(* Title: Provers/classical.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Theorem prover for classical reasoning, including predicate calculus, set
theory, etc.
Rules must be classified as intro, elim, safe, hazardous (unsafe).
A rule is unsafe unless it can be applied blindly without harmful results.
For a rule to be safe, its premises and conclusion should be logically
equivalent. There should be no variables in the premises that are not in
the conclusion.
*)
(*higher precedence than := facilitates use of references*)
infix 4 addSIs addSEs addSDs addIs addEs addDs delrules
addSWrapper delSWrapper addWrapper delWrapper
addSbefore addSafter addbefore addafter
addD2 addE2 addSD2 addSE2;
signature CLASSICAL_DATA =
sig
val imp_elim: thm (* P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R *)
val not_elim: thm (* ~P ==> P ==> R *)
val swap: thm (* ~ P ==> (~ R ==> P) ==> R *)
val classical: thm (* (~ P ==> P) ==> P *)
val sizef: thm -> int (* size function for BEST_FIRST, typically size_of_thm *)
val hyp_subst_tacs: (Proof.context -> int -> tactic) list (* optional tactics for
substitution in the hypotheses; assumed to be safe! *)
end;
signature BASIC_CLASSICAL =
sig
type wrapper = (int -> tactic) -> int -> tactic
type claset
val print_claset: Proof.context -> unit
val addDs: Proof.context * thm list -> Proof.context
val addEs: Proof.context * thm list -> Proof.context
val addIs: Proof.context * thm list -> Proof.context
val addSDs: Proof.context * thm list -> Proof.context
val addSEs: Proof.context * thm list -> Proof.context
val addSIs: Proof.context * thm list -> Proof.context
val delrules: Proof.context * thm list -> Proof.context
val addSWrapper: Proof.context * (string * (Proof.context -> wrapper)) -> Proof.context
val delSWrapper: Proof.context * string -> Proof.context
val addWrapper: Proof.context * (string * (Proof.context -> wrapper)) -> Proof.context
val delWrapper: Proof.context * string -> Proof.context
val addSbefore: Proof.context * (string * (Proof.context -> int -> tactic)) -> Proof.context
val addSafter: Proof.context * (string * (Proof.context -> int -> tactic)) -> Proof.context
val addbefore: Proof.context * (string * (Proof.context -> int -> tactic)) -> Proof.context
val addafter: Proof.context * (string * (Proof.context -> int -> tactic)) -> Proof.context
val addD2: Proof.context * (string * thm) -> Proof.context
val addE2: Proof.context * (string * thm) -> Proof.context
val addSD2: Proof.context * (string * thm) -> Proof.context
val addSE2: Proof.context * (string * thm) -> Proof.context
val appSWrappers: Proof.context -> wrapper
val appWrappers: Proof.context -> wrapper
val claset_of: Proof.context -> claset
val put_claset: claset -> Proof.context -> Proof.context
val map_theory_claset: (Proof.context -> Proof.context) -> theory -> theory
val fast_tac: Proof.context -> int -> tactic
val slow_tac: Proof.context -> int -> tactic
val astar_tac: Proof.context -> int -> tactic
val slow_astar_tac: Proof.context -> int -> tactic
val best_tac: Proof.context -> int -> tactic
val first_best_tac: Proof.context -> int -> tactic
val slow_best_tac: Proof.context -> int -> tactic
val depth_tac: Proof.context -> int -> int -> tactic
val deepen_tac: Proof.context -> int -> int -> tactic
val contr_tac: Proof.context -> int -> tactic
val dup_elim: Proof.context -> thm -> thm
val dup_intr: thm -> thm
val dup_step_tac: Proof.context -> int -> tactic
val eq_mp_tac: Proof.context -> int -> tactic
val haz_step_tac: Proof.context -> int -> tactic
val joinrules: thm list * thm list -> (bool * thm) list
val mp_tac: Proof.context -> int -> tactic
val safe_tac: Proof.context -> tactic
val safe_steps_tac: Proof.context -> int -> tactic
val safe_step_tac: Proof.context -> int -> tactic
val clarify_tac: Proof.context -> int -> tactic
val clarify_step_tac: Proof.context -> int -> tactic
val step_tac: Proof.context -> int -> tactic
val slow_step_tac: Proof.context -> int -> tactic
val swapify: thm list -> thm list
val swap_res_tac: Proof.context -> thm list -> int -> tactic
val inst_step_tac: Proof.context -> int -> tactic
val inst0_step_tac: Proof.context -> int -> tactic
val instp_step_tac: Proof.context -> int -> tactic
end;
signature CLASSICAL =
sig
include BASIC_CLASSICAL
val classical_rule: Proof.context -> thm -> thm
type netpair = (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net
val rep_cs: claset ->
{safeIs: thm Item_Net.T,
safeEs: thm Item_Net.T,
hazIs: thm Item_Net.T,
hazEs: thm Item_Net.T,
swrappers: (string * (Proof.context -> wrapper)) list,
uwrappers: (string * (Proof.context -> wrapper)) list,
safe0_netpair: netpair,
