From 9ebf44d84754adc5b64fcf612c6816c02c80462d Mon Sep 17 00:00:00 2001 From: Benjamin Barenblat Date: Sat, 2 Feb 2019 19:29:23 -0500 Subject: Imported Upstream version 8.9.0 --- plugins/micromega/sos.ml | 616 ++++------------------------------------------- 1 file changed, 52 insertions(+), 564 deletions(-) (limited to 'plugins/micromega/sos.ml') diff --git a/plugins/micromega/sos.ml b/plugins/micromega/sos.ml index e1ceabe9..42a41e17 100644 --- a/plugins/micromega/sos.ml +++ b/plugins/micromega/sos.ml @@ -95,7 +95,7 @@ let dim (v:vector) = fst v;; let vector_const c n = if c =/ Int 0 then vector_0 n - else (n,itlist (fun k -> k |-> c) (1--n) undefined :vector);; + else (n,List.fold_right (fun k -> k |-> c) (1--n) undefined :vector);; let vector_cmul c (v:vector) = let n = dim v in @@ -104,7 +104,7 @@ let vector_cmul c (v:vector) = let vector_of_list l = let n = List.length l in - (n,itlist2 (|->) (1--n) l undefined :vector);; + (n,List.fold_right2 (|->) (1--n) l undefined :vector);; (* ------------------------------------------------------------------------- *) (* Matrices; again rows and columns indexed from 1. *) @@ -242,7 +242,7 @@ let string_of_monomial m = if m = monomial_1 then "1" else let vps = List.fold_right (fun (x,k) a -> string_of_varpow x k :: a) (sort humanorder_varpow (graph m)) [] in - end_itlist (fun s t -> s^"*"^t) vps;; + String.concat "*" vps;; let string_of_cmonomial (c,m) = if m = monomial_1 then string_of_num c @@ -310,7 +310,7 @@ let rec poly_of_term t = match t with let sdpa_of_vector (v:vector) = let n = dim v in let strs = List.map (o (decimalize 20) (element v)) (1--n) in - end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";; + String.concat " " strs ^ "\n";; (* ------------------------------------------------------------------------- *) (* String for a matrix numbered k, in SDPA sparse format. *) @@ -321,7 +321,7 @@ let sdpa_of_matrix k (m:matrix) = let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a) (snd m) [] in let mss = sort (increasing fst) ms in - itlist (fun ((i,j),c) a -> + List.fold_right (fun ((i,j),c) a -> pfx ^ string_of_int i ^ " " ^ string_of_int j ^ " " ^ decimalize 20 c ^ "\n" ^ a) mss "";; @@ -340,7 +340,7 @@ let sdpa_of_problem comment obj mats = "1\n" ^ string_of_int n ^ "\n" ^ sdpa_of_vector obj ^ - itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a) + List.fold_right2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a) (1--List.length mats) mats "";; (* ------------------------------------------------------------------------- *) @@ -489,11 +489,11 @@ let scale_then = and maximal_element amat acc = foldl (fun maxa m c -> max_num maxa (abs_num c)) acc amat in fun solver obj mats -> - let cd1 = itlist common_denominator mats (Int 1) + let cd1 = List.fold_right common_denominator mats (Int 1) and cd2 = common_denominator (snd obj) (Int 1) in let mats' = List.map (mapf (fun x -> cd1 */ x)) mats and obj' = vector_cmul cd2 obj in - let max1 = itlist maximal_element mats' (Int 0) + let max1 = List.fold_right maximal_element mats' (Int 0) and max2 = maximal_element (snd obj') (Int 0) in let scal1 = pow2 (20-int_of_float(log(float_of_num max1) /. log 2.0)) and scal2 = pow2 (20-int_of_float(log(float_of_num max2) /. log 2.0)) in @@ -551,7 +551,7 @@ let minimal_convex_hull = | (m::ms) -> if in_convex_hull ms m then ms else ms@[m] in let augment m ms = funpow 3 