+++ /dev/null
-(* Copyright (C) 2002, HELM Team.
- *
- * This file is part of HELM, an Hypertextual, Electronic
- * Library of Mathematics, developed at the Computer Science
- * Department, University of Bologna, Italy.
- *
- * HELM is free software; you can redistribute it and/or
- * modify it under the terms of the GNU General Public License
- * as published by the Free Software Foundation; either version 2
- * of the License, or (at your option) any later version.
- *
- * HELM is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU General Public License for more details.
- *
- * You should have received a copy of the GNU General Public License
- * along with HELM; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
- * MA 02111-1307, USA.
- *
- * For details, see the HELM World-Wide-Web page,
- * http://cs.unibo.it/helm/.
- *)
-
-open CicReduction
-open PrimitiveTactics
-open ProofEngineTypes
-open UriManager
-
-(** DEBUGGING *)
-
- (** perform debugging output? *)
-let debug = false
-
- (** debugging print *)
-let warn s =
- if debug then
- prerr_endline ("RING WARNING: " ^ s)
-
-(** CIC URIS *)
-
-(**
- Note: For constructors URIs aren't really URIs but rather triples of
- the form (uri, typeno, consno). This discrepancy is to preserver an
- uniformity of invocation of "mkXXX" functions.
-*)
-
-let eqt_uri = uri_of_string "cic:/Coq/Init/Logic_Type/eqT.ind"
-let refl_eqt_uri = (eqt_uri, 0, 1)
-let sym_eqt_uri =
- uri_of_string "cic:/Coq/Init/Logic_Type/Equality_is_a_congruence/sym_eqT.con"
-let bool_uri = uri_of_string "cic:/Coq/Init/Datatypes/bool.ind"
-let true_uri = (bool_uri, 0, 1)
-let false_uri = (bool_uri, 0, 2)
-
-let r_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/R.con"
-let rplus_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/Rplus.con"
-let rmult_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/Rmult.con"
-let ropp_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/Ropp.con"
-let r0_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/R0.con"
-let r1_uri = uri_of_string "cic:/Coq/Reals/Rdefinitions/R1.con"
-let rtheory_uri = uri_of_string "cic:/Coq/Reals/Rbase/RTheory.con"
-
-let apolynomial_uri =
- uri_of_string
- "cic:/Coq/ring/Ring_abstract/abstract_rings/apolynomial.ind"
-let apvar_uri = (apolynomial_uri, 0, 1)
-let ap0_uri = (apolynomial_uri, 0, 2)
-let ap1_uri = (apolynomial_uri, 0, 3)
-let applus_uri = (apolynomial_uri, 0, 4)
-let apmult_uri = (apolynomial_uri, 0, 5)
-let apopp_uri = (apolynomial_uri, 0, 6)
-
-let varmap_uri =
- uri_of_string "cic:/Coq/ring/Quote/variables_map/varmap.ind"
-let empty_vm_uri = (varmap_uri, 0, 1)
-let node_vm_uri = (varmap_uri, 0, 2)
-let varmap_find_uri =
- uri_of_string "cic:/Coq/ring/Quote/variables_map/varmap_find.con"
-let index_uri =
- uri_of_string "cic:/Coq/ring/Quote/variables_map/index.ind"
-let left_idx_uri = (index_uri, 0, 1)
-let right_idx_uri = (index_uri, 0, 2)
-let end_idx_uri = (index_uri, 0, 3)
-
-let interp_ap_uri =
- uri_of_string "cic:/Coq/ring/Ring_abstract/abstract_rings/interp_ap.con"
-let interp_sacs_uri =
- uri_of_string "cic:/Coq/ring/Ring_abstract/abstract_rings/interp_sacs.con"
-let apolynomial_normalize_uri =
- uri_of_string
- "cic:/Coq/ring/Ring_abstract/abstract_rings/apolynomial_normalize.con"
-let apolynomial_normalize_ok_uri =
- uri_of_string
- "cic:/Coq/ring/Ring_abstract/abstract_rings/apolynomial_normalize_ok.con"
-
-(** CIC PREDICATES *)
-
- (**
- check whether a term is a constant or not, if argument "uri" is given and is
- not "None" also check if the constant correspond to the given one or not
- *)
-let cic_is_const ?(uri: uri option = None) term =
- match uri with
- | None ->
- (match term with
- | Cic.Const _ -> true
- | _ -> false)
- | Some realuri ->
- (match term with