safep_netpair: netpair,
haz_netpair: netpair,
dup_netpair: netpair,
extra_netpair: Context_Rules.netpair}
val get_cs: Context.generic -> claset
val map_cs: (claset -> claset) -> Context.generic -> Context.generic
val safe_dest: int option -> attribute
val safe_elim: int option -> attribute
val safe_intro: int option -> attribute
val haz_dest: int option -> attribute
val haz_elim: int option -> attribute
val haz_intro: int option -> attribute
val rule_del: attribute
val cla_modifiers: Method.modifier parser list
val cla_method:
(Proof.context -> tactic) -> (Proof.context -> Proof.method) context_parser
val cla_method':
(Proof.context -> int -> tactic) -> (Proof.context -> Proof.method) context_parser
end;
functor Classical(Data: CLASSICAL_DATA): CLASSICAL =
struct
(** classical elimination rules **)
(*
Classical reasoning requires stronger elimination rules. For
instance, make_elim of Pure transforms the HOL rule injD into
[| inj f; f x = f y; x = y ==> PROP W |] ==> PROP W
Such rules can cause fast_tac to fail and blast_tac to report "PROOF
FAILED"; classical_rule will strengthen this to
[| inj f; ~ W ==> f x = f y; x = y ==> W |] ==> W
*)
fun classical_rule ctxt rule =
if is_some (Object_Logic.elim_concl ctxt rule) then
let
val thy = Proof_Context.theory_of ctxt;
val rule' = rule RS Data.classical;
val concl' = Thm.concl_of rule';
fun redundant_hyp goal =
concl' aconv Logic.strip_assums_concl goal orelse
(case Logic.strip_assums_hyp goal of
hyp :: hyps => exists (fn t => t aconv hyp) hyps
| _ => false);
val rule'' =
rule' |> ALLGOALS (SUBGOAL (fn (goal, i) =>
if i = 1 orelse redundant_hyp goal
then eresolve_tac ctxt [thin_rl] i
else all_tac))
|> Seq.hd
|> Drule.zero_var_indexes;
in if Thm.equiv_thm thy (rule, rule'') then rule else rule'' end
else rule;
(*flatten nested meta connectives in prems*)
fun flat_rule ctxt =
Conv.fconv_rule (Conv.prems_conv ~1 (Object_Logic.atomize_prems ctxt));
(*** Useful tactics for classical reasoning ***)
(*Prove goal that assumes both P and ~P.
No backtracking if it finds an equal assumption. Perhaps should call
ematch_tac instead of eresolve_tac, but then cannot prove ZF/cantor.*)
fun contr_tac ctxt =
eresolve_tac ctxt [Data.not_elim] THEN' (eq_assume_tac ORELSE' assume_tac ctxt);
(*Finds P-->Q and P in the assumptions, replaces implication by Q.
Could do the same thing for P<->Q and P... *)
fun mp_tac ctxt i =
eresolve_tac ctxt [Data.not_elim, Data.imp_elim] i THEN assume_tac ctxt i;
(*Like mp_tac but instantiates no variables*)
fun eq_mp_tac ctxt i = ematch_tac ctxt [Data.not_elim, Data.imp_elim] i THEN eq_assume_tac i;
(*Creates rules to eliminate ~A, from rules to introduce A*)
fun swapify intrs = intrs RLN (2, [Data.swap]);
val swapped = Thm.rule_attribute (fn _ => fn th => th RSN (2, Data.swap));
(*Uses introduction rules in the normal way, or on negated assumptions,
trying rules in order. *)
fun swap_res_tac ctxt rls =
let fun addrl rl brls = (false, rl) :: (true, rl RSN (2, Data.swap)) :: brls in
assume_tac ctxt ORELSE'
contr_tac ctxt ORELSE'
biresolve_tac ctxt (fold_rev addrl rls [])
end;
(*Duplication of hazardous rules, for complete provers*)
fun dup_intr th = zero_var_indexes (th RS Data.classical);
fun dup_elim ctxt th =
let val rl = (th RSN (2, revcut_rl)) |> Thm.assumption (SOME ctxt) 2 |> Seq.hd;
in rule_by_tactic ctxt (TRYALL (eresolve_tac ctxt [revcut_rl])) rl end;
(**** Classical rule sets ****)
type netpair = (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net;
type wrapper = (int -> tactic) -> int -> tactic;
datatype claset =
CS of
{safeIs : thm Item_Net.T, (*safe introduction rules*)
safeEs : thm Item_Net.T, (*safe elimination rules*)
hazIs : thm Item_Net.T, (*unsafe introduction rules*)
hazEs : thm Item_Net.T, (*unsafe elimination rules*)
swrappers : (string * (Proof.context -> wrapper)) list, (*for transforming safe_step_tac*)
uwrappers : (string * (Proof.context -> wrapper)) list, (*for transforming step_tac*)
safe0_netpair : netpair, (*nets for trivial cases*)
safep_netpair : netpair, (*nets for >0 subgoals*)
haz_netpair : netpair, (*nets for unsafe rules*)
dup_netpair : netpair, (*nets for duplication*)
extra_netpair : Context_Rules.netpair}; (*nets for extra rules*)
val empty_netpair = (Net.empty, Net.empty);
val empty_cs =
CS
{safeIs = Thm.full_rules,
safeEs = Thm.full_rules,
hazIs = Thm.full_rules,
hazEs = Thm.full_rules,
swrappers = [],
uwrappers = [],
safe0_netpair = empty_netpair,
safep_netpair = empty_netpair,
haz_netpair = empty_netpair,
dup_netpair = empty_netpair,
extra_netpair = empty_netpair};
fun rep_cs (CS args) = args;
(*** Adding (un)safe introduction or elimination rules.