augment1 (m::ms) in fun mons -> - let mons' = itlist augment (List.tl mons) [List.hd mons] in + let mons' = List.fold_right augment (List.tl mons) [List.hd mons] in funpow (List.length mons') augment1 mons';; (* ------------------------------------------------------------------------- *) @@ -612,11 +612,11 @@ let newton_polytope pol = let vars = poly_variables pol in let mons = List.map (fun m -> List.map (fun x -> monomial_degree x m) vars) (dom pol) and ds = List.map (fun x -> (degree x pol + 1) / 2) vars in - let all = itlist (fun n -> allpairs (fun h t -> h::t) (0--n)) ds [[]] + let all = List.fold_right (fun n -> allpairs (fun h t -> h::t) (0--n)) ds [[]] and mons' = minimal_convex_hull mons in let all' = List.filter (fun m -> in_convex_hull mons' (List.map (fun x -> 2 * x) m)) all in - List.map (fun m -> itlist2 (fun v i a -> if i = 0 then a else (v |-> i) a) + List.map (fun m -> List.fold_right2 (fun v i a -> if i = 0 then a else (v |-> i) a) vars m monomial_1) (List.rev all');; (* ------------------------------------------------------------------------- *) @@ -657,8 +657,8 @@ let deration d = foldl (fun a i c -> gcd_num a (numerator c)) (Int 0) (snd l) in (c // (a */ a)),mapa (fun x -> a */ x) l in let d' = List.map adj d in - let a = itlist ((o) lcm_num ( (o) denominator fst)) d' (Int 1) // - itlist ((o) gcd_num ( (o) numerator fst)) d' (Int 0) in + let a = List.fold_right ((o) lcm_num ( (o) denominator fst)) d' (Int 1) // + List.fold_right ((o) gcd_num ( (o) numerator fst)) d' (Int 0) in (Int 1 // a),List.map (fun (c,l) -> (a */ c,l)) d';; (* ------------------------------------------------------------------------- *) @@ -719,7 +719,7 @@ let sdpa_of_blockdiagonal k m = let ents = foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in let entss = sort (increasing fst) ents in - itlist (fun ((b,i,j),c) a -> + List.fold_right (fun ((b,i,j),c) a -> pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^ " " ^ decimalize 20 c ^ "\n" ^ a) entss "";; @@ -732,10 +732,10 @@ let sdpa_of_blockproblem comment nblocks blocksizes obj mats = "\"" ^ comment ^ "\"\n" ^ string_of_int m ^ "\n" ^ string_of_int nblocks ^ "\n" ^ - (end_itlist (fun s t -> s^" "^t) (List.map string_of_int blocksizes)) ^ + (String.concat " " (List.map string_of_int blocksizes)) ^ "\n" ^ sdpa_of_vector obj ^ - itlist2 (fun k m a -> sdpa_of_blockdiagonal (k - 1) m ^ a) + List.fold_right2 (fun k m a -> sdpa_of_blockdiagonal (k - 1) m ^ a) (1--List.length mats) mats "";; (* ------------------------------------------------------------------------- *) @@ -791,14 +791,14 @@ let blocks blocksizes bm = (fun a (b,i,j) c -> if b = b0 then ((i,j) |-> c) a else a) undefined bm in (((bs,bs),m):matrix)) - (zip blocksizes (1--List.length blocksizes));; + (List.combine blocksizes (1--List.length blocksizes));; (* ------------------------------------------------------------------------- *) (* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *) (* ------------------------------------------------------------------------- *) let real_positivnullstellensatz_general linf d eqs leqs pol = - let vars = itlist ((o) union poly_variables) (pol::eqs @ List.map fst leqs) [] in + let vars = List.fold_right ((o) union poly_variables) (pol::eqs @ List.map fst leqs) [] in let monoid = if linf then (poly_const num_1,Rational_lt num_1):: @@ -808,16 +808,16 @@ let real_positivnullstellensatz_general