- | Cic.Const (u, _) when (eq u realuri) -> true
- | _ -> false)
-
-(** PROOF AND GOAL ACCESSORS *)
-
- (**
- @param proof a proof
- @return the uri of a given proof
- *)
-let uri_of_proof ~proof:(uri, _, _, _) = uri
-
- (**
- @param metano meta list index
- @param metasenv meta list (environment)
- @raise Failure if metano is not found in metasenv
- @return conjecture corresponding to metano in metasenv
- *)
-let conj_of_metano metano =
- try
- List.find (function (m, _, _) -> m = metano)
- with Not_found ->
- failwith (
- "Ring.conj_of_metano: " ^
- (string_of_int metano) ^ " no such meta")
-
- (**
- @param status current proof engine status
- @raise Failure if proof is None
- @return current goal's metasenv
- *)
-let metasenv_of_status ~status:((_,m,_,_), _) = m
-
- (**
- @param status a proof engine status
- @raise Failure when proof or goal are None
- @return context corresponding to current goal
- *)
-let context_of_status ~status:(proof, goal as status) =
- let metasenv = metasenv_of_status ~status:status in
- let _, context, _ = List.find (function (m,_,_) -> m=goal) metasenv in
- context
-
-(** CIC TERM CONSTRUCTORS *)
-
- (**
- Create a Cic term consisting of a constant
- @param uri URI of the constant
- @proof current proof
- *)
-let mkConst ~uri ~proof =
- let cookingsno = relative_depth (uri_of_proof ~proof) uri 0 in
- Cic.Const (uri, cookingsno)
-
- (**
- Create a Cic term consisting of a constructor
- @param uri triple <uri, typeno, consno> where uri is the uri of an inductive
- type, typeno is the type number in a mutind structure (0 based), consno is
- the constructor number (1 based)
- @param proof current proof
- *)
-let mkCtor ~uri:(uri, typeno, consno) ~proof =
- let cookingsno = relative_depth (uri_of_proof ~proof) uri 0 in
- Cic.MutConstruct (uri, cookingsno, typeno, consno)
-
- (**
- Create a Cic term consisting of a type member of a mutual induction
- @param uri pair <uri, typeno> where uri is the uri of a mutual inductive
- type and typeno is the type number (0 based) in the mutual induction
- *)
-let mkMutInd ~uri:(uri, typeno) ~proof =
- let cookingsno = relative_depth (uri_of_proof ~proof) uri 0 in
- Cic.MutInd (uri, cookingsno, typeno)
-
-(** EXCEPTIONS *)
-
- (**
- raised when the current goal is not ringable; a goal is ringable when is an
- equality on reals (@see r_uri)
- *)
-exception GoalUnringable
-
-(** RING's FUNCTIONS LIBRARY *)
-
- (**
- Check whether the ring tactic can be applied on a given term (i.e. that is
- an equality on reals)
- @param term term to be tested
- @return true if the term is ringable, false otherwise
- *)
-let ringable =
- let is_equality = function
- | Cic.MutInd uri _ 0 when (eq uri eqt_uri) -> true
- | _ -> false
- in
- let is_real = function
- | Cic.Const (uri, _) when (eq uri r_uri) -> true
- | _ -> false
- in
- function
- | Cic.Appl (app::set::_::_::[]) when (is_equality app && is_real set) ->
- warn "Goal Ringable!";
- true
- | _ ->
- warn "Goal Not Ringable :-((";
- false
-
- (**
- split an equality goal of the form "t1 = t2" in its two subterms t1 and t2
- after checking that the goal is ringable
- @param goal the current goal
- @return a pair (t1,t2) that are two sides of the equality goal
- @raise GoalUnringable if the goal isn't ringable
- *)
-let split_eq = function
- | (Cic.Appl (_::_::t1::t2::[])) as term when ringable term ->
- warn ("<term1>" ^ (CicPp.ppterm t1) ^ "</term1>");
- warn ("<term2>" ^ (CicPp.ppterm t2) ^ "</term2>");
- (t1, t2)
- | _ -> raise GoalUnringable
-
- (**
- @param i an integer index representing a 1 based number of node in a binary
- search tree counted in a fbs manner (i.e.: 1 is the root, 2 is the left
- child of the root (if any), 3 is the right child of the root (if any), 4 is
- the left child of the left child of the root (if any), ....)