In case of overlap, new rules are tried BEFORE old ones!!
***)
fun default_context (SOME context) _ = Context.proof_of context
| default_context NONE th = Proof_Context.init_global (Thm.theory_of_thm th);
(*For use with biresolve_tac. Combines intro rules with swap to handle negated
assumptions. Pairs elim rules with true. *)
fun joinrules (intrs, elims) =
(map (pair true) (elims @ swapify intrs)) @ map (pair false) intrs;
fun joinrules' (intrs, elims) =
map (pair true) elims @ map (pair false) intrs;
(*Priority: prefer rules with fewest subgoals,
then rules added most recently (preferring the head of the list).*)
fun tag_brls k [] = []
| tag_brls k (brl::brls) =
(1000000*subgoals_of_brl brl + k, brl) ::
tag_brls (k+1) brls;
fun tag_brls' _ _ [] = []
| tag_brls' w k (brl::brls) = ((w, k), brl) :: tag_brls' w (k + 1) brls;
fun insert_tagged_list rls = fold_rev Tactic.insert_tagged_brl rls;
(*Insert into netpair that already has nI intr rules and nE elim rules.
Count the intr rules double (to account for swapify). Negate to give the
new insertions the lowest priority.*)
fun insert (nI, nE) = insert_tagged_list o (tag_brls (~(2*nI+nE))) o joinrules;
fun insert' w (nI, nE) = insert_tagged_list o tag_brls' w (~(nI + nE)) o joinrules';
fun delete_tagged_list rls = fold_rev Tactic.delete_tagged_brl rls;
fun delete x = delete_tagged_list (joinrules x);
fun delete' x = delete_tagged_list (joinrules' x);
fun bad_thm (SOME context) msg th =
error (msg ^ "\n" ^ Display.string_of_thm (Context.proof_of context) th)
| bad_thm NONE msg th = raise THM (msg, 0, [th]);
fun make_elim opt_context th =
if has_fewer_prems 1 th then bad_thm opt_context "Ill-formed destruction rule" th
else Tactic.make_elim th;
fun warn_thm (SOME (Context.Proof ctxt)) msg th =
if Context_Position.is_really_visible ctxt
then warning (msg ^ Display.string_of_thm ctxt th) else ()
| warn_thm _ _ _ = ();
fun warn_rules opt_context msg rules th =
Item_Net.member rules th andalso (warn_thm opt_context msg th; true);
fun warn_claset opt_context th (CS {safeIs, safeEs, hazIs, hazEs, ...}) =
warn_rules opt_context "Rule already declared as safe introduction (intro!)\n" safeIs th orelse
warn_rules opt_context "Rule already declared as safe elimination (elim!)\n" safeEs th orelse
warn_rules opt_context "Rule already declared as introduction (intro)\n" hazIs th orelse
warn_rules opt_context "Rule already declared as elimination (elim)\n" hazEs th;
(*** Safe rules ***)
fun addSI w opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if warn_rules opt_context "Ignoring duplicate safe introduction (intro!)\n" safeIs th then cs
else
let
val ctxt = default_context opt_context th;
val th' = flat_rule ctxt th;
val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
List.partition Thm.no_prems [th'];
val nI = Item_Net.length safeIs + 1;
val nE = Item_Net.length safeEs;
val _ = warn_claset opt_context th cs;
in
CS
{safeIs = Item_Net.update th safeIs,
safe0_netpair = insert (nI,nE) (safe0_rls, []) safe0_netpair,
safep_netpair = insert (nI,nE) (safep_rls, []) safep_netpair,
safeEs = safeEs,
hazIs = hazIs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
haz_netpair = haz_netpair,
dup_netpair = dup_netpair,
extra_netpair = insert' (the_default 0 w) (nI,nE) ([th], []) extra_netpair}
end;
fun addSE w opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if warn_rules opt_context "Ignoring duplicate safe elimination (elim!)\n" safeEs th then cs
else if has_fewer_prems 1 th then bad_thm opt_context "Ill-formed elimination rule" th
else
let
val ctxt = default_context opt_context th;
val th' = classical_rule ctxt (flat_rule ctxt th);
val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
List.partition (fn rl => Thm.nprems_of rl=1) [th'];
val nI = Item_Net.length safeIs;
val nE = Item_Net.length safeEs + 1;
val _ = warn_claset opt_context th cs;
in
CS
{safeEs = Item_Net.update th safeEs,
safe0_netpair = insert (nI,nE) ([], safe0_rls) safe0_netpair,
safep_netpair = insert (nI,nE) ([], safep_rls) safep_netpair,
safeIs = safeIs,
hazIs = hazIs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
haz_netpair = haz_netpair,
dup_netpair = dup_netpair,
extra_netpair = insert' (the_default 0 w) (nI,nE) ([], [th]) extra_netpair}