linf d eqs leqs pol = let mk_idmultiplier k p = let e = d - multidegree p in let mons = enumerate_monomials e vars in - let nons = zip mons (1--List.length mons) in + let nons = List.combine mons (1--List.length mons) in mons, - itlist (fun (m,n) -> (m |-> ((-k,-n,n) |=> Int 1))) nons undefined in + List.fold_right (fun (m,n) -> (m |-> ((-k,-n,n) |=> Int 1))) nons undefined in let mk_sqmultiplier k (p,c) = let e = (d - multidegree p) / 2 in let mons = enumerate_monomials e vars in - let nons = zip mons (1--List.length mons) in + let nons = List.combine mons (1--List.length mons) in mons, - itlist (fun (m1,n1) -> - itlist (fun (m2,n2) a -> + List.fold_right (fun (m1,n1) -> + List.fold_right (fun (m2,n2) a -> let m = monomial_mul m1 m2 in if n1 > n2 then a else let c = if n1 = n2 then Int 1 else Int 2 in @@ -825,17 +825,17 @@ let real_positivnullstellensatz_general linf d eqs leqs pol = (m |-> equation_add ((k,n1,n2) |=> c) e) a) nons) nons undefined in - let sqmonlist,sqs = unzip(List.map2 mk_sqmultiplier (1--List.length monoid) monoid) - and idmonlist,ids = unzip(List.map2 mk_idmultiplier (1--List.length eqs) eqs) in + let sqmonlist,sqs = List.split(List.map2 mk_sqmultiplier (1--List.length monoid) monoid) + and idmonlist,ids = List.split(List.map2 mk_idmultiplier (1--List.length eqs) eqs) in let blocksizes = List.map List.length sqmonlist in let bigsum = - itlist2 (fun p q a -> epoly_pmul p q a) eqs ids - (itlist2 (fun (p,c) s a -> epoly_pmul p s a) monoid sqs + List.fold_right2 (fun p q a -> epoly_pmul p q a) eqs ids + (List.fold_right2 (fun (p,c) s a -> epoly_pmul p s a) monoid sqs (epoly_of_poly(poly_neg pol))) in let eqns = foldl (fun a m e -> e::a) [] bigsum in let pvs,assig = eliminate_all_equations (0,0,0) eqns in let qvars = (0,0,0)::pvs in - let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in + let allassig = List.fold_right (fun v -> (v |-> (v |=> Int 1))) pvs assig in let mk_matrix v = foldl (fun m (b,i,j) ass -> if b < 0 then m else let c = tryapplyd ass v (Int 0) in @@ -858,8 +858,8 @@ let real_positivnullstellensatz_general linf d eqs leqs pol = else ()); let vec = nice_vector d raw_vec in let blockmat = iter (1,dim vec) - (fun i a -> bmatrix_add (bmatrix_cmul (element vec i) (el i mats)) a) - (bmatrix_neg (el 0 mats)) in + (fun i a -> bmatrix_add (bmatrix_cmul (element vec i) (List.nth mats i)) a) + (bmatrix_neg (List.nth mats 0)) in let allmats = blocks blocksizes blockmat in vec,List.map diag allmats in let vec,ratdias = @@ -867,7 +867,7 @@ let real_positivnullstellensatz_general linf d eqs leqs pol = else tryfind find_rounding (List.map Num.num_of_int (1--31) @ List.map pow2 (5--66)) in let newassigs = - itlist (fun k -> el (k - 1) pvs |-> element vec k) + List.fold_right (fun k -> List.nth pvs (k - 1) |-> element vec k) (1--dim vec) ((0,0,0) |=> Int(-1)) in let finalassigs = foldl (fun a v e -> (v |-> equation_eval newassigs e) a) newassigs @@ -877,17 +877,17 @@ let real_positivnullstellensatz_general linf d eqs leqs pol = undefined p in let mk_sos mons = let mk_sq (c,m) = - c,itlist (fun k a -> (el (k - 1) mons |--> element m k) a) + c,List.fold_right (fun k a -> (List.nth mons (k - 1) |--> element m k) a) (1--List.length mons) undefined in List.map mk_sq in let sqs = List.map2 mk_sos sqmonlist ratdias and cfs = List.map poly_of_epoly ids in let msq = List.filter (fun (a,b) -> b <> []) (List.map2 (fun a b -> a,b) monoid sqs) in - let eval_sq sqs = itlist + let eval_sq sqs = List.fold_right (fun (c,q) -> poly_add (poly_cmul c (poly_mul q q))) sqs poly_0 in let sanity = - itlist (fun ((p,c),s) -> poly_add (poly_mul p (eval_sq s))) msq - (itlist2 (fun p q -> poly_add (poly_mul p q)) cfs eqs + List.fold_right (fun ((p,c),s) -> poly_add (poly_mul p (eval_sq s))) msq + (List.fold_right2 (fun p q -> poly_add (poly_mul p q)) cfs eqs (poly_neg pol)) in if not(is_undefined sanity) then raise Sanity else cfs,List.map (fun (a,b) -> snd a,b) msq;; @@ -913,8 +913,8 @@ let monomial_order = fun m1 m2 -> if m2 = monomial_1 then true else if m1 = monomial_1 then false else let mon1 = dest_monomial m1 and mon2 = dest_monomial m2 in - let deg1 = itlist ((o) (+) snd) mon1 0 - and deg2 = itlist ((o) (+) snd) mon2 0 in + let deg1 = List.fold_right ((o) (+) snd) mon1 0 + and deg2 = List.fold_right ((o) (+) snd) mon2 0 in if deg1 < deg2 then false else if deg1 > deg2 then true else lexorder mon1 mon2;; @@ -929,7 +929,7 @@ let term_of_varpow = let term_of_monomial = fun m -> if m = monomial_1 then Const num_1 else let m' = dest_monomial m in - let vps = itlist (fun (x,k) a -> term_of_varpow x k :: a) m' [] in + let vps = List.fold_right (fun (x,k) a -> term_of_varpow x k :: a) m' [] in end_itlist (fun s t -> Mul (s,t)) vps;; let term_of_cmonomial = @@ -952,203 +952,13 @@ let term_of_sos (pr,sqs) = if sqs = [] then pr else Product(pr,end_itlist (fun a b -> Sum(a,b)) (List.map term_of_sqterm sqs));; -(* ------------------------------------------------------------------------- *) -(* Interface to HOL. *) -(* ------------------------------------------------------------------------- *) -(* -let REAL_NONLINEAR_PROVER translator (eqs,les,lts) = - let eq0 = map (poly_of_term o lhand o concl) eqs - and le0 = map (poly_of_term o lhand o concl) les - and lt0 = map (poly_of_term o lhand o concl) lts in - let eqp0 = map (fun (t,i) -> t,Axiom_eq i) (zip eq0 (0--(length eq0 - 1))) - and lep0 = map (fun (t,i) -> t,Axiom_le i) (zip le0 (0--(length le0 - 1))) - and ltp0 = map (fun (t,i) -> t,Axiom_lt i) (zip lt0 (0--(length lt0 - 1))) in - let keq,eq = partition (fun (p,_) -> multidegree p = 0) eqp0 - and klep,lep = partition (fun (p,_) -> multidegree p = 0) lep0 - and kltp,ltp = partition (fun (p,_) -> multidegree p = 0) ltp0 in - let trivial_axiom (p,ax) = - match ax with - Axiom_eq n when eval undefined p <>/ num_0 -> el n eqs - | Axiom_le n when eval undefined p el n les - | Axiom_lt n when eval undefined p <=/ num_0 -> el n lts - | _ -> failwith "not a trivial axiom" in - try let th = tryfind trivial_axiom (keq @ klep @ kltp) in - CONV_RULE (LAND_CONV REAL_POLY_CONV THENC REAL_RAT_RED_CONV) th - with Failure _ -> - let pol = itlist poly_mul (map fst ltp) (poly_const num_1) in - let leq = lep @ ltp in - let tryall d = - let e = multidegree pol in - let k = if e = 0 then 0 else d / e in - let eq' = map fst eq in - tryfind (fun i -> d,i,real_positivnullstellensatz_general false d eq' leq - (poly_neg(poly_pow pol i))) - (0--k) in - let d,i,(cert_ideal,cert_cone) = deepen tryall 0 in - let proofs_ideal = - map2 (fun q (p,ax) -> Eqmul(term_of_poly q,ax)) cert_ideal eq - and proofs_cone = map term_of_sos cert_cone - and proof_ne = - if ltp = [] then Rational_lt num_1 else - let p = end_itlist (fun s t -> Product(s,t)) (map snd ltp) in - funpow