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(******************************************************************************)
(* FILE : waterfall.ml *)
(* DESCRIPTION : `Waterfall' of heuristics. Clauses pour over. *)
(* Some evaporate. Others collect in a pool to be cleaned up. *)
(* Heuristics that act on a clause send the new clauses to *)
(* the top of the waterfall. *)
(* *)
(* READS FILES : <none> *)
(* WRITES FILES : <none> *)
(* *)
(* AUTHOR : R.J.Boulton *)
(* DATE : 9th May 1991 *)
(* *)
(* LAST MODIFIED : R.J.Boulton *)
(* DATE : 12th August 1992 *)
(* *)
(* LAST MODIFIED : P. Papapanagiotou (University of Edinburgh) *)
(* DATE : July 2009 *)
(******************************************************************************)
(*============================================================================*)
(* Some auxiliary functions *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* proves : thm -> term -> bool *)
(* *)
(* Returns true if and only if the theorem proves the term without making any *)
(* assumptions. *)
(*----------------------------------------------------------------------------*)
let proves th tm =
(let (hyp,concl) = dest_thm th
in (hyp = []) && (concl = tm));;
(*----------------------------------------------------------------------------*)
(* fproves : thm -> term -> bool *)
(* *)
(* Returns true if and only if the theorem proves the term without making any *)
(* assumptions except _FALSITY_. *)
(* This alternative is used for the HOL Light tactic validation. *)
(*----------------------------------------------------------------------------*)
let fproves th tm =
(let (hyp,concl) = dest_thm th
in ((subtract hyp [`_FALSITY_`]) = []) && (concl = tm));;
(*----------------------------------------------------------------------------*)
(* apply_proof : proof -> term list -> proof *)
(* *)
(* Converts a proof into a new proof that checks that the theorems it is *)
(* given have no hypotheses and have conclusions equal to the specified *)
(* terms. Used to make a proof robust. *)
(*----------------------------------------------------------------------------*)
let apply_proof f tms ths =
try
(if (itlist (fun (tm,th) b -> (fproves th tm) && b) (List.combine tms ths) true)
then (f ths)
else failwith ""
) with Failure _ -> failwith "apply_proof";;
(*----------------------------------------------------------------------------*)
(* apply_fproof : proof -> term list -> proof *)
(* *)
(* Converts a proof into a new proof that checks that the theorems it is *)
(* given have no hypotheses and have conclusions equal to the specified *)
(* terms. Used to make a proof robust. *)
(* Ignores _FALSITY_ assumptions so as to be used in the validation of the *)
(* HOL Light tactics. *)
(*----------------------------------------------------------------------------*)
let apply_fproof s f tms ths =
try (
(* print_string "Terms:" ; print_newline() ;
ignore (map (fun x -> (print_term x ; print_newline())) tms);
print_string "Theorems:"; print_newline();
ignore (map (fun x -> (print_thm x ; print_newline())) ths);
print_string "===" ; print_newline();*)
if (itlist (fun (tm,th) b -> (fproves th tm) && b) (List.combine tms ths) true)
then (f ths)
else failwith ""
) with Failure _ -> failwith ("apply_fproof: " ^ s);;
(*============================================================================*)
(* The `waterfall' for heuristics *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* proof_printing : bool *)
(* *)
(* Assignable variable. If true, clauses are printed as they are `poured' *)
(* over the waterfall. *)
(*----------------------------------------------------------------------------*)
let proof_printing = ref false;;
(*----------------------------------------------------------------------------*)
(* proof_printer : bool -> bool *)
(* *)
(* Function for setting the flag that controls the printing of clauses as *)
(* are `poured' over the waterfall. *)
(*----------------------------------------------------------------------------*)
let proof_printer state =
let old_state = !proof_printing
in proof_printing := state; old_state;;
(*----------------------------------------------------------------------------*)