end;
fun addSD w opt_context th = addSE w opt_context (make_elim opt_context th);
(*** Hazardous (unsafe) rules ***)
fun addI w opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if warn_rules opt_context "Ignoring duplicate introduction (intro)\n" hazIs th then cs
else
let
val ctxt = default_context opt_context th;
val th' = flat_rule ctxt th;
val nI = Item_Net.length hazIs + 1;
val nE = Item_Net.length hazEs;
val _ = warn_claset opt_context th cs;
in
CS
{hazIs = Item_Net.update th hazIs,
haz_netpair = insert (nI, nE) ([th'], []) haz_netpair,
dup_netpair = insert (nI, nE) ([dup_intr th'], []) dup_netpair,
safeIs = safeIs,
safeEs = safeEs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
safe0_netpair = safe0_netpair,
safep_netpair = safep_netpair,
extra_netpair = insert' (the_default 1 w) (nI, nE) ([th], []) extra_netpair}
end
handle THM ("RSN: no unifiers", _, _) => (*from dup_intr*) (* FIXME !? *)
bad_thm opt_context "Ill-formed introduction rule" th;
fun addE w opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if warn_rules opt_context "Ignoring duplicate elimination (elim)\n" hazEs th then cs
else if has_fewer_prems 1 th then bad_thm opt_context "Ill-formed elimination rule" th
else
let
val ctxt = default_context opt_context th;
val th' = classical_rule ctxt (flat_rule ctxt th);
val nI = Item_Net.length hazIs;
val nE = Item_Net.length hazEs + 1;
val _ = warn_claset opt_context th cs;
in
CS
{hazEs = Item_Net.update th hazEs,
haz_netpair = insert (nI, nE) ([], [th']) haz_netpair,
dup_netpair = insert (nI, nE) ([], [dup_elim ctxt th']) dup_netpair,
safeIs = safeIs,
safeEs = safeEs,
hazIs = hazIs,
swrappers = swrappers,
uwrappers = uwrappers,
safe0_netpair = safe0_netpair,
safep_netpair = safep_netpair,
extra_netpair = insert' (the_default 1 w) (nI, nE) ([], [th]) extra_netpair}
end;
fun addD w opt_context th = addE w opt_context (make_elim opt_context th);
(*** Deletion of rules
Working out what to delete, requires repeating much of the code used
to insert.
***)
fun delSI opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if Item_Net.member safeIs th then
let
val ctxt = default_context opt_context th;
val th' = flat_rule ctxt th;
val (safe0_rls, safep_rls) = List.partition Thm.no_prems [th'];
in
CS
{safe0_netpair = delete (safe0_rls, []) safe0_netpair,
safep_netpair = delete (safep_rls, []) safep_netpair,
safeIs = Item_Net.remove th safeIs,
safeEs = safeEs,
hazIs = hazIs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
haz_netpair = haz_netpair,
dup_netpair = dup_netpair,
extra_netpair = delete' ([th], []) extra_netpair}
end
else cs;
fun delSE opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if Item_Net.member safeEs th then
let
val ctxt = default_context opt_context th;
val th' = classical_rule ctxt (flat_rule ctxt th);
val (safe0_rls, safep_rls) = List.partition (fn rl => Thm.nprems_of rl = 1) [th'];
in
CS
{safe0_netpair = delete ([], safe0_rls) safe0_netpair,
safep_netpair = delete ([], safep_rls) safep_netpair,
safeIs = safeIs,
safeEs = Item_Net.remove th safeEs,
hazIs = hazIs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
haz_netpair = haz_netpair,
dup_netpair = dup_netpair,
extra_netpair = delete' ([], [th]) extra_netpair}
end
else cs;
fun delI opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if Item_Net.member hazIs th then
let
val ctxt = default_context opt_context th;
val th' = flat_rule ctxt th;
in
CS
{haz_netpair = delete ([th'], []) haz_netpair,
dup_netpair = delete ([dup_intr th'], []) dup_netpair,
safeIs = safeIs,
safeEs = safeEs,
hazIs = Item_Net.remove th hazIs,
hazEs = hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
safe0_netpair = safe0_netpair,
safep_netpair = safep_netpair,
extra_netpair = delete' ([th], []) extra_netpair}
end
else cs
handle THM ("RSN: no unifiers", _, _) => (*from dup_intr*) (* FIXME !? *)
bad_thm opt_context "Ill-formed introduction rule" th;
fun delE opt_context th
(cs as CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
if Item_Net.member hazEs th then
let
val ctxt = default_context opt_context th;
val th' = classical_rule ctxt (flat_rule ctxt th);
in
CS
{haz_netpair = delete ([], [th']) haz_netpair,