i (fun q -> Product(p,q)) (Rational_lt num_1) in - let proof = end_itlist (fun s t -> Sum(s,t)) - (proof_ne :: proofs_ideal @ proofs_cone) in - print_string("Translating proof certificate to HOL"); - print_newline(); - translator (eqs,les,lts) proof;; -*) -(* ------------------------------------------------------------------------- *) -(* A wrapper that tries to substitute away variables first. *) -(* ------------------------------------------------------------------------- *) -(* -let REAL_NONLINEAR_SUBST_PROVER = - let zero = `&0:real` - and mul_tm = `( * ):real->real->real` - and shuffle1 = - CONV_RULE(REWR_CONV(REAL_ARITH `a + x = (y:real) <=> x = y - a`)) - and shuffle2 = - CONV_RULE(REWR_CONV(REAL_ARITH `x + a = (y:real) <=> x = y - a`)) in - let rec substitutable_monomial fvs tm = - match tm with - Var(_,Tyapp("real",[])) when not (mem tm fvs) -> Int 1,tm - | Comb(Comb(Const("real_mul",_),c),(Var(_,_) as t)) - when is_ratconst c && not (mem t fvs) - -> rat_of_term c,t - | Comb(Comb(Const("real_add",_),s),t) -> - (try substitutable_monomial (union (frees t) fvs) s - with Failure _ -> substitutable_monomial (union (frees s) fvs) t) - | _ -> failwith "substitutable_monomial" - and isolate_variable v th = - match lhs(concl th) with - x when x = v -> th - | Comb(Comb(Const("real_add",_),(Var(_,Tyapp("real",[])) as x)),t) - when x = v -> shuffle2 th - | Comb(Comb(Const("real_add",_),s),t) -> - isolate_variable v(shuffle1 th) in - let make_substitution th = - let (c,v) = substitutable_monomial [] (lhs(concl th)) in - let th1 = AP_TERM (mk_comb(mul_tm,term_of_rat(Int 1 // c))) th in - let th2 = CONV_RULE(BINOP_CONV REAL_POLY_MUL_CONV) th1 in - CONV_RULE (RAND_CONV REAL_POLY_CONV) (isolate_variable v th2) in - fun translator -> - let rec substfirst(eqs,les,lts) = - try let eth = tryfind make_substitution eqs in - let modify = - CONV_RULE(LAND_CONV(SUBS_CONV[eth] THENC REAL_POLY_CONV)) in - substfirst(filter (fun t -> lhand(concl t) <> zero) (map modify eqs), - map modify les,map modify lts) - with Failure _ -> REAL_NONLINEAR_PROVER translator (eqs,les,lts) in - substfirst;; -*) -(* ------------------------------------------------------------------------- *) -(* Overall function. *) -(* ------------------------------------------------------------------------- *) -(* -let REAL_SOS = - let init = GEN_REWRITE_CONV ONCE_DEPTH_CONV [DECIMAL] - and pure = GEN_REAL_ARITH REAL_NONLINEAR_SUBST_PROVER in - fun tm -> let th = init tm in EQ_MP (SYM th) (pure(rand(concl th)));; -*) -(* ------------------------------------------------------------------------- *) -(* Add hacks for division. *) -(* ------------------------------------------------------------------------- *) -(* -let REAL_SOSFIELD = - let inv_tm = `inv:real->real` in - let prenex_conv = - TOP_DEPTH_CONV BETA_CONV THENC - PURE_REWRITE_CONV[FORALL_SIMP; EXISTS_SIMP; real_div; - REAL_INV_INV; REAL_INV_MUL; GSYM REAL_POW_INV] THENC - NNFC_CONV THENC DEPTH_BINOP_CONV `(/\)` CONDS_CELIM_CONV THENC - PRENEX_CONV - and setup_conv = NNF_CONV THENC WEAK_CNF_CONV THENC CONJ_CANON_CONV - and core_rule t = - try REAL_ARITH t - with Failure _ -> try REAL_RING t - with Failure _ -> REAL_SOS t - and is_inv = - let is_div = is_binop `(/):real->real->real` in - fun tm -> (is_div tm or (is_comb tm && rator tm = inv_tm)) && - not(is_ratconst(rand tm)) in - let BASIC_REAL_FIELD tm = - let