(* proof_print_depth : int *)
(* *)
(* Assignable variable. A number indicating the `depth' of the proof and more *)
(* practically the number of spaces printed before a term. *)
(*----------------------------------------------------------------------------*)
let proof_print_depth = ref 0;;
(*----------------------------------------------------------------------------*)
(* inc_print_depth : * -> * *)
(* *)
(* Identity function that has the side-effect of incrementing the *)
(* print_proof_depth. *)
(*----------------------------------------------------------------------------*)
let inc_print_depth x = (proof_print_depth := !proof_print_depth + 1; x);;
(*----------------------------------------------------------------------------*)
(* dec_print_depth : * -> * *)
(* *)
(* Identity function that has the side-effect of decrementing the *)
(* print_proof_depth. *)
(*----------------------------------------------------------------------------*)
let dec_print_depth x =
if (!proof_print_depth < 1)
then (proof_print_depth := 0; x)
else (proof_print_depth := !proof_print_depth - 1; x);;
(*----------------------------------------------------------------------------*)
(* proof_print_term : term -> term *)
(* *)
(* Identity function on terms that has the side effect of printing the term *)
(* if the `proof_printing' flag is set to true. *)
(*----------------------------------------------------------------------------*)
let proof_print_term tm =
if !proof_printing
then (print_string (implode (replicate " " !proof_print_depth));
print_term tm; print_newline () ; tm)
else tm;;
let proof_print_thm thm =
if !proof_printing
then ( print_thm thm; print_newline (); print_newline());;
let proof_print_tmi (tm,i) =
if !proof_printing
then ( proof_print_term tm; print_string " ("; print_bool i; print_string ")"; (tm,i) )
else (tm,i);;
(*
let proof_print_clause cl =
if !proof_printing
then (
match cl with
| Clause_proved thm -> (print_thm thm; print_newline (); cl)
| _ -> cl
)
else cl;;
*)
(*----------------------------------------------------------------------------*)
(* proof_print_newline : * -> * *)
(* *)
(* Identity function that has the side effect of printing a newline if the *)
(* `proof_printing' flag is set to true. *)
(*----------------------------------------------------------------------------*)
let proof_print_newline x =
if !proof_printing
then (print_newline (); x)
else x;;
let proof_print_string s x =
if !proof_printing
then (print_string s; x)
else x;;
let proof_print_string_l s x =
if !proof_printing
then (print_string s; print_newline (); x)
else x;;
(*----------------------------------------------------------------------------*)
(* Recursive type for holding partly processed clauses. *)
(* *)
(* A clause is either still to be proved, has been proved, or can be proved *)
(* once sub-clauses have been. A clause that is still to be proved carries a *)
(* flag indicating whether or not it is an induction step. *)
(*----------------------------------------------------------------------------*)
type clause_tree = Clause of (term * bool)
| Clause_proved of thm
| Clause_split of clause_tree list * (thm list -> thm);;
let rec proof_print_clausetree cl =
if !proof_printing
then ( proof_print_string_l "Clause tree:" (); match cl with
| Clause (tm,bool) -> (print_term tm; print_newline ())
| Clause_proved thm -> (print_thm thm; print_newline())
| Clause_split (tlist,proof) -> (print_string "Split -> "; print_int (length tlist); print_newline () ;
let void = map proof_print_clausetree tlist in () ));;
(*----------------------------------------------------------------------------*)
(* waterfall : ((term # bool) -> ((term # bool) list # proof)) list -> *)
(* (term # bool) -> *)
(* clause_tree *)
(* *)
(* `Waterfall' of heuristics. Takes a list of heuristics and a term as *)
(* arguments. Each heuristic should fail if it can do nothing with its input. *)
(* Otherwise the heuristic should return a list of new clauses to be proved *)
(* together with a proof of the original clause from these new clauses. *)
(* *)
(* Clauses that are not processed by any of the heuristics are placed in a *)
(* leaf node of the tree, to be proved later. The heuristics are attempted in *)