dup_netpair = delete ([], [dup_elim ctxt th']) dup_netpair,
safeIs = safeIs,
safeEs = safeEs,
hazIs = hazIs,
hazEs = Item_Net.remove th hazEs,
swrappers = swrappers,
uwrappers = uwrappers,
safe0_netpair = safe0_netpair,
safep_netpair = safep_netpair,
extra_netpair = delete' ([], [th]) extra_netpair}
end
else cs;
(*Delete ALL occurrences of "th" in the claset (perhaps from several lists)*)
fun delrule opt_context th (cs as CS {safeIs, safeEs, hazIs, hazEs, ...}) =
let val th' = Tactic.make_elim th in
if Item_Net.member safeIs th orelse Item_Net.member safeEs th orelse
Item_Net.member hazIs th orelse Item_Net.member hazEs th orelse
Item_Net.member safeEs th' orelse Item_Net.member hazEs th'
then
cs
|> delE opt_context th'
|> delSE opt_context th'
|> delE opt_context th
|> delI opt_context th
|> delSE opt_context th
|> delSI opt_context th
else (warn_thm opt_context "Undeclared classical rule\n" th; cs)
end;
(** claset data **)
(* wrappers *)
fun map_swrappers f
(CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
CS {safeIs = safeIs, safeEs = safeEs, hazIs = hazIs, hazEs = hazEs,
swrappers = f swrappers, uwrappers = uwrappers,
safe0_netpair = safe0_netpair, safep_netpair = safep_netpair,
haz_netpair = haz_netpair, dup_netpair = dup_netpair, extra_netpair = extra_netpair};
fun map_uwrappers f
(CS {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers,
safe0_netpair, safep_netpair, haz_netpair, dup_netpair, extra_netpair}) =
CS {safeIs = safeIs, safeEs = safeEs, hazIs = hazIs, hazEs = hazEs,
swrappers = swrappers, uwrappers = f uwrappers,
safe0_netpair = safe0_netpair, safep_netpair = safep_netpair,
haz_netpair = haz_netpair, dup_netpair = dup_netpair, extra_netpair = extra_netpair};
(* merge_cs *)
(*Merge works by adding all new rules of the 2nd claset into the 1st claset,
in order to preserve priorities reliably.*)
fun merge_thms add thms1 thms2 =
fold_rev (fn thm => if Item_Net.member thms1 thm then I else add thm) (Item_Net.content thms2);
fun merge_cs (cs as CS {safeIs, safeEs, hazIs, hazEs, ...},
cs' as CS {safeIs = safeIs2, safeEs = safeEs2, hazIs = hazIs2, hazEs = hazEs2,
swrappers, uwrappers, ...}) =
if pointer_eq (cs, cs') then cs
else
cs
|> merge_thms (addSI NONE NONE) safeIs safeIs2
|> merge_thms (addSE NONE NONE) safeEs safeEs2
|> merge_thms (addI NONE NONE) hazIs hazIs2
|> merge_thms (addE NONE NONE) hazEs hazEs2
|> map_swrappers (fn ws => AList.merge (op =) (K true) (ws, swrappers))
|> map_uwrappers (fn ws => AList.merge (op =) (K true) (ws, uwrappers));
(* data *)
structure Claset = Generic_Data
(
type T = claset;
val empty = empty_cs;
val extend = I;
val merge = merge_cs;
);
val claset_of = Claset.get o Context.Proof;
val rep_claset_of = rep_cs o claset_of;
val get_cs = Claset.get;
val map_cs = Claset.map;
fun map_theory_claset f thy =
let
val ctxt' = f (Proof_Context.init_global thy);
val thy' = Proof_Context.theory_of ctxt';
in Context.theory_map (Claset.map (K (claset_of ctxt'))) thy' end;
fun map_claset f = Context.proof_map (map_cs f);
fun put_claset cs = map_claset (K cs);
fun print_claset ctxt =
let
val {safeIs, safeEs, hazIs, hazEs, swrappers, uwrappers, ...} = rep_claset_of ctxt;
val pretty_thms = map (Display.pretty_thm_item ctxt) o Item_Net.content;
in
[Pretty.big_list "safe introduction rules (intro!):" (pretty_thms safeIs),
Pretty.big_list "introduction rules (intro):" (pretty_thms hazIs),
Pretty.big_list "safe elimination rules (elim!):" (pretty_thms safeEs),
Pretty.big_list "elimination rules (elim):" (pretty_thms hazEs),
Pretty.strs ("safe wrappers:" :: map #1 swrappers),
Pretty.strs ("unsafe wrappers:" :: map #1 uwrappers)]
|> Pretty.writeln_chunks
end;
(* old-style declarations *)
fun decl f (ctxt, ths) = map_claset (fold_rev (f (SOME (Context.Proof ctxt))) ths) ctxt;
val op addSIs = decl (addSI NONE);
val op addSEs = decl (addSE NONE);
val op addSDs = decl (addSD NONE);
val op addIs = decl (addI NONE);
val op addEs = decl (addE NONE);
val op addDs = decl (addD NONE);
val op delrules = decl delrule;
(*** Modifying the wrapper tacticals ***)
fun appSWrappers ctxt = fold (fn (_, w) => w ctxt) (#swrappers (rep_claset_of ctxt));
fun appWrappers ctxt = fold (fn (_, w) => w ctxt) (#uwrappers (rep_claset_of ctxt));
