--- a/src/HOL/Tools/ATP/atp_redirect.ML Thu Feb 09 19:34:23 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,223 +0,0 @@
-(* Title: HOL/Tools/ATP/atp_redirect.ML
- Author: Jasmin Blanchette, TU Muenchen
-
-Transformation of a proof by contradiction into a direct proof.
-*)
-
-signature ATP_ATOM =
-sig
- type key
- val ord : key * key -> order
- val string_of : key -> string
-end;
-
-signature ATP_REDIRECT =
-sig
- type atom
-
- structure Atom_Graph : GRAPH
-
- type ref_sequent = atom list * atom
- type ref_graph = unit Atom_Graph.T
-
- type clause = atom list
- type direct_sequent = atom list * clause
- type direct_graph = unit Atom_Graph.T
-
- type rich_sequent = clause list * clause
-
- datatype direct_inference =
- Have of rich_sequent |
- Hence of rich_sequent |
- Cases of (clause * direct_inference list) list
-
- type direct_proof = direct_inference list
-
- val make_ref_graph : (atom list * atom) list -> ref_graph
- val axioms_of_ref_graph : ref_graph -> atom list -> atom list
- val tainted_atoms_of_ref_graph : ref_graph -> atom list -> atom list
- val sequents_of_ref_graph : ref_graph -> ref_sequent list
- val redirect_sequent : atom list -> atom -> ref_sequent -> direct_sequent
- val direct_graph : direct_sequent list -> direct_graph
- val redirect_graph : atom list -> atom list -> ref_graph -> direct_proof
- val succedent_of_cases : (clause * direct_inference list) list -> clause
- val chain_direct_proof : direct_proof -> direct_proof
- val string_of_direct_proof : direct_proof -> string
-end;
-
-functor ATP_Redirect(Atom : ATP_ATOM): ATP_REDIRECT =
-struct
-
-type atom = Atom.key
-
-structure Atom_Graph = Graph(Atom)
-
-type ref_sequent = atom list * atom
-type ref_graph = unit Atom_Graph.T
-
-type clause = atom list
-type direct_sequent = atom list * clause
-type direct_graph = unit Atom_Graph.T
-
-type rich_sequent = clause list * clause
-
-datatype direct_inference =
- Have of rich_sequent |
- Hence of rich_sequent |
- Cases of (clause * direct_inference list) list
-
-type direct_proof = direct_inference list
-
-fun atom_eq p = (Atom.ord p = EQUAL)
-fun clause_eq (c, d) = (length c = length d andalso forall atom_eq (c ~~ d))
-fun direct_sequent_eq ((gamma, c), (delta, d)) =
- clause_eq (gamma, delta) andalso clause_eq (c, d)
-
-fun make_ref_graph infers =
- let
- fun add_edge to from =
- Atom_Graph.default_node (from, ())
- #> Atom_Graph.default_node (to, ())
- #> Atom_Graph.add_edge_acyclic (from, to)
- fun add_infer (froms, to) = fold (add_edge to) froms
- in Atom_Graph.empty |> fold add_infer infers end
-
-fun axioms_of_ref_graph ref_graph conjs =
- subtract atom_eq conjs (Atom_Graph.minimals ref_graph)
-fun tainted_atoms_of_ref_graph ref_graph = Atom_Graph.all_succs ref_graph
-
-fun sequents_of_ref_graph ref_graph =
- map (`(Atom_Graph.immediate_preds ref_graph))
- (filter_out (Atom_Graph.is_minimal ref_graph) (Atom_Graph.keys ref_graph))
-
-fun redirect_sequent tainted bot (gamma, c) =
- if member atom_eq tainted c then
- gamma |> List.partition (not o member atom_eq tainted)
- |>> not (atom_eq (c, bot)) ? cons c
- else
- (gamma, [c])
-
-fun direct_graph seqs =
- let
- fun add_edge from to =
- Atom_Graph.default_node (from, ())
- #> Atom_Graph.default_node (to, ())
- #> Atom_Graph.add_edge_acyclic (from, to)
- fun add_seq (gamma, c) = fold (fn l => fold (add_edge l) c) gamma
- in Atom_Graph.empty |> fold add_seq seqs end
-
-fun disj cs = fold (union atom_eq) cs [] |> sort Atom.ord
-
-fun succedent_of_inference (Have (_, c)) = c
- | succedent_of_inference (Hence (_, c)) = c
- | succedent_of_inference (Cases cases) = succedent_of_cases cases
-and succedent_of_case (c, []) = c