(* turn. If a heuristic returns an empty list of new clauses, the proof is *)
(* applied to an empty list, and the resultant theorem is placed in the tree *)
(* as a leaf node. Otherwise, the tree is split, and each of the new clauses *)
(* is passed to ALL of the heuristics. *)
(*----------------------------------------------------------------------------*)
let nth_tail n l = if (n > length l) then []
else let rec repeattl l i =
if ( i = 0 ) then l
else tl (repeattl l (i-1))
in repeattl l n;;
let rec waterfall heuristics tmi =
bm_steps := hashI ((+) 1) !bm_steps;
let rec flow_on_down rest_of_heuristics tmi =
if (is_F (fst tmi)) then (failwith "cannot prove")
else if (rest_of_heuristics = []) then (Clause tmi)
else try ((let (tms,f) = hd rest_of_heuristics tmi
in if (tms = []) then (proof_print_string "Proven:" (); proof_print_thm (f []) ; Clause_proved (f []))
else if ((tl tms) = []) then
(Clause_split ([waterfall heuristics (hd tms)],f))
else Clause_split
((dec_print_depth o
map (waterfall heuristics o proof_print_newline) o
inc_print_depth) tms,
f))
)with Failure s -> (if (s = "cannot prove")
then failwith s
else (flow_on_down (tl rest_of_heuristics) tmi)
)
in flow_on_down heuristics (proof_print_tmi tmi);;
let rec filtered_waterfall heuristics warehouse tmi =
bm_steps := hashI ((+) 1) !bm_steps;
if (max_var_depth (fst tmi) > 12) then (warn true "Reached maximum depth!" ; Clause tmi) (*failwith "cannot prove"*)
else
let heurn = try (assoc (fst tmi) warehouse) with Failure _ -> 0 in
let warehouse = (if (heurn > 0) then
( proof_print_string ("Warehouse kicking in! Skipping " ^ string_of_int(heurn) ^ " heuristic(s)") () ;
proof_print_newline () ;
List.remove_assoc (fst tmi) warehouse) else (warehouse)) in
let rec flow_on_down rest_of_heuristics tmi it =
if (is_F (fst tmi)) then (failwith "cannot prove")
else let rest_of_heuristics = nth_tail heurn rest_of_heuristics in
if (rest_of_heuristics = []) then (Clause tmi)
else try (
let (tms,f) = hd rest_of_heuristics tmi in
if (tms = []) then (proof_print_string "Proven:" (); proof_print_thm (f []) ; Clause_proved (f []))
else if ((tl tms) = []) then
Clause_split ([filtered_waterfall heuristics (((fst tmi),it)::warehouse) (hd tms)],f)
else
( proof_print_string (string_of_int(length tms) ^ " new clauses") () ;
proof_print_newline () ;
Clause_split
((dec_print_depth o
map (filtered_waterfall heuristics (((fst tmi),it)::warehouse) o proof_print_newline) o
inc_print_depth) tms,
f))
)with Failure s -> (
if (s = "cannot prove")
then failwith s
else (flow_on_down (tl rest_of_heuristics) tmi (it+1))
)
in flow_on_down heuristics ((hashI proof_print_term) tmi) 1;;
(* in
let fringe = fringe_of_clause_tree restree in
if (fringe = []) then restree
else ( waterfall_warehouse := ((fst tmi),it)::(!waterfall_warehouse) ; restree ) *)
(*----------------------------------------------------------------------------*)
(* fringe_of_clause_tree : clause_tree -> (term # bool) list *)
(* *)
(* Computes the fringe of a clause_tree, including in the resultant list only *)
(* those clauses that remain to be proved. *)
(*----------------------------------------------------------------------------*)
let rec fringe_of_clause_tree tree =
match tree with
| (Clause tmi) -> [tmi]
| (Clause_proved _) -> []
| (Clause_split (trees,_)) -> (flat (map fringe_of_clause_tree trees));;
(*----------------------------------------------------------------------------*)
(* prove_clause_tree : clause_tree -> proof *)
(* *)
(* Given a clause_tree, returns a proof that if given theorems for the *)
(* unproved clauses in the tree, returns a theorem for the clause at the root *)
(* of the tree. The list of theorems must be in the same order as the clauses *)
(* appear in the fringe of the tree. *)
(*----------------------------------------------------------------------------*)
let prove_clause_tree tree ths =
try(
let rec prove_clause_trees trees ths =
if (trees = []) then ([],ths)
else let (th,ths') = prove_clause_tree' (hd trees) ths
in let (thl,ths'') = prove_clause_trees (tl trees) ths'
in (th::thl,ths'')
and prove_clause_tree' tree ths =
match tree with
| (Clause (tm,_)) ->
(let th = hd ths
in if (fproves th tm)
then (th,tl ths)