- @param proof the current proof
- @return an index representing the same node in a varmap (@see varmap_uri),
- the returned index is as defined in index (@see index_uri)
- *)
-let path_of_int n proof =
- let rec digits_of_int n =
- if n=1 then [] else (n mod 2 = 1)::(digits_of_int (n lsr 1))
- in
- List.fold_right
- (fun digit path ->
- Cic.Appl [
- mkCtor (if (digit = true) then right_idx_uri else left_idx_uri) proof;
- path])
- (List.rev (digits_of_int n)) (* remove leading true (i.e. digit 1) *)
- (mkCtor end_idx_uri proof)
-
- (**
- Build a variable map (@see varmap_uri) from a variables array.
- A variable map is almost a binary tree so this function receiving a var list
- like [v;w;x;y;z] will build a varmap of shape: v
- / \
- w x
- / \
- y z
- @param vars variables array
- @param proof current proof
- @return a cic term representing the variable map containing vars variables
- *)
-let btree_of_array ~vars ~proof =
- let r = mkConst r_uri proof in (* cic objects *)
- let empty_vm = mkCtor empty_vm_uri proof in
- let empty_vm_r = Cic.Appl [empty_vm; r] in
- let node_vm = mkCtor node_vm_uri proof in
- let node_vm_r = Cic.Appl [node_vm; r] in
- let size = Array.length vars in
- let halfsize = size lsr 1 in
- let rec aux n = (* build the btree starting from position n *)
- (*
- n is the position in the vars array _1_based_ in order to access
- left and right child using (n*2, n*2+1) trick
- *)
- if n > size then
- empty_vm_r
- else if n > halfsize then (* no more children *)
- Cic.Appl [node_vm; r; vars.(n-1); empty_vm_r; empty_vm_r]
- else (* still children *)
- Cic.Appl [node_vm; r; vars.(n-1); aux (n*2); aux (n*2+1)]
- in
- aux 1
-
- (**
- abstraction function:
- concrete polynoms -----> (abstract polynoms, varmap)
- @param terms list of conrete polynoms
- @param proof current proof
- @return a pair <aterms, varmap> where aterms is a list of abstract polynoms
- and varmap is the variable map needed to interpret them
- *)
-let abstract_poly ~terms ~proof =
- let varhash = Hashtbl.create 19 in (* vars hash, to speed up lookup *)
- let varlist = ref [] in (* vars list in reverse order *)
- let counter = ref 1 in (* index of next new variable *)
- let rec aux = function (* TODO not tail recursive *)
- (* "bop" -> binary operator | "uop" -> unary operator *)
- | Cic.Appl (bop::t1::t2::[])
- when (cic_is_const ~uri:(Some rplus_uri) bop) -> (* +. *)
- Cic.Appl [mkCtor applus_uri proof; aux t1; aux t2]
- | Cic.Appl (bop::t1::t2::[])
- when (cic_is_const ~uri:(Some rmult_uri) bop) -> (* *. *)
- Cic.Appl [mkCtor apmult_uri proof; aux t1; aux t2]
- | Cic.Appl (uop::t::[])
- when (cic_is_const ~uri:(Some ropp_uri) uop) -> (* ~-. *)
- Cic.Appl [mkCtor apopp_uri proof; aux t]
- | t when (cic_is_const ~uri:(Some r0_uri) t) -> (* 0. *)
- mkCtor ap0_uri proof
- | t when (cic_is_const ~uri:(Some r1_uri) t) -> (* 1. *)
- mkCtor ap1_uri proof
- | t -> (* variable *)
- try
- Hashtbl.find varhash t (* use an old var *)
- with Not_found -> begin (* create a new var *)
- let newvar =
- Cic.Appl [mkCtor apvar_uri proof; path_of_int !counter proof]
- in
- incr counter;
- varlist := t :: !varlist;
- Hashtbl.add varhash t newvar;
- newvar
- end
- in
- let aterms = List.map aux terms in (* abstract vars *)
- let varmap = (* build varmap *)
- btree_of_array ~vars:(Array.of_list (List.rev !varlist)) ~proof
- in
- (aterms, varmap)
-
- (**
- given a list of abstract terms (i.e. apolynomials) build the ring "segments"
- that is triples like (t', t'', t''') where
- t' = interp_ap(varmap, at)
- t'' = interp_sacs(varmap, (apolynomial_normalize at))
