rel ((v1, v2), T) =
case AList.lookup (op =) (newTs ~~ Rs) T of
NONE => mk_eq (v1, v2) | SOME r => r $ v1 $ v2
val eqs = foldr1 mk_conj (map rel (vs1 ~~ vs2 ~~ Ts) @ [eq1, eq2])
in
Library.foldr mk_ex (vs1 @ vs2, eqs)
end
fun one_eq ((T, R), cons) =
let
val x = Free ("x", T)
val y = Free ("y", T)
val disj1 = mk_conj (mk_eq (x, mk_bottom T), mk_eq (y, mk_bottom T))
val disjs = disj1 :: map (one_con T) cons
in
mk_all (x, mk_all (y, mk_imp (R $ x $ y, foldr1 mk_disj disjs)))
end
val conjs = map one_eq (newTs ~~ Rs ~~ map #con_specs constr_infos)
val bisim_rhs = lambdas Rs (Library.foldr1 mk_conj conjs)
val bisim_eqn = Logic.mk_equals (bisim_const, bisim_rhs)
in
val (bisim_def_thm, thy) = thy |>
yield_singleton (Global_Theory.add_defs false)
((Binding.qualified true "bisim_def" comp_dbind, bisim_eqn), [])
end (* local *)
(* prove coinduction lemma *)
val coind_lemma =
let
val assm = mk_trp (list_comb (bisim_const, Rs))
fun one ((T, R), take_const) =
let
val x = Free ("x", T)
val y = Free ("y", T)
val lhs = mk_capply (take_const $ n, x)
val rhs = mk_capply (take_const $ n, y)
in
mk_all (x, mk_all (y, mk_imp (R $ x $ y, mk_eq (lhs, rhs))))
end
val goal =
mk_trp (foldr1 mk_conj (map one (newTs ~~ Rs ~~ take_consts)))
val rules = @{thm Rep_cfun_strict1} :: take_0_thms
fun tacf {prems, context = ctxt} =
let
val prem' = rewrite_rule ctxt [bisim_def_thm] (hd prems)
val prems' = Project_Rule.projections ctxt prem'
val dests = map (fn th => th RS spec RS spec RS mp) prems'
fun one_tac (dest, rews) =
dresolve_tac ctxt [dest] 1 THEN safe_tac (put_claset HOL_cs ctxt) THEN
ALLGOALS (asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps rews))
in
resolve_tac ctxt @{thms nat.induct} 1 THEN
simp_tac (put_simpset HOL_ss ctxt addsimps rules) 1 THEN
safe_tac (put_claset HOL_cs ctxt) THEN
EVERY (map one_tac (dests ~~ take_rews))
end
in
Goal.prove_global thy [] [assm] goal tacf
end
(* prove individual coinduction rules *)
fun prove_coind ((T, R), take_lemma) =
let
val x = Free ("x", T)
val y = Free ("y", T)
val assm1 = mk_trp (list_comb (bisim_const, Rs))
val assm2 = mk_trp (R $ x $ y)
val goal = mk_trp (mk_eq (x, y))
fun tacf {prems, context = ctxt} =
let
val rule = hd prems RS coind_lemma
in
resolve_tac ctxt [take_lemma] 1 THEN
asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps (rule :: prems)) 1
end
in
Goal.prove_global thy [] [assm1, assm2] goal tacf
end
val coinds = map prove_coind (newTs ~~ Rs ~~ take_lemma_thms)
val coind_binds = map (Binding.qualified true "coinduct") dbinds
in
thy |> snd o Global_Theory.add_thms
(map Thm.no_attributes (coind_binds ~~ coinds))
end (* let *)
(******************************************************************************)
(******************************* main function ********************************)
(******************************************************************************)
fun comp_theorems
(dbinds : binding list)
(take_info : Domain_Take_Proofs.take_induct_info)
(constr_infos : Domain_Constructors.constr_info list)
(thy : theory) =
let
val comp_dname = space_implode "_" (map Binding.name_of dbinds)
val comp_dbind = Binding.name comp_dname
(* Test for emptiness *)
(* FIXME: reimplement emptiness test
local
open Domain_Library
val dnames = map (fst o fst) eqs
val conss = map snd eqs
fun rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
is_rec arg andalso not (member (op =) ns (rec_of arg)) andalso
((rec_of arg = n andalso not (lazy_rec orelse is_lazy arg)) orelse
rec_of arg <> n andalso rec_to (rec_of arg::ns)
(lazy_rec orelse is_lazy arg) (n, nth conss (rec_of arg)))
) o snd) cons
fun warn (n,cons) =
if rec_to [] false (n,cons)
then (warning ("domain " ^ nth dnames n ^ " is empty!") true)
else false
in
val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs
val is_emptys = map warn n__eqs
end
*)
(* Test for indirect recursion *)
local
val newTs = map (#absT o #iso_info) constr_infos
fun indirect_typ (Type (_, Ts)) =
exists (fn T => member (op =) newTs T orelse indirect_typ T) Ts
| indirect_typ _ = false
fun indirect_arg (_, T) = indirect_typ T
fun indirect_con (_, args) = exists indirect_arg args
fun indirect_eq cons = exists indirect_con cons
in
val is_indirect = exists indirect_eq (map #con_specs constr_infos)
val _ =
if is_indirect
then message "Indirect recursion detected, skipping proofs of (co)induction rules"
else message ("Proving induction properties of domain "^comp_dname^" ...")
end
(* theorems about take *)
val (take_rewss, thy) =
take_theorems dbinds take_info constr_infos thy
val {take_0_thms, take_strict_thms, ...} = take_info
val take_rews = take_0_thms @ take_strict_thms @ flat take_rewss
(* prove induction rules, unless definition is indirect recursive *)
val thy =
if is_indirect then thy else
prove_induction comp_dbind constr_infos take_info take_rews thy
val thy =
if is_indirect then thy else
prove_coinduction (comp_dbind, dbinds) constr_infos take_info take_rewss thy
in
(take_rews, thy)
end (* let *)
end (* struct *)