else failwith "prove_clause_tree")
| (Clause_proved th) -> (th,ths)
| (Clause_split (trees,f)) ->
(let (thl,ths') = prove_clause_trees trees ths
in (f thl,ths'))
in (let (th,[]) = (prove_clause_tree' tree ths) in th)
) with Failure _ -> failwith "prove_clause_tree";;
let fprove_clause_tree tree ths =
try(
let rec prove_clause_trees trees ths =
if (trees = []) then ([],ths)
else let (th,ths') = prove_clause_tree' (hd trees) ths
in let (thl,ths'') = prove_clause_trees (tl trees) ths'
in (th::thl,ths'')
and prove_clause_tree' tree ths =
match tree with
| (Clause (tm,_)) ->
(let th = hd ths
in if (fproves th tm)
then (th,tl ths)
else failwith "fprove_clause_tree")
| (Clause_proved th) -> (th,ths)
| (Clause_split (trees,f)) ->
(let (thl,ths') = prove_clause_trees trees ths
in (f thl,ths'))
in (let (th,[]) = (prove_clause_tree' tree ths) in th)
) with Failure s -> failwith ("fprove_clause_tree: " ^ s);;
(*============================================================================*)
(* Eliminating instances in the `pool' of clauses remaining to be proved *)
(* *)
(* Constructing partial orderings is probably overkill. It may only be *)
(* necessary to split the clauses into two sets, one of most general clauses *)
(* and the rest. This would still be computationally intensive, but it would *)
(* avoid comparing two clauses that are both instances of some other clause. *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* inst_of : term -> term -> thm -> thm *)
(* *)
(* Takes two terms and computes whether the first is an instance of the *)
(* second. If this is the case, a proof of the first term from the second *)
(* (assuming they are formulae) is returned. Otherwise the function fails. *)
(*----------------------------------------------------------------------------*)
let inst_of tm patt =
try(
(let (_,tm_bind,ty_bind) = term_match [] patt tm
in let (insts,vars) = List.split tm_bind
in let f = (SPECL insts) o (GENL vars) o (INST_TYPE ty_bind)
in fun th -> apply_fproof "inst_of" (f o hd) [patt] [th]
)) with Failure _ -> failwith "inst_of";;
(*----------------------------------------------------------------------------*)
(* Recursive datatype for a partial ordering of terms using the *)
(* `is an instance of' relation. *)
(* *)
(* The leaf nodes of the tree are terms that have no instances. The other *)
(* nodes have a list of instance trees and proofs of each instance from the *)
(* term at that node. *)
(* *)
(* Each term carries a number along with it. This is used to keep track of *)
(* where the term came from in a list. The idea is to take the fringe of a *)
(* clause tree, number the elements, then form partial orderings so that *)
(* only the most general theorems have to be proved. *)
(*----------------------------------------------------------------------------*)
type inst_tree = No_insts of (term * int)
| Insts of (term * int * (inst_tree * (thm -> thm)) list);;
(*----------------------------------------------------------------------------*)
(* insert_into_inst_tree : (term # int) -> inst_tree -> inst_tree *)
(* insert_into_inst_trees : (term # int # (thm -> thm)) -> *)
(* (inst_tree # (thm -> thm)) list -> *)
(* (inst_tree # (thm -> thm)) list *)
(* *)
(* Mutually recursive functions for constructing partial orderings, ordered *)
(* by `is an instance of' relation. The algorithm is grossly inefficient. *)
(* Structures are repeatedly destroyed and rebuilt. Reference variables *)
(* should be used for efficiency. *)
(* *)
(* Inserting into a single tree: *)
(* *)
(* If tm' is an instance of tm, tm is put in the root node, with the old tree *)
(* as its single child. If tm is not an instance of tm', the function fails. *)
(* Assume now that tm is an instance of tm'. If the tree is a leaf, it is *)
(* made into a branch and tm is inserted as the one and only child. If the *)
(* tree is a branch, an attempt is made to insert tm in the list of *)
(* sub-trees. If this fails, tm is added as a leaf to the list of instances. *)
(* Note that if tm is not an instance of tm', then it cannot be an instance *)
(* of the instances of tm'. *)
(* *)
(* Inserting into a list of trees: *)
(* *)
(* The list of trees carry proofs with them. The list is split into those *)