- t''' = apolynomial_normalize_ok(varmap, at)
- at is the abstract term built from t, t is a single member of aterms
- *)
-let build_segments ~terms ~proof =
- let r = mkConst r_uri proof in (* cic objects *)
- let rplus = mkConst rplus_uri proof in
- let rmult = mkConst rmult_uri proof in
- let ropp = mkConst ropp_uri proof in
- let r0 = mkConst r0_uri proof in
- let r1 = mkConst r1_uri proof in
- let interp_ap = mkConst interp_ap_uri proof in
- let interp_sacs = mkConst interp_sacs_uri proof in
- let apolynomial_normalize = mkConst apolynomial_normalize_uri proof in
- let apolynomial_normalize_ok = mkConst apolynomial_normalize_ok_uri proof in
- let rtheory = mkConst rtheory_uri proof in
- let lxy_false = (** Cic funcion "fun (x,y):R -> false" *)
- Cic.Lambda (Cic.Anonimous, r,
- Cic.Lambda (Cic.Anonimous, r,
- mkCtor false_uri proof))
- in
- let theory_args = [r; rplus; rmult; r1; r0; ropp] in
- let (aterms, varmap) = abstract_poly ~terms ~proof in (* abstract polys *)
- List.map (* build ring segments *)
- (fun t ->
- Cic.Appl ((interp_ap :: theory_args) @ [varmap; t]),
- Cic.Appl (
- interp_sacs ::
- (theory_args @
- [varmap; Cic.Appl [apolynomial_normalize; t]])),
- Cic.Appl (
- (apolynomial_normalize_ok :: theory_args) @
- [lxy_false; varmap; rtheory; t]))
- aterms
-
-let id_tac ~status:(proof,goal) =
- (proof,[goal])
-
-let status_of_single_goal_tactic_result =
- function
- proof,[goal] -> proof,goal
- | _ ->
- raise (Fail "status_of_single_goal_tactic_result: the tactic did not produce exactly a new goal")
-
- (**
- auxiliary tactic "elim_type"
- @param status current proof engine status
- @param term term to cut
- *)
-let elim_type_tac ~term ~status =
- warn "in Ring.elim_type_tac";
- Tacticals.thens ~start:(cut_tac ~term)
- ~continuations:[elim_intros_simpl_tac ~term:(Cic.Rel 1) ; id_tac] ~status
-
- (**
- auxiliary tactic, use elim_type and try to close 2nd subgoal using proof
- @param status current proof engine status
- @param term term to cut
- @param proof term used to prove second subgoal generated by elim_type
- *)
-let elim_type2_tac ~term ~proof ~status =
- warn "in Ring.elim_type2";
- Tacticals.thens ~start:(elim_type_tac ~term)
- ~continuations:[id_tac ; exact_tac ~term:proof] ~status
-
- (**
- Reflexivity tactic, try to solve current goal using "refl_eqT"
- Warning: this isn't equale to the coq's Reflexivity because this one tries
- only refl_eqT, coq's one also try "refl_equal"
- @param status current proof engine status
- *)
-let reflexivity_tac ~status:(proof, goal) =
- warn "in Ring.reflexivity_tac";
- let refl_eqt = mkCtor ~uri:refl_eqt_uri ~proof:proof in
- try
- apply_tac ~status:(proof, goal) ~term:refl_eqt
- with (Fail _) as e ->
- let e_str = Printexc.to_string e in
- raise (Fail ("Reflexivity failed with exception: " ^ e_str))
-
- (** lift an 8-uple of debrujins indexes of n *)
-let lift ~n (a,b,c,d,e,f,g,h) =
- match (List.map (CicSubstitution.lift n) [a;b;c;d;e;f;g;h]) with
- | [a;b;c;d;e;f;g;h] -> (a,b,c,d,e,f,g,h)
- | _ -> assert false
-
- (**
- remove hypothesis from a given status starting from the last one
- @param count number of hypotheses to remove
- @param status current proof engine status
- *)
-let purge_hyps_tac ~count ~status:(proof, goal as status) =
- let module S = ProofEngineStructuralRules in
- let rec aux n context status =
- assert(n>=0);
- match (n, context) with
- | (0, _) -> status
- | (n, hd::tl) ->
- aux (n-1) tl
- (status_of_single_goal_tactic_result (S.clear ~hyp:hd ~status))
- | (_, []) -> failwith "Ring.purge_hyps_tac: no hypotheses left"
- in
- let (_, metasenv, _, _) = proof in