- | succedent_of_case (_, infs) = succedent_of_inference (List.last infs)
-and succedent_of_cases cases = disj (map succedent_of_case cases)
-
-fun dest_Have (Have z) = z
- | dest_Have _ = raise Fail "non-Have"
-
-fun enrich_Have nontrivs trivs (cs, c) =
- (cs |> map (fn c => if member clause_eq nontrivs c then disj (c :: trivs)
- else c),
- disj (c :: trivs))
- |> Have
-
-fun s_cases cases =
- case cases |> List.partition (null o snd) of
- (trivs, nontrivs as [(nontriv0, proof)]) =>
- if forall (can dest_Have) proof then
- let val seqs = proof |> map dest_Have in
- seqs |> map (enrich_Have (nontriv0 :: map snd seqs) (map fst trivs))
- end
- else
- [Cases nontrivs]
- | (_, nontrivs) => [Cases nontrivs]
-
-fun descendants direct_graph =
- these o try (Atom_Graph.all_succs direct_graph) o single
-
-fun zones_of 0 _ = []
- | zones_of n (bs :: bss) =
- (fold (subtract atom_eq) bss) bs :: zones_of (n - 1) (bss @ [bs])
-
-fun redirect_graph axioms tainted ref_graph =
- let
- val [bot] = Atom_Graph.maximals ref_graph
- val seqs =
- map (redirect_sequent tainted bot) (sequents_of_ref_graph ref_graph)
- val direct_graph = direct_graph seqs
-
- fun redirect c proved seqs =
- if null seqs then
- []
- else if length c < 2 then
- let
- val proved = c @ proved
- val provable =
- filter (fn (gamma, _) => subset atom_eq (gamma, proved)) seqs
- val horn_provable = filter (fn (_, [_]) => true | _ => false) provable
- val seq as (gamma, c) = hd (horn_provable @ provable)
- in
- Have (map single gamma, c) ::
- redirect c proved (filter (curry (not o direct_sequent_eq) seq) seqs)
- end
- else
- let
- fun subsequents seqs zone =
- filter (fn (gamma, _) => subset atom_eq (gamma, zone @ proved)) seqs
- val zones = zones_of (length c) (map (descendants direct_graph) c)
- val subseqss = map (subsequents seqs) zones
- val seqs = fold (subtract direct_sequent_eq) subseqss seqs
- val cases =
- map2 (fn l => fn subseqs => ([l], redirect [l] proved subseqs))
- c subseqss
- in s_cases cases @ redirect (succedent_of_cases cases) proved seqs end
- in redirect [] axioms seqs end
-
-val chain_direct_proof =
- let
- fun chain_inf cl0 (seq as Have (cs, c)) =
- if member clause_eq cs cl0 then
- Hence (filter_out (curry clause_eq cl0) cs, c)
- else
- seq
- | chain_inf _ (Cases cases) = Cases (map chain_case cases)
- and chain_case (c, is) = (c, chain_proof (SOME c) is)
- and chain_proof _ [] = []
- | chain_proof (SOME prev) (i :: is) =
- chain_inf prev i :: chain_proof (SOME (succedent_of_inference i)) is
- | chain_proof _ (i :: is) =
- i :: chain_proof (SOME (succedent_of_inference i)) is
- in chain_proof NONE end
-
-fun indent 0 = ""
- | indent n = " " ^ indent (n - 1)
-
-fun string_of_clause [] = "\<bottom>"
- | string_of_clause ls = space_implode " \<or> " (map Atom.string_of ls)
-
-fun string_of_rich_sequent ch ([], c) = ch ^ " " ^ string_of_clause c
- | string_of_rich_sequent ch (cs, c) =
- commas (map string_of_clause cs) ^ " " ^ ch ^ " " ^ string_of_clause c
-
-fun string_of_case depth (c, proof) =
- indent (depth + 1) ^ "[" ^ string_of_clause c ^ "]"
- |> not (null proof) ? suffix ("\n" ^ string_of_subproof (depth + 1) proof)
-
-and string_of_inference depth (Have seq) =
- indent depth ^ string_of_rich_sequent "\<triangleright>" seq
- | string_of_inference depth (Hence seq) =
- indent depth ^ string_of_rich_sequent "\<guillemotright>" seq
- | string_of_inference depth (Cases cases) =
- indent depth ^ "[\n" ^
- space_implode ("\n" ^ indent depth ^ "|\n")
- (map (string_of_case depth) cases) ^ "\n" ^
- indent depth ^ "]"
-
-and string_of_subproof depth = cat_lines o map (string_of_inference depth)
-
-val string_of_direct_proof = string_of_subproof 0
-
-end;