(* whose root is an instance of tm, and those whose root is not. The proof *)
(* associated with a tree that is an instance is replaced by a proof of the *)
(* term from tm. If the list of trees that are instances is non-empty, they *)
(* are made children of a node containing tm, and this new tree is added to *)
(* the list of trees that are n't instances. If tm has instances in the list, *)
(* it cannot be the case that tm is an instance of one of the other trees in *)
(* the list, for the trees in a list must be unrelated. *)
(* *)
(* If there are no instances of tm in the list of trees, an attempt is made *)
(* to insert tm into the list. If it is unrelated to all of the trees, this *)
(* attempt fails, in which case tm is made into a leaf and added to the list. *)
(*----------------------------------------------------------------------------*)
let rec insert_into_inst_tree (tm,n) tree =
match tree with
| (No_insts (tm',n')) ->
(try ( (let f = inst_of tm' tm
in Insts (tm,n,[No_insts (tm',n'),f]))
) with Failure _ -> try( let f = inst_of tm tm'
in Insts (tm',n',[No_insts (tm,n),f])) with Failure _ -> failwith "insert_into_inst_tree"
)
| (Insts (tm',n',insts)) ->
(try (let f = inst_of tm' tm
in Insts (tm,n,[Insts (tm',n',insts),f]))
with Failure _ -> try(let f = inst_of tm tm'
in try( Insts (tm',n',insert_into_inst_trees (tm,n,f) insts))
with Failure _ -> (Insts (tm',n',(No_insts (tm,n),f)::insts))
)
with Failure _ -> failwith "insert_into_inst_tree" )
and insert_into_inst_trees (tm,n,f) insts =
let rec instances (tm,n) insts =
if (insts = [])
then ([],[])
else let (not_instl,instl) = instances (tm,n) (tl insts)
and (h,f) = hd insts
in let (tm',n') =
match h with
| (No_insts (tm',n')) -> (tm',n')
| (Insts (tm',n',_)) -> (tm',n')
in (try( (let f' = inst_of tm' tm
in (not_instl,(h,f')::instl))
) with Failure _ -> ((h,f)::not_instl,instl)
)
and insert_into_inst_trees' (tm,n) trees =
if (trees = [])
then failwith "insert_into_inst_trees'"
else (try( ((insert_into_inst_tree (tm,n) (hd trees))::(tl trees))
) with Failure _ -> ((hd trees)::(insert_into_inst_trees' (tm,n) (tl trees)))
)
in let (not_instl,instl) = instances (tm,n) insts
in if (instl = [])
then try( (lcombinep o (hashI (insert_into_inst_trees' (tm,n)))) (List.split insts)
) with Failure _ -> ((No_insts (tm,n),f)::insts)
else (Insts (tm,n,instl),f)::not_instl;;
(*----------------------------------------------------------------------------*)
(* mk_inst_trees : (term # int) list -> inst_tree list *)
(* *)
(* Constructs a partial ordering of terms under the `is an instance of' *)
(* relation from a list of numbered terms. *)
(* *)
(* A dummy proof is passed to the call of insert_into_inst_trees. The result *)
(* of the call has a proof associated with the root of each tree. These *)
(* proofs are dummies and so are discarded before the final result is *)
(* returned. *)
(*----------------------------------------------------------------------------*)
let mk_inst_trees tmnl =
let rec mk_inst_trees' insts tmnl =
if (tmnl = [])
then insts
else let (tm,n) = hd tmnl
in mk_inst_trees'
(insert_into_inst_trees (tm,n,I) insts) (tl tmnl)
in map fst (mk_inst_trees' [] tmnl);;
(*----------------------------------------------------------------------------*)
(* roots_of_inst_trees : inst_tree list -> term list *)
(* *)
(* Computes the terms at the roots of a list of partial orderings. *)
(*----------------------------------------------------------------------------*)
let rec roots_of_inst_trees trees =
if (trees = [])
then []
else let tm =
match (hd trees) with
| (No_insts (tm,_)) -> tm
| (Insts (tm,_,_)) -> tm
in tm::(roots_of_inst_trees (tl trees));;
(*----------------------------------------------------------------------------*)
(* prove_inst_tree : inst_tree -> thm -> (thm # int) list *)
(* *)
(* Given a partial ordering of terms and a theorem for its root, returns a *)
(* list of theorems for the terms in the tree. *)
(*----------------------------------------------------------------------------*)
let rec prove_inst_tree tree th =
match tree with
| (No_insts (tm,n)) ->
(if (fproves th tm) then [(th,n)] else failwith "prove_inst_tree")
| (Insts (tm,n,insts)) ->