- let (_, context, _) = conj_of_metano goal metasenv in
- let proof',goal' = aux count context status in
- assert (goal = goal') ;
- proof',[goal']
-
-(** THE TACTIC! *)
-
- (**
- Ring tactic, does associative and commutative rewritings in Reals ring
- @param status current proof engine status
- *)
-let ring_tac ~status:(proof, goal) =
- warn "in Ring tactic";
- let status = (proof, goal) in
- let eqt = mkMutInd (eqt_uri, 0) proof in
- let r = mkConst r_uri proof in
- let metasenv = metasenv_of_status ~status in
- let (metano, context, ty) = conj_of_metano goal metasenv in
- let (t1, t2) = split_eq ty in (* goal like t1 = t2 *)
- match (build_segments ~terms:[t1; t2] ~proof) with
- | (t1', t1'', t1'_eq_t1'')::(t2', t2'', t2'_eq_t2'')::[] -> begin
- List.iter (* debugging, feel free to remove *)
- (fun (descr, term) ->
- warn (descr ^ " " ^ (CicPp.ppterm term)))
- (List.combine
- ["t1"; "t1'"; "t1''"; "t1'_eq_t1''";
- "t2"; "t2'"; "t2''"; "t2'_eq_t2''"]
- [t1; t1'; t1''; t1'_eq_t1'';
- t2; t2'; t2''; t2'_eq_t2'']);
- try
- let new_hyps = ref 0 in (* number of new hypotheses created *)
- Tacticals.try_tactics
- ~status
- ~tactics:[
- "reflexivity", reflexivity_tac;
- "exact t1'_eq_t1''", exact_tac ~term:t1'_eq_t1'';
- "exact t2'_eq_t2''", exact_tac ~term:t2'_eq_t2'';
- "exact sym_eqt su t1 ...", exact_tac ~term:(
- Cic.Appl [
- mkConst sym_eqt_uri proof; mkConst r_uri proof;
- t1''; t1; t1'_eq_t1'']);
- "elim_type eqt su t1 ...", (fun ~status ->
- let status' = (* status after 1st elim_type use *)
- let context = context_of_status ~status in
- if not (are_convertible context t1'' t1) then begin
- warn "t1'' and t1 are NOT CONVERTIBLE";
- let newstatus =
- elim_type2_tac (* 1st elim_type use *)
- ~status ~proof:t1'_eq_t1''
- ~term:(Cic.Appl [eqt; r; t1''; t1])
- in
- incr new_hyps; (* elim_type add an hyp *)
- match newstatus with
- (proof,[goal]) -> proof,goal
- | _ -> assert false
- end else begin
- warn "t1'' and t1 are CONVERTIBLE";
- status
- end
- in
- let (t1,t1',t1'',t1'_eq_t1'',t2,t2',t2'',t2'_eq_t2'') =
- lift 1 (t1,t1',t1'',t1'_eq_t1'', t2,t2',t2'',t2'_eq_t2'')
- in
- let status'' =
- Tacticals.try_tactics (* try to solve 1st subgoal *)
- ~status:status'
- ~tactics:[
- "exact t2'_eq_t2''", exact_tac ~term:t2'_eq_t2'';
- "exact sym_eqt su t2 ...",
- exact_tac ~term:(
- Cic.Appl [
- mkConst sym_eqt_uri proof;
- mkConst r_uri proof;
- t2''; t2; t2'_eq_t2'']);
- "elim_type eqt su t2 ...", (fun ~status ->
- let status' =
- let context = context_of_status ~status in
- if not (are_convertible context t2'' t2) then begin
- warn "t2'' and t2 are NOT CONVERTIBLE";
- let newstatus =
- elim_type2_tac (* 2nd elim_type use *)
- ~status ~proof:t2'_eq_t2''
- ~term:(Cic.Appl [eqt; r; t2''; t2])
- in
- incr new_hyps; (* elim_type add an hyp *)
- match newstatus with
- (proof,[goal]) -> proof,goal
- | _ -> assert false
- end else begin
- warn "t2'' and t2 are CONVERTIBLE";
- status
- end
- in
- try (* try to solve main goal *)
- warn "trying reflexivity ....";
- reflexivity_tac ~status:status'
- with (Fail _) -> (* leave conclusion to the user *)
- warn "reflexivity failed, solution's left as an ex :-)";
- purge_hyps_tac ~count:!new_hyps ~status:status')]
- in
- status'')]
- with (Fail s) ->
- raise (Fail ("Ring failure: " ^ s))
- end
- | _ -> (* impossible: we are applying ring exacty to 2 terms *)
- assert false
-
- (* wrap ring_tac catching GoalUnringable and raising Fail *)
-let ring_tac ~status =
- try
- ring_tac ~status
- with GoalUnringable ->
- raise (Fail "goal unringable")
-
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