Source file mfourier.ml
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open NumCompat
open Q.Notations
open Util
open Polynomial
open Vect
let debug = false
let compare_float (p : float) q = pervasives_compare p q
(** Implementation of intervals *)
open Itv
type vector = Vect.t
(** 'cstr' is the type of constraints.
{coeffs = v ; bound = (l,r) } models the constraints l <= v <= r
**)
module ISet = Set.Make (Int)
module System = Hashtbl.Make (Vect)
type proof = Assum of int | Elim of var * proof * proof | And of proof * proof
type system = {sys : cstr_info ref System.t; vars : ISet.t}
and cstr_info = {bound : interval; prf : proof; pos : int; neg : int}
(** A system of constraints has the form [\{sys = s ; vars = v\}].
[s] is a hashtable mapping a normalised vector to a [cstr_info] record where
- [bound] is an interval
- [prf_idx] is the set of hypothesis indexes (i.e. constraints in the initial system) used to obtain the current constraint.
In the initial system, each constraint is given an unique singleton proof_idx.
When a new constraint c is computed by a function f(c1,...,cn), its proof_idx is ISet.fold union (List.map (fun x -> x.proof_idx) [c1;...;cn]
- [pos] is the number of positive values of the vector
- [neg] is the number of negative values of the vector
( [neg] + [pos] is therefore the length of the vector)
[v] is an upper-bound of the set of variables which appear in [s].
*)
(** To be thrown when a system has no solution *)
exception SystemContradiction of proof
(** Pretty printing *)
let rec pp_proof o prf =
match prf with
| Assum i -> Printf.fprintf o "H%i" i
| Elim (v, prf1, prf2) ->
Printf.fprintf o "E(%i,%a,%a)" v pp_proof prf1 pp_proof prf2
| And (prf1, prf2) -> Printf.fprintf o "A(%a,%a)" pp_proof prf1 pp_proof prf2
let pp_cstr o (vect, bnd) =
let l, r = bnd in
( match l with
| None -> ()
| Some n -> Printf.fprintf o "%s <= " (Q.to_string n) );
Vect.pp o vect;
match r with
| None -> output_string o "\n"
| Some n -> Printf.fprintf o "<=%s\n" (Q.to_string n)
let pp_system o sys =
System.iter (fun vect ibnd -> pp_cstr o (vect, !ibnd.bound)) sys
(** [merge_cstr_info] takes:
- the intersection of bounds and
- the union of proofs
- [pos] and [neg] fields should be identical *)
let merge_cstr_info i1 i2 =
let {pos = p1; neg = n1; bound = i1; prf = prf1} = i1
and {pos = p2; neg = n2; bound = i2; prf = prf2} = i2 in
assert (Int.equal p1 p2 && Int.equal n1 n2);
match inter i1 i2 with
| None -> None
| Some bnd -> Some {pos = p1; neg = n1; bound = bnd; prf = And (prf1, prf2)}
(** [xadd_cstr vect cstr_info] loads an constraint into the system.
The constraint is neither redundant nor contradictory.
@raise SystemContradiction if [cstr_info] returns [None]
*)
let xadd_cstr vect cstr_info sys =
try
let info = System.find sys vect in
match merge_cstr_info cstr_info !info with
| None -> raise (SystemContradiction (And (cstr_info.prf, !info.prf)))
| Some info' -> info := info'
with Not_found -> System.replace sys vect (ref cstr_info)
exception TimeOut
let xadd_cstr vect cstr_info sys =
if debug && Int.equal (System.length sys mod 1000) 0 then (
print_string "*"; flush stdout );
if System.length sys < !max_nb_cstr then xadd_cstr vect cstr_info sys
else raise TimeOut
type cstr_ext =
| Contradiction
(** The constraint is contradictory.
Typically, a [SystemContradiction] exception will be raised. *)
| Redundant
(** The constrain is redundant.
Typically, the constraint will be dropped *)
| Cstr of vector * cstr_info
(** Taken alone, the constraint is neither contradictory nor redundant.