(if (fproves th tm)
then (th,n)::(flat (map (fun (tr,f) -> prove_inst_tree tr (f th)) insts))
else failwith "prove_inst_tree");;
(*----------------------------------------------------------------------------*)
(* prove_inst_trees : inst_tree list -> thm list -> thm list *)
(* *)
(* Given a list of partial orderings of terms and a list of theorems for *)
(* their roots, returns a sorted list of theorems for the terms in the trees. *)
(* The sorting is done based on the integer labels attached to each term in *)
(* the trees. *)
(*----------------------------------------------------------------------------*)
let prove_inst_trees trees ths =
try ( map fst
(sort_on_snd (flat (map (uncurry prove_inst_tree) (lcombinep (trees,ths)))))
) with Failure _ -> failwith "prove_inst_trees";;
(*----------------------------------------------------------------------------*)
(* prove_pool : conv -> term list -> thm list *)
(* *)
(* Attempts to prove the pool of clauses left over from the waterfall, by *)
(* applying the conversion, conv, to the most general clauses. *)
(*----------------------------------------------------------------------------*)
let prove_pool conv tml =
let tmnl = number_list tml
in let trees = mk_inst_trees tmnl
in let most_gen_terms = roots_of_inst_trees trees
in let ths = map conv most_gen_terms
in prove_inst_trees trees ths;;
(*============================================================================*)
(* Boyer-Moore prover *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* WATERFALL : ((term # bool) -> ((term # bool) list # proof)) list -> *)
(* ((term # bool) -> ((term # bool) list # proof)) -> *)
(* (term # bool) -> *)
(* thm *)
(* *)
(* Boyer-Moore style automatic proof procedure. Takes a list of heuristics, *)
(* and a single heuristic that does induction, as arguments. The result is a *)
(* function that, given a term and a Boolean, attempts to prove the term. The *)
(* Boolean is used to indicate whether the term is the step of an induction. *)
(* It will normally be set to false for an external call. *)
(*----------------------------------------------------------------------------*)
let rec WATERFALL heuristics induction (tm,(ind:bool)) =
let conv tm =
proof_print_string "Doing induction on:" () ; bm_steps := hash ((+) 1) ((+) 1) !bm_steps ;
let void = proof_print_term tm ; proof_print_newline ()
in let (tmil,proof) = induction (tm,false)
in dec_print_depth
(proof
(map (WATERFALL heuristics induction) (inc_print_depth tmil)))
in let void = proof_print_newline ()
in let tree = waterfall heuristics (tm,ind)
in let tmil = fringe_of_clause_tree tree
in let thl = prove_pool conv (map fst tmil)
in prove_clause_tree tree thl;;
let rec FILTERED_WATERFALL heuristics induction warehouse (tm,(ind:bool)) =
let conv tm =
(* let constr_check = ind && not((find_bm_terms (fun t -> count_constructors t > 5) tm) = []) in *)
let constr_check = (max_var_depth tm > 12) in
if (constr_check) then let void = (warn true "Reached maximum depth!") in failwith "cannot prove"
else
let heurn = try (assoc tm warehouse) with Failure _ -> 0 in
let warehouse = (if (heurn > 0) then (List.remove_assoc tm warehouse) else (warehouse)) in
if (heurn > length heuristics) then ( warn true "Induction loop detected."; failwith "cannot prove" )
else
proof_print_string "Doing induction on:" (); bm_steps := hash ((+) 1) ((+) 1) !bm_steps ;
let void = proof_print_term tm ; proof_print_newline () in
let (tmil,proof) = induction (tm,false)
in dec_print_depth
(proof
(map (FILTERED_WATERFALL heuristics induction ((tm,(length heuristics) + 1)::warehouse)) (inc_print_depth tmil)))
in let void = proof_print_newline ()
in let tree = filtered_waterfall heuristics [] (tm,ind)
(* in let void = proof_print_clausetree tree *)
in let tmil = fringe_of_clause_tree tree
in let thl = prove_pool conv (map fst tmil)
in prove_clause_tree tree thl;;
(*============================================================================*)
(* Some sample heuristics *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* conjuncts_heuristic : (term # bool) -> ((term # bool) list # proof) *)
(* *)
(* `Heuristic' for splitting a conjunction into a list of conjuncts. *)