fun update_warn msg (p as (key : string, _)) xs =
(if AList.defined (op =) xs key then warning msg else (); AList.update (op =) p xs);
fun delete_warn msg (key : string) xs =
if AList.defined (op =) xs key then AList.delete (op =) key xs
else (warning msg; xs);
(*Add/replace a safe wrapper*)
fun ctxt addSWrapper new_swrapper = ctxt |> map_claset
(map_swrappers (update_warn ("Overwriting safe wrapper " ^ fst new_swrapper) new_swrapper));
(*Add/replace an unsafe wrapper*)
fun ctxt addWrapper new_uwrapper = ctxt |> map_claset
(map_uwrappers (update_warn ("Overwriting unsafe wrapper " ^ fst new_uwrapper) new_uwrapper));
(*Remove a safe wrapper*)
fun ctxt delSWrapper name = ctxt |> map_claset
(map_swrappers (delete_warn ("No such safe wrapper in claset: " ^ name) name));
(*Remove an unsafe wrapper*)
fun ctxt delWrapper name = ctxt |> map_claset
(map_uwrappers (delete_warn ("No such unsafe wrapper in claset: " ^ name) name));
(* compose a safe tactic alternatively before/after safe_step_tac *)
fun ctxt addSbefore (name, tac1) =
ctxt addSWrapper (name, fn ctxt => fn tac2 => tac1 ctxt ORELSE' tac2);
fun ctxt addSafter (name, tac2) =
ctxt addSWrapper (name, fn ctxt => fn tac1 => tac1 ORELSE' tac2 ctxt);
(*compose a tactic alternatively before/after the step tactic *)
fun ctxt addbefore (name, tac1) =
ctxt addWrapper (name, fn ctxt => fn tac2 => tac1 ctxt APPEND' tac2);
fun ctxt addafter (name, tac2) =
ctxt addWrapper (name, fn ctxt => fn tac1 => tac1 APPEND' tac2 ctxt);
fun ctxt addD2 (name, thm) =
ctxt addafter (name, fn ctxt' => dresolve_tac ctxt' [thm] THEN' assume_tac ctxt');
fun ctxt addE2 (name, thm) =
ctxt addafter (name, fn ctxt' => eresolve_tac ctxt' [thm] THEN' assume_tac ctxt');
fun ctxt addSD2 (name, thm) =
ctxt addSafter (name, fn ctxt' => dmatch_tac ctxt' [thm] THEN' eq_assume_tac);
fun ctxt addSE2 (name, thm) =
ctxt addSafter (name, fn ctxt' => ematch_tac ctxt' [thm] THEN' eq_assume_tac);
(**** Simple tactics for theorem proving ****)
(*Attack subgoals using safe inferences -- matching, not resolution*)
fun safe_step_tac ctxt =
let val {safe0_netpair, safep_netpair, ...} = rep_claset_of ctxt in
appSWrappers ctxt
(FIRST'
[eq_assume_tac,
eq_mp_tac ctxt,
bimatch_from_nets_tac ctxt safe0_netpair,
FIRST' (map (fn tac => tac ctxt) Data.hyp_subst_tacs),
bimatch_from_nets_tac ctxt safep_netpair])
end;
(*Repeatedly attack a subgoal using safe inferences -- it's deterministic!*)
fun safe_steps_tac ctxt =
REPEAT_DETERM1 o (fn i => COND (has_fewer_prems i) no_tac (safe_step_tac ctxt i));
(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
fun safe_tac ctxt = REPEAT_DETERM1 (FIRSTGOAL (safe_steps_tac ctxt));
(*** Clarify_tac: do safe steps without causing branching ***)
fun nsubgoalsP n (k, brl) = (subgoals_of_brl brl = n);
(*version of bimatch_from_nets_tac that only applies rules that
create precisely n subgoals.*)
fun n_bimatch_from_nets_tac ctxt n =
biresolution_from_nets_tac ctxt (order_list o filter (nsubgoalsP n)) true;
fun eq_contr_tac ctxt i = ematch_tac ctxt [Data.not_elim] i THEN eq_assume_tac i;
fun eq_assume_contr_tac ctxt = eq_assume_tac ORELSE' eq_contr_tac ctxt;
(*Two-way branching is allowed only if one of the branches immediately closes*)
fun bimatch2_tac ctxt netpair i =
n_bimatch_from_nets_tac ctxt 2 netpair i THEN
(eq_assume_contr_tac ctxt i ORELSE eq_assume_contr_tac ctxt (i + 1));
(*Attack subgoals using safe inferences -- matching, not resolution*)
fun clarify_step_tac ctxt =
let val {safe0_netpair, safep_netpair, ...} = rep_claset_of ctxt in
appSWrappers ctxt
(FIRST'
[eq_assume_contr_tac ctxt,
bimatch_from_nets_tac ctxt safe0_netpair,
FIRST' (map (fn tac => tac ctxt) Data.hyp_subst_tacs),
n_bimatch_from_nets_tac ctxt 1 safep_netpair,
bimatch2_tac ctxt safep_netpair])
end;
fun clarify_tac ctxt = SELECT_GOAL (REPEAT_DETERM (clarify_step_tac ctxt 1));
(*** Unsafe steps instantiate variables or lose information ***)
(*Backtracking is allowed among the various these unsafe ways of
proving a subgoal. *)
fun inst0_step_tac ctxt =
assume_tac ctxt APPEND'
contr_tac ctxt APPEND'
biresolve_from_nets_tac ctxt (#safe0_netpair (rep_claset_of ctxt));
(*These unsafe steps could generate more subgoals.*)
fun instp_step_tac ctxt =