Typically, it will be added to the constraint system. *)
(** [normalise_cstr] : vector -> cstr_info -> cstr_ext *)
let normalise_cstr vect cinfo =
match norm_itv cinfo.bound with
| None -> Contradiction
| Some (l, r) -> (
match Vect.choose vect with
| None -> if Itv.in_bound (l, r) Q.zero then Redundant else Contradiction
| Some (_, n, _) ->
Cstr
( Vect.div n vect
, let divn x = x // n in
if Int.equal (Q.sign n) 1 then
{cinfo with bound = (Option.map divn l, Option.map divn r)}
else
{ cinfo with
pos = cinfo.neg
; neg = cinfo.pos
; bound = (Option.map divn r, Option.map divn l) } ) )
(** For compatibility, there is an external representation of constraints *)
let count v =
Vect.fold
(fun (n, p) _ vl ->
let sg = Q.sign vl in
assert (sg <> 0);
if Int.equal sg 1 then (n, p + 1) else (n + 1, p))
(0, 0) v
let norm_cstr {coeffs = v; op = o; cst = c} idx =
let n, p = count v in
normalise_cstr v
{ pos = p
; neg = n
; bound =
( match o with
| Eq -> (Some c, Some c)
| Ge -> (Some c, None)
| Gt -> raise Polynomial.Strict )
; prf = Assum idx }
(** [load_system l] takes a list of constraints of type [cstr_compat]
@return a system of constraints
@raise SystemContradiction if a contradiction is found
*)
let load_system l =
let sys = System.create 1000 in
let li = List.mapi (fun i e -> (e, i)) l in
let vars =
List.fold_left
(fun vrs (cstr, i) ->
match norm_cstr cstr i with
| Contradiction -> raise (SystemContradiction (Assum i))
| Redundant -> vrs
| Cstr (vect, info) ->
xadd_cstr vect info sys;
Vect.fold (fun s v _ -> ISet.add v s) vrs cstr.coeffs)
ISet.empty li
in
{sys; vars}
let system_list sys =
let {sys = s; vars = v} = sys in
System.fold (fun k bi l -> (k, !bi) :: l) s []
(** [add (v1,c1) (v2,c2) ]
precondition: (c1 <>/ Q.zero && c2 <>/ Q.zero)
@return a pair [(v,ln)] such that
[v] is the sum of vector [v1] divided by [c1] and vector [v2] divided by [c2]
Note that the resulting vector is not normalised.
*)
let add (v1, c1) (v2, c2) =
assert (c1 <>/ Q.zero && c2 <>/ Q.zero);
let res = mul_add (Q.one // c1) v1 (Q.one // c2) v2 in
(res, count res)
let add (v1, c1) (v2, c2) =
let res = add (v1, c1) (v2, c2) in
res
(** To perform Fourier elimination, constraints are categorised depending on the sign of the variable to eliminate. *)
(** [split x vect info (l,m,r)]
@param v is the variable to eliminate
@param l contains constraints such that (e + a*x) // a >= c / a
@param r contains constraints such that (e + a*x) // - a >= c / -a
@param m contains constraints which do not mention [x]
*)
let split x (vect : vector) info (l, m, r) =
let vl = get x vect in
if Q.zero =/ vl then
(l, (vect, info) :: m, r)
else
let cons_bound lst bd =
match bd with
| None -> lst
| Some bnd -> (vl, vect, {info with bound = (Some bnd, None)}) :: lst
in
let lb, rb = info.bound in
if Int.equal (Q.sign vl) 1 then (cons_bound l lb, m, cons_bound r rb)
else
(cons_bound l rb, m, cons_bound r lb)
(** [project vr sys] projects system [sys] over the set of variables [ISet.remove vr sys.vars ].
This is a one step Fourier elimination.
*)
let project vr sys =
let l, m, r =
System.fold
(fun vect rf l_m_r -> split vr vect !rf l_m_r)
sys.sys ([], [], [])
in
let new_sys = System.create (System.length sys.sys) in
List.iter (fun (vect, info) -> System.replace new_sys vect (ref info)) m;
let elim (v1, vect1, info1) (v2, vect2, info2) =
let {neg = n1; pos = p1; bound = bound1; prf = prf1} = info1
and {neg = n2; pos = p2; bound = bound2; prf = prf2} = info2 in
let bnd1 = Option.get (fst bound1) and bnd2 = Option.get (fst bound2) in
let bound = (bnd1 // v1) +/ (bnd2 // Q.neg v2) in
let vres, (n, p) = add (vect1, v1) (vect2, Q.neg v2) in
( vres
, { neg = n
; pos = p
; bound = (Some bound, None)
; prf = Elim (vr, info1.prf, info2.prf) } )
in
List.iter
(fun l_elem ->
List.iter
(fun r_elem ->
let vect, info = elim l_elem r_elem in
match normalise_cstr vect info with
| Redundant -> ()
| Contradiction -> raise (SystemContradiction info.prf)
| Cstr (vect, info) -> xadd_cstr vect info new_sys)
r)
l;
{sys = new_sys; vars = ISet.remove vr sys.vars}
(** [project_using_eq] performs elimination by pivoting using an equation.