(* Right conjuncts are split recursively. *)
(* Fails if the argument term is not a conjunction. *)
(*----------------------------------------------------------------------------*)
let conjuncts_heuristic (tm,(i:bool)) =
let tms = conj_list tm
in if (length tms = 1)
then failwith "conjuncts_heuristic"
else (map (fun tm -> (tm,i)) tms,apply_fproof "conjuncts_heuristic" LIST_CONJ tms);;
(*----------------------------------------------------------------------------*)
(* refl_heuristic : (term # bool) -> ((term # bool) list # proof) *)
(* *)
(* `Heuristic' for proving that terms of the form "x = x" are true. Fails if *)
(* the argument term is not of this form. Otherwise it returns an empty list *)
(* of new clauses, and a proof of the original term. *)
(*----------------------------------------------------------------------------*)
let refl_heuristic (tm,(i:bool)) =
try(if (lhs tm = rhs tm)
then (([]:(term * bool) list),apply_fproof "refl_heuristic" (fun ths -> REFL (lhs tm)) [])
else failwith ""
) with Failure _ -> failwith "refl_heuristic";;
(*----------------------------------------------------------------------------*)
(* clausal_form_heuristic : (term # bool) -> ((term # bool) list # proof) *)
(* *)
(* `Heuristic' that tests a term to see if it is in clausal form, and if not *)
(* converts it to clausal form and returns the resulting clauses as new *)
(* `goals'. It is critical for efficiency that the normalization is only done *)
(* if the term is not in clausal form. Note that the functions `conjuncts' *)
(* and `disjuncts' are not used for the testing because they split trees of *)
(* conjuncts (disjuncts) rather than simply `linear' con(dis)junctions. *)
(* If the term is in clausal form, but is not a single clause, it is split *)
(* into single clauses. If the term is in clausal form but contains Boolean *)
(* constants, the normalizer is applied to it. A single new goal will be *)
(* produced in this case unless the result of the normalization was true. *)
(*----------------------------------------------------------------------------*)
let clausal_form_heuristic (tm,(i:bool)) =
try (let is_atom tm =
(not (has_boolean_args_and_result tm)) || (is_var tm) || (is_const tm)
in let is_literal tm =
(is_atom tm) || ((is_neg tm) && (try (is_atom (rand tm)) with Failure _ -> false))
in let is_clause tm = forall is_literal (disj_list tm)
in let result_string = fun tms -> let s = length tms
in let plural = if (s=1) then "" else "s"
in ("-> Clausal Form Heuristic (" ^ string_of_int(s) ^ " clause" ^ plural ^ ")")
in if (forall is_clause (conj_list tm)) &&
(not (free_in `T` tm)) && (not (free_in `F` tm))
then if (is_conj tm)
then let tms = conj_list tm
in (proof_print_string_l (result_string tms) () ;
(map (fun tm -> (tm,i)) tms,apply_fproof "clausal_form_heuristic" LIST_CONJ tms))
else failwith ""
else let th = CLAUSAL_FORM_CONV tm
in let tm' = rhs (concl th)
in if (is_T tm')
then (proof_print_string_l "-> Clausal Form Heuristic" () ; ([],apply_fproof "clausal_form_heuristic" (fun _ -> EQT_ELIM th) []))
else let tms = conj_list tm'
in (proof_print_string_l (result_string tms) () ;
(map (fun tm -> (tm,i)) tms,
apply_fproof "clausal_form_heuristic" ((EQ_MP (SYM th)) o LIST_CONJ) tms))
) with Failure _ -> failwith "clausal_form_heuristic";;
let meson_heuristic l (tm,(i:bool)) =
try( let meth = MESON (l @ rewrite_rules()) tm in
(([]:(term * bool) list),apply_fproof "meson_heuristic" (fun ths -> meth) [])
) with Failure _ -> failwith "meson_heuristic";;
let taut_heuristic (tm,(i:bool)) =
try( let tautthm = TAUT tm in (proof_print_string_l "-> Tautology Heuristic" () ;
(([]:(term * bool) list),apply_fproof "taut_heuristic" (fun ths -> tautthm) []))
) with Failure _ -> failwith "taut_heuristic";;
let setify_heuristic (tm,(i:bool)) =
try (
if (not (is_disj tm)) then failwith ""
else
let tms = disj_list tm
in let tms' = setify tms
in let tm' = list_mk_disj tms'
in if ((length tms) = (length tms')) then failwith ""
else let th = TAUT (mk_imp (tm',tm))
in (proof_print_string_l "-> Setify Heuristic" () ;
([tm',i],apply_fproof "setify_heuristic" ((MP th) o hd) [tm']))
)
with Failure _ -> failwith "setify_heuristic";;
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