biresolve_from_nets_tac ctxt (#safep_netpair (rep_claset_of ctxt));
(*These steps could instantiate variables and are therefore unsafe.*)
fun inst_step_tac ctxt = inst0_step_tac ctxt APPEND' instp_step_tac ctxt;
fun haz_step_tac ctxt =
biresolve_from_nets_tac ctxt (#haz_netpair (rep_claset_of ctxt));
(*Single step for the prover. FAILS unless it makes progress. *)
fun step_tac ctxt i =
safe_tac ctxt ORELSE appWrappers ctxt (inst_step_tac ctxt ORELSE' haz_step_tac ctxt) i;
(*Using a "safe" rule to instantiate variables is unsafe. This tactic
allows backtracking from "safe" rules to "unsafe" rules here.*)
fun slow_step_tac ctxt i =
safe_tac ctxt ORELSE appWrappers ctxt (inst_step_tac ctxt APPEND' haz_step_tac ctxt) i;
(**** The following tactics all fail unless they solve one goal ****)
(*Dumb but fast*)
fun fast_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN' SELECT_GOAL (DEPTH_SOLVE (step_tac ctxt 1));
(*Slower but smarter than fast_tac*)
fun best_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, Data.sizef) (step_tac ctxt 1));
(*even a bit smarter than best_tac*)
fun first_best_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, Data.sizef) (FIRSTGOAL (step_tac ctxt)));
fun slow_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL (DEPTH_SOLVE (slow_step_tac ctxt 1));
fun slow_best_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, Data.sizef) (slow_step_tac ctxt 1));
(***ASTAR with weight weight_ASTAR, by Norbert Voelker*)
val weight_ASTAR = 5;
fun astar_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL
(ASTAR (has_fewer_prems 1, fn lev => fn thm => Data.sizef thm + weight_ASTAR * lev)
(step_tac ctxt 1));
fun slow_astar_tac ctxt =
Object_Logic.atomize_prems_tac ctxt THEN'
SELECT_GOAL
(ASTAR (has_fewer_prems 1, fn lev => fn thm => Data.sizef thm + weight_ASTAR * lev)
(slow_step_tac ctxt 1));
(**** Complete tactic, loosely based upon LeanTaP. This tactic is the outcome
of much experimentation! Changing APPEND to ORELSE below would prove
easy theorems faster, but loses completeness -- and many of the harder
theorems such as 43. ****)
(*Non-deterministic! Could always expand the first unsafe connective.
That's hard to implement and did not perform better in experiments, due to
greater search depth required.*)
fun dup_step_tac ctxt =
biresolve_from_nets_tac ctxt (#dup_netpair (rep_claset_of ctxt));
(*Searching to depth m. A variant called nodup_depth_tac appears in clasimp.ML*)
local
fun slow_step_tac' ctxt = appWrappers ctxt (instp_step_tac ctxt APPEND' dup_step_tac ctxt);
in
fun depth_tac ctxt m i state = SELECT_GOAL
(safe_steps_tac ctxt 1 THEN_ELSE
(DEPTH_SOLVE (depth_tac ctxt m 1),
inst0_step_tac ctxt 1 APPEND COND (K (m = 0)) no_tac
(slow_step_tac' ctxt 1 THEN DEPTH_SOLVE (depth_tac ctxt (m - 1) 1)))) i state;
end;
(*Search, with depth bound m.
This is the "entry point", which does safe inferences first.*)
fun safe_depth_tac ctxt m = SUBGOAL (fn (prem, i) =>
let
val deti = (*No Vars in the goal? No need to backtrack between goals.*)
if exists_subterm (fn Var _ => true | _ => false) prem then DETERM else I;
in
SELECT_GOAL (TRY (safe_tac ctxt) THEN DEPTH_SOLVE (deti (depth_tac ctxt m 1))) i
end);
fun deepen_tac ctxt = DEEPEN (2, 10) (safe_depth_tac ctxt);
(* attributes *)
fun attrib f =
Thm.declaration_attribute (fn th => fn opt_context =>
map_cs (f (SOME opt_context) th) opt_context);
val safe_elim = attrib o addSE;
val safe_intro = attrib o addSI;
val safe_dest = attrib o addSD;
val haz_elim = attrib o addE;
val haz_intro = attrib o addI;
val haz_dest = attrib o addD;
val rule_del =
Thm.declaration_attribute (fn th => fn opt_context =>
opt_context |> map_cs (delrule (SOME opt_context) th) |>
Thm.attribute_declaration Context_Rules.rule_del th);
(** concrete syntax of attributes **)
val introN = "intro";
val elimN = "elim";
val destN = "dest";
val _ =
Theory.setup
(Attrib.setup @{binding swapped} (Scan.succeed swapped)
"classical swap of introduction rule" #>
Attrib.setup @{binding dest} (Context_Rules.add safe_dest haz_dest Context_Rules.dest_query)
"declaration of Classical destruction rule" #>
Attrib.setup @{binding elim} (Context_Rules.add safe_elim haz_elim Context_Rules.elim_query)