This is the counter_part of the [elim] sub-function of [!project].
@param vr is the variable to be used as pivot
@param c is the coefficient of variable [vr] in vector [vect]
@param len is the length of the equation
@param bound is the bound of the equation
@param prf is the proof of the equation
*)
let project_using_eq vr c vect bound prf (vect', info') =
let c2 = get vr vect' in
if Q.zero =/ c2 then (vect', info')
else
let c1 = if c2 >=/ Q.zero then Q.neg c else c in
let c2 = Q.abs c2 in
let vres, (n, p) = add (vect, c1) (vect', c2) in
let cst = bound // c1 in
let bndres =
let f x = cst +/ (x // c2) in
let l, r = info'.bound in
(Option.map f l, Option.map f r)
in
(vres, {neg = n; pos = p; bound = bndres; prf = Elim (vr, prf, info'.prf)})
let elim_var_using_eq vr vect cst prf sys =
let c = get vr vect in
let elim_var = project_using_eq vr c vect cst prf in
let new_sys = System.create (System.length sys.sys) in
System.iter
(fun vect iref ->
let vect', info' = elim_var (vect, !iref) in
match normalise_cstr vect' info' with
| Redundant -> ()
| Contradiction -> raise (SystemContradiction info'.prf)
| Cstr (vect, info') -> xadd_cstr vect info' new_sys)
sys.sys;
{sys = new_sys; vars = ISet.remove vr sys.vars}
(** [size sys] computes the number of entries in the system of constraints *)
let size sys = System.fold (fun v iref s -> s + !iref.neg + !iref.pos) sys 0
module IMap = CMap.Make (Int)
(** [eval_vect map vect] evaluates vector [vect] using the values of [map].
If [map] binds all the variables of [vect], we get
[eval_vect map [(x1,v1);...;(xn,vn)] = (IMap.find x1 map * v1) + ... + (IMap.find xn map) * vn , []]
The function returns as second argument, a sub-vector consisting in the variables that are not in [map]. *)
let eval_vect map vect =
Vect.fold
(fun (sum, rst) v vl ->
try
let val_v = IMap.find v map in
(sum +/ (val_v */ vl), rst)
with Not_found -> (sum, Vect.set v vl rst))
(Q.zero, Vect.null) vect
(** [restrict_bound n sum itv] returns the interval of [x]
given that (fst itv) <= x * n + sum <= (snd itv) *)
let restrict_bound n sum (itv : interval) =
let f x = (x -/ sum) // n in
let l, r = itv in
match Q.sign n with
| 0 ->
if in_bound itv sum then (None, None)
else failwith "SystemContradiction"
| 1 -> (Option.map f l, Option.map f r)
| _ -> (Option.map f r, Option.map f l)
(** [bound_of_variable map v sys] computes the interval of [v] in
[sys] given a mapping [map] binding all the other variables *)
let bound_of_variable map v sys =
System.fold
(fun vect iref bnd ->
let sum, rst = eval_vect map vect in
let vl = Vect.get v rst in
match inter bnd (restrict_bound vl sum !iref.bound) with
| None ->
Printf.fprintf stdout "bound_of_variable: eval_vecr %a = %s,%a\n"
Vect.pp vect (Q.to_string sum) Vect.pp rst;
Printf.fprintf stdout "current interval: %a\n" Itv.pp !iref.bound;
failwith "bound_of_variable: impossible"
| Some itv -> itv)
sys (None, None)
(** [pick_small_value bnd] picks a value being closed to zero within the interval *)
let pick_small_value bnd =
match bnd with
| None, None -> Q.zero
| None, Some i -> if Q.zero <=/ Q.floor i then Q.zero else Q.floor i