is_freeinv t = is_inv t && free_in t tm in - let itms = setify(map rand (find_terms is_freeinv tm)) in - let hyps = map (fun t -> SPEC t REAL_MUL_RINV) itms in - let tm' = itlist (fun th t -> mk_imp(concl th,t)) hyps tm in - let itms' = map (curry mk_comb inv_tm) itms in - let gvs = map (genvar o type_of) itms' in - let tm'' = subst (zip gvs itms') tm' in - let th1 = setup_conv tm'' in - let cjs = conjuncts(rand(concl th1)) in - let ths = map core_rule cjs in - let th2 = EQ_MP (SYM th1) (end_itlist CONJ ths) in - rev_itlist (C MP) hyps (INST (zip itms' gvs) th2) in - fun tm -> - let th0 = prenex_conv tm in - let tm0 = rand(concl th0) in - let avs,bod = strip_forall tm0 in - let th1 = setup_conv bod in - let ths = map BASIC_REAL_FIELD (conjuncts(rand(concl th1))) in - EQ_MP (SYM th0) (GENL avs (EQ_MP (SYM th1) (end_itlist CONJ ths)));; -*) -(* ------------------------------------------------------------------------- *) -(* Integer version. *) -(* ------------------------------------------------------------------------- *) -(* -let INT_SOS = - let atom_CONV = - let pth = prove - (`(~(x <= y) <=> y + &1 <= x:int) /\ - (~(x < y) <=> y <= x) /\ - (~(x = y) <=> x + &1 <= y \/ y + &1 <= x) /\ - (x < y <=> x + &1 <= y)`, - REWRITE_TAC[INT_NOT_LE; INT_NOT_LT; INT_NOT_EQ; INT_LT_DISCRETE]) in - GEN_REWRITE_CONV I [pth] - and bub_CONV = GEN_REWRITE_CONV TOP_SWEEP_CONV - [int_eq; int_le; int_lt; int_ge; int_gt; - int_of_num_th; int_neg_th; int_add_th; int_mul_th; - int_sub_th; int_pow_th; int_abs_th; int_max_th; int_min_th] in - let base_CONV = TRY_CONV atom_CONV THENC bub_CONV in - let NNF_NORM_CONV = GEN_NNF_CONV false - (base_CONV,fun t -> base_CONV t,base_CONV(mk_neg t)) in - let init_CONV = - GEN_REWRITE_CONV DEPTH_CONV [FORALL_SIMP; EXISTS_SIMP] THENC - GEN_REWRITE_CONV DEPTH_CONV [INT_GT; INT_GE] THENC - CONDS_ELIM_CONV THENC NNF_NORM_CONV in - let p_tm = `p:bool` - and not_tm = `(~)` in - let pth = TAUT(mk_eq(mk_neg(mk_neg p_tm),p_tm)) in - fun tm -> - let th0 = INST [tm,p_tm] pth - and th1 = NNF_NORM_CONV(mk_neg tm) in - let th2 = REAL_SOS(mk_neg(rand(concl th1))) in - EQ_MP th0 (EQ_MP (AP_TERM not_tm (SYM th1)) th2);; -*) -(* ------------------------------------------------------------------------- *) -(* Natural number version. *) -(* ------------------------------------------------------------------------- *) -(* -let SOS_RULE tm = - let avs = frees tm in - let tm' = list_mk_forall(avs,tm) in - let th1 = NUM_TO_INT_CONV tm' in - let th2 = INT_SOS (rand(concl th1)) in - SPECL avs (EQ_MP (SYM th1) th2);; -*) -(* ------------------------------------------------------------------------- *) -(* Now pure SOS stuff. *) -(* ------------------------------------------------------------------------- *) - -(*prioritize_real();;*) - (* ------------------------------------------------------------------------- *) (* Some combinatorial helper functions. *) (* ------------------------------------------------------------------------- *) let rec allpermutations l = if l = [] then [[]] else - itlist (fun h acc -> List.map (fun t -> h::t) + List.fold_right (fun h acc -> List.map (fun t -> h::t) (allpermutations (subtract l [h])) @ acc) l [];; let changevariables_monomial zoln (m:monomial) = @@ -1165,14 +975,14 @@ let changevariables zoln pol = let sdpa_of_vector (v:vector) = let n = dim v in let strs = List.map (o (decimalize 20) (element v)) (1--n) in - end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";; + String.concat " " strs ^ "\n";; let sdpa_of_matrix k (m:matrix) = let pfx = string_of_int k ^ " 1 " in let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a) (snd m) [] in let mss = sort (increasing fst) ms in - itlist (fun ((i,j),c) a -> + List.fold_right (fun ((i,j),c) a -> pfx ^ string_of_int i ^ " " ^ string_of_int j ^ " " ^ decimalize 20 c ^ "\n" ^ a) mss "";; @@ -1184,7 +994,7 @@ let sdpa_of_problem comment obj mats = "1\n" ^ string_of_int n ^ "\n" ^ sdpa_of_vector obj ^ - itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a) + List.fold_right2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a) (1--List.length mats) mats "";; let run_csdp dbg obj mats = @@ -1224,9 +1034,9 @@ let sumofsquares_general_symmetry tool pol = let sym_eqs = let invariants = List.filter (fun vars' -> - is_undefined(poly_sub pol (changevariables (zip vars vars') pol))) + is_undefined(poly_sub pol (changevariables (List.combine vars vars') pol))) (allpermutations vars) in - let lpns = zip lpps (1--List.length lpps) in + let lpns = List.combine lpps (1--List.length lpps) in let lppcs = List.filter (fun (m,(n1,n2)) -> n1 <= n2) (allpairs @@ -1234,8 +1044,8 @@ let sumofsquares_general_symmetry tool pol = let clppcs = end_itlist (@) (List.map (fun ((m1,m2),(n1,n2)) -> List.map (fun vars' -> - (changevariables_monomial (zip vars vars') m1, - changevariables_monomial (zip vars vars') m2),(n1,n2)) + (changevariables_monomial (List.combine vars vars') m1, + changevariables_monomial (List.combine vars vars') m2),(n1,n2)) invariants) lppcs) in let clppcs_dom = setify(List.map fst clppcs) in @@ -1247,7 +1057,7 @@ let sumofsquares_general_symmetry tool pol = [] -> raise Sanity | [h] -> acc | h::t -> List.map (fun k -> (k |-> Int(-1)) (h |=> Int 1)) t @ acc in - itlist mk_eq eqvcls [] in + List.fold_right mk_eq eqvcls [] in let eqs = foldl (fun a x y -> y::a) [] (itern 1 lpps (fun m1 n1 -> itern 1 lpps (fun m2 n2 f -> @@ -1259,7 +1069,7 @@ let sumofsquares_general_symmetry tool pol = undefined pol)) @ sym_eqs in let pvs,assig = eliminate_all_equations (0,0) eqs in - let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in + let allassig = List.fold_right (fun v -> (v |-> (v |=> Int 1))) pvs assig in let qvars = (0,0)::pvs in let diagents = end_itlist equation_add (List.map (fun i -> apply allassig (i,i)) (1--n)) in @@ -1281,18 +1091,18 @@ let sumofsquares_general_symmetry tool pol = else ()); let vec = nice_vector d raw_vec in let mat = iter (1,dim vec) - (fun i a -> matrix_add (matrix_cmul (element vec i) (el i mats)) a) - (matrix_neg (el 0 mats)) in + (fun i a -> matrix_add (matrix_cmul (element vec i) (List.nth mats i)) a) + (matrix_neg (List.nth mats 0)) in deration(diag mat) in let rat,dia = if pvs = [] then - let mat = matrix_neg (el 0 mats) in + let mat = matrix_neg (List.nth mats 0) in deration(diag mat) else tryfind find_rounding (List.map Num.num_of_int (1--31) @ List.map pow2 (5--66)) in let poly_of_lin(d,v) = - d,foldl(fun a i c -> (el (i - 1) lpps |-> c) a) undefined (snd v) in + d,foldl(fun a i c -> (List.nth lpps (i - 1) |-> c) a) undefined (snd v) in let lins = List.map poly_of_lin dia in let sqs = List.map (fun (d,l) -> poly_mul (poly_const d) (poly_pow l 2)) lins in let sos = poly_cmul rat (end_itlist poly_add sqs) in @@ -1300,325 +1110,3 @@ let sumofsquares