"declaration of Classical elimination rule" #>
Attrib.setup @{binding intro} (Context_Rules.add safe_intro haz_intro Context_Rules.intro_query)
"declaration of Classical introduction rule" #>
Attrib.setup @{binding rule} (Scan.lift Args.del >> K rule_del)
"remove declaration of intro/elim/dest rule");
(** proof methods **)
local
fun some_rule_tac ctxt facts = SUBGOAL (fn (goal, i) =>
let
val [rules1, rules2, rules4] = Context_Rules.find_rules false facts goal ctxt;
val {extra_netpair, ...} = rep_claset_of ctxt;
val rules3 = Context_Rules.find_rules_netpair true facts goal extra_netpair;
val rules = rules1 @ rules2 @ rules3 @ rules4;
val ruleq = Drule.multi_resolves (SOME ctxt) facts rules;
val _ = Method.trace ctxt rules;
in
fn st => Seq.maps (fn rule => resolve_tac ctxt [rule] i st) ruleq
end)
THEN_ALL_NEW Goal.norm_hhf_tac ctxt;
in
fun rule_tac ctxt [] facts = some_rule_tac ctxt facts
| rule_tac ctxt rules facts = Method.rule_tac ctxt rules facts;
fun standard_tac ctxt facts =
HEADGOAL (some_rule_tac ctxt facts) ORELSE
Class.standard_intro_classes_tac ctxt facts;
fun default_tac ctxt facts st =
(if Method.detect_closure_state st then
legacy_feature
("Old method \"default\"; use \"standard\" instead" ^ Position.here (Position.thread_data ()))
else ();
standard_tac ctxt facts st);
end;
(* automatic methods *)
val cla_modifiers =
[Args.$$$ destN -- Args.bang_colon >> K (Method.modifier (safe_dest NONE) @{here}),
Args.$$$ destN -- Args.colon >> K (Method.modifier (haz_dest NONE) @{here}),
Args.$$$ elimN -- Args.bang_colon >> K (Method.modifier (safe_elim NONE) @{here}),
Args.$$$ elimN -- Args.colon >> K (Method.modifier (haz_elim NONE) @{here}),
Args.$$$ introN -- Args.bang_colon >> K (Method.modifier (safe_intro NONE) @{here}),
Args.$$$ introN -- Args.colon >> K (Method.modifier (haz_intro NONE) @{here}),
Args.del -- Args.colon >> K (Method.modifier rule_del @{here})];
fun cla_method tac = Method.sections cla_modifiers >> K (SIMPLE_METHOD o tac);
fun cla_method' tac = Method.sections cla_modifiers >> K (SIMPLE_METHOD' o tac);
(** method setup **)
val _ =
Theory.setup
(Method.setup @{binding standard} (Scan.succeed (METHOD o standard_tac))
"standard proof step: classical intro/elim rule or class introduction" #>
Method.setup @{binding default} (Scan.succeed (METHOD o default_tac))
"standard proof step: classical intro/elim rule or class introduction (legacy)" #>
Method.setup @{binding rule}
(Attrib.thms >> (fn rules => fn ctxt => METHOD (HEADGOAL o rule_tac ctxt rules)))
"apply some intro/elim rule (potentially classical)" #>
Method.setup @{binding contradiction}
(Scan.succeed (fn ctxt => Method.rule ctxt [Data.not_elim, Drule.rotate_prems 1 Data.not_elim]))
"proof by contradiction" #>
Method.setup @{binding clarify} (cla_method' (CHANGED_PROP oo clarify_tac))
"repeatedly apply safe steps" #>
Method.setup @{binding fast} (cla_method' fast_tac) "classical prover (depth-first)" #>
Method.setup @{binding slow} (cla_method' slow_tac) "classical prover (slow depth-first)" #>
Method.setup @{binding best} (cla_method' best_tac) "classical prover (best-first)" #>
Method.setup @{binding deepen}
(Scan.lift (Scan.optional Parse.nat 4) --| Method.sections cla_modifiers
>> (fn n => fn ctxt => SIMPLE_METHOD' (deepen_tac ctxt n)))
"classical prover (iterative deepening)" #>
Method.setup @{binding safe} (cla_method (CHANGED_PROP o safe_tac))
"classical prover (apply safe rules)" #>
Method.setup @{binding safe_step} (cla_method' safe_step_tac)
"single classical step (safe rules)" #>
Method.setup @{binding inst_step} (cla_method' inst_step_tac)
"single classical step (safe rules, allow instantiations)" #>
Method.setup @{binding step} (cla_method' step_tac)
"single classical step (safe and unsafe rules)" #>
Method.setup @{binding slow_step} (cla_method' slow_step_tac)
"single classical step (safe and unsafe rules, allow backtracking)" #>
Method.setup @{binding clarify_step} (cla_method' clarify_step_tac)
"single classical step (safe rules, without splitting)");
(** outer syntax **)
val _ =
Outer_Syntax.command @{command_keyword print_claset} "print context of Classical Reasoner"
(Scan.succeed (Toplevel.keep (print_claset o Toplevel.context_of)));
end;