| Some i, None -> if i <=/ Q.zero then Q.zero else Q.ceiling i
| Some i, Some j ->
if i <=/ Q.zero && Q.zero <=/ j then Q.zero
else if Q.ceiling i <=/ Q.floor j then Q.ceiling i
else i
(** [solution s1 sys_l = Some(sn,\[(vn-1,sn-1);...; (v1,s1)\]\@sys_l)]
then [sn] is a system which contains only [black_v] -- if it existed in [s1]
and [sn+1] is obtained by projecting [vn] out of [sn]
@raise SystemContradiction if system [s] has no solution
*)
let solve_sys black_v choose_eq choose_variable sys sys_l =
let rec solve_sys sys sys_l =
if debug then
Printf.printf "S #%i size %i\n" (System.length sys.sys) (size sys.sys);
if debug then Printf.printf "solve_sys :\n %a" pp_system sys.sys;
let eqs = choose_eq sys in
try
let v, vect, cst, ln =
fst (List.find (fun ((v, _, _, _), _) -> v <> black_v) eqs)
in
if debug then (
Printf.printf "\nE %a = %s variable %i\n" Vect.pp vect (Q.to_string cst)
v;
flush stdout );
let sys' = elim_var_using_eq v vect cst ln sys in
solve_sys sys' ((v, sys) :: sys_l)
with Not_found -> (
let vars = choose_variable sys in
try
let v, est = List.find (fun (v, _) -> v <> black_v) vars in
if debug then (
Printf.printf "\nV : %i estimate %f\n" v est;
flush stdout );
let sys' = project v sys in
solve_sys sys' ((v, sys) :: sys_l)
with Not_found -> Inl (sys, sys_l) )
in
solve_sys sys sys_l
let solve black_v choose_eq choose_variable cstrs =
try
let sys = load_system cstrs in
if debug then Printf.printf "solve :\n %a" pp_system sys.sys;
solve_sys black_v choose_eq choose_variable sys []
with SystemContradiction prf -> Inr prf
(** The purpose of module [EstimateElimVar] is to try to estimate the cost of eliminating a variable.
The output is an ordered list of (variable,cost).
*)
module EstimateElimVar = struct
type sys_list = (vector * cstr_info) list
let abstract_partition (v : int) (l : sys_list) =
let rec xpart (l : sys_list) (ltl : sys_list) (n : int list) (z : int)
(p : int list) =
match l with
| [] -> (ltl, n, z, p)
| (l1, info) :: rl -> (
match Vect.choose l1 with
| None ->
xpart rl ((Vect.null, info) :: ltl) n (info.neg + info.pos + z) p
| Some (vr, vl, rl1) ->
if Int.equal v vr then
let cons_bound lst bd =
match bd with
| None -> lst
| Some bnd -> (info.neg + info.pos) :: lst
in
let lb, rb = info.bound in
if Int.equal (Q.sign vl) 1 then
xpart rl ((rl1, info) :: ltl) (cons_bound n lb) z
(cons_bound p rb)
else
xpart rl ((rl1, info) :: ltl) (cons_bound n rb) z
(cons_bound p lb)
else
xpart rl ((l1, info) :: ltl) n (info.neg + info.pos + z) p )
in
let sys', n, z, p = xpart l [] [] 0 [] in
let ln = float_of_int (List.length n) in
let sn = float_of_int (List.fold_left ( + ) 0 n) in
let lp = float_of_int (List.length p) in
let sp = float_of_int (List.fold_left ( + ) 0 p) in
(sys', float_of_int z +. (lp *. sn) +. (ln *. sp) -. (lp *. ln))
let choose_variable sys =
let {sys = s; vars = v} = sys in
let sl = system_list sys in
let evals =
fst
(ISet.fold
(fun v (eval, s) ->
let ts, vl = abstract_partition v s in
((v, vl) :: eval, ts))
v ([], sl))
in
List.sort (fun x y -> compare_float (snd x) (snd y)) evals
end
open EstimateElimVar
(** The module [EstimateElimEq] is similar to [EstimateElimVar] but it orders equations.
*)
module EstimateElimEq = struct
let itv_point bnd = match bnd with Some a, Some b -> a =/ b | _ -> false
let rec unroll_until v l =
match Vect.choose l with
| None -> (false, Vect.null)
| Some (i, _, rl) ->
if Int.equal i v then (true, rl)
else if i < v then unroll_until v rl
else (false, l)
let rec choose_simple_equation eqs =
match eqs with
| [] -> None
| (vect, a, prf, ln) :: eqs -> (
match Vect.choose vect with
| Some (i, v, rst) ->
if Vect.is_null rst then Some (i, vect, a, prf, ln)
else choose_simple_equation eqs
| _ -> choose_simple_equation eqs )
let choose_primal_equation eqs (sys_l : (Vect.t * cstr_info) list) =
let is_primal_equation_var v =
List.fold_left
(fun nb_eq (vect, info) ->
if fst (unroll_until v vect) then
if itv_point info.bound then nb_eq + 1 else nb_eq
else nb_eq)
0 sys_l
in
let rec find_var vect =
match Vect.choose vect with
| None -> None
| Some (i, _, vect) ->
let nb_eq = is_primal_equation_var i in
if Int.equal nb_eq 2 then Some i else find_var vect
in
let rec find_eq_var eqs =
match eqs with
| [] -> None
| (vect, a, prf, ln) :: l -> (
match find_var vect with
| None -> find_eq_var l
| Some r -> Some (r, vect, a, prf, ln) )
in
match choose_simple_equation eqs with
| None -> find_eq_var eqs
| Some res -> Some res
let choose_equality_var sys =
let sys_l = system_list sys in
let equalities =
List.fold_left
(fun l (vect, info) ->
match info.bound with
| Some a, Some b ->
if a =/ b then
(vect, a, info.prf, info.neg + info.pos) :: l
else l
| _ -> l)
[] sys_l
in
let rec estimate_cost v ct sysl acc tlsys =
match sysl with
| [] -> (acc, tlsys)
| (l, info) :: rsys -> (
let ln = info.pos + info.neg in
let b, l = unroll_until v l in
match b with
| true ->
if itv_point info.bound then
estimate_cost v ct rsys (acc + ln) ((l, info) :: tlsys)
else estimate_cost v ct rsys (acc + ln + ct) ((l, info) :: tlsys)
| false -> estimate_cost v ct rsys (acc + ln) ((l, info) :: tlsys) )
in
match choose_primal_equation equalities sys_l with
| None ->
let cost_eq eq const prf ln acc_costs =
let rec cost_eq eqr sysl costs =
match Vect.choose eqr with
| None -> costs
| Some (v, _, eqr) ->
let cst, tlsys = estimate_cost v (ln - 1) sysl 0 [] in
cost_eq eqr tlsys (((v, eq, const, prf), cst) :: costs)
in
cost_eq eq sys_l acc_costs
in
let all_costs =
List.fold_left
(fun all_costs (vect, const, prf, ln) ->
cost_eq vect const prf ln all_costs)
[] equalities
in
List.sort (fun x y -> Int.compare (snd x) (snd y)) all_costs
| Some (v, vect, const, prf, _) -> [((v, vect, const, prf), 0)]
end
open EstimateElimEq
module Fourier = struct
let optimise vect l =
let fresh = List.fold_left (fun fr c -> max fr (Vect.fresh c.coeffs)) 0 l in
let cstr =
{coeffs = Vect.set fresh Q.minus_one vect; op = Eq; cst = Q.zero}
in
match solve fresh choose_equality_var choose_variable (cstr :: l) with
| Inr prf -> None
| Inl (s, _) -> (
try Some (bound_of_variable IMap.empty fresh s.sys)
with x when CErrors.noncritical x ->
Printf.printf "optimise Exception : %s" (Printexc.to_string x);
None )
let find_point cstrs =
match solve max_int choose_equality_var choose_variable cstrs with
| Inr prf -> Inr prf
| Inl (_, l) ->
let rec rebuild_solution l map =
match l with
| [] -> map
| (v, e) :: l ->
let itv = bound_of_variable map v e.sys in
let map = IMap.add v (pick_small_value itv) map in
rebuild_solution l map
in
let map = rebuild_solution l IMap.empty in
let vect = IMap.fold (fun v i vect -> Vect.set v i vect) map Vect.null in
if debug then Printf.printf "SOLUTION %a" Vect.pp vect;
let res = Inl vect in
res
end
module Proof = struct
(** A proof term in the sense of a ZMicromega.RatProof is a positive combination of the hypotheses which leads to a contradiction.
The proofs constructed by Fourier elimination are more like execution traces:
- certain facts are recorded but are useless
- certain inferences are implicit.
The following code implements proof reconstruction.
*)
let add x y = fst (add x y)
let forall_pairs f l1 l2 =
List.fold_left
(fun acc e1 ->
List.fold_left
(fun acc e2 -> match f e1 e2 with None -> acc | Some v -> v :: acc)
acc l2)
[] l1
let add_op x y = match (x, y) with Eq, Eq -> Eq | _ -> Ge
let pivot v (p1, c1) (p2, c2) =
let {coeffs = v1; op = op1; cst = n1} = c1
and {coeffs = v2; op = op2; cst = n2} = c2 in
let a, b = (Vect.get v v1, Vect.get v v2) in
if Q.zero =/ a || Q.zero =/ b then None
else if Int.equal (Q.sign a * Q.sign b) (-1) then
Some
( add (p1, Q.abs a) (p2, Q.abs b)
, { coeffs = add (v1, Q.abs a) (v2, Q.abs b)
; op = add_op op1 op2
; cst = (n1 // Q.abs a) +/ (n2 // Q.abs b) } )
else if op1 == Eq then
Some
( add (p1, Q.neg (a // b)) (p2, Q.one)
, { coeffs = add (v1, Q.neg (a // b)) (v2, Q.one)
; op = add_op op1 op2
; cst = (n1 // Q.neg (a // b)) +/ (n2 // Q.one) } )
else if op2 == Eq then
Some
( add (p2, Q.neg (b // a)) (p1, Q.one)
, { coeffs = add (v2, Q.neg (b // a)) (v1, Q.one)
; op = add_op op1 op2
; cst = (n2 // Q.neg (b // a)) +/ (n1 // Q.one) } )
else None
let normalise_proofs l =
List.fold_left
(fun acc (prf, cstr) ->
match acc with
| Inr _ -> acc
| Inl acc -> (
match norm_cstr cstr 0 with
| Redundant -> Inl acc
| Contradiction -> Inr (prf, cstr)
| Cstr (v, info) -> Inl ((prf, cstr, v, info) :: acc) ))
(Inl []) l
type oproof = (vector * cstr * Q.t) option
let merge_proof (oleft : oproof) (prf, cstr, v, info) (oright : oproof) =
let l, r = info.bound in
let keep p ob bd =
match (ob, bd) with
| None, None -> None
| None, Some b -> Some (prf, cstr, b)
| Some _, None -> ob
| Some (prfl, cstrl, bl), Some b ->
if p bl b then Some (prf, cstr, b) else ob
in
let oleft = keep ( <=/ ) oleft l in
let oright = keep ( >=/ ) oright r in
match (oleft, oright) with
| None, _ | _, None -> Inl (oleft, oright)
| Some (prfl, cstrl, l), Some (prfr, cstrr, r) -> (
if l <=/ r then Inl (oleft, oright)
else
match Vect.choose cstrr.coeffs with
| None ->
Inr (add (prfl, Q.one) (prfr, Q.one), cstrr)
| Some (v, _, _) -> (
match pivot v (prfl, cstrl) (prfr, cstrr) with
| None -> failwith "merge_proof : pivot is not possible"
| Some x -> Inr x ) )
let mk_proof hyps prf =
let rec mk_proof prf =
match prf with
| Assum i -> [(Vect.set i Q.one Vect.null, List.nth hyps i)]
| Elim (v, prf1, prf2) ->
let prfsl = mk_proof prf1 and prfsr = mk_proof prf2 in
forall_pairs (pivot v) prfsl prfsr
| And (prf1, prf2) -> (
let prfsl1 = mk_proof prf1 and prfsl2 = mk_proof prf2 in
match normalise_proofs (prfsl1 @ prfsl2) with
| Inr x -> [x]
| Inl l -> (
let prfs =
List.fold_left
(fun acc e ->
match acc with
| Inr _ -> acc
| Inl (oleft, oright) -> merge_proof oleft e oright)
(Inl (None, None))
l
in
match prfs with
| Inr x -> [x]
| Inl (oleft, oright) -> (
match (oleft, oright) with
| None, None -> []
| None, Some (prf, cstr, _) | Some (prf, cstr, _), None ->
[(prf, cstr)]
| Some (prf1, cstr1, _), Some (prf2, cstr2, _) ->
[(prf1, cstr1); (prf2, cstr2)] ) ) )
in
mk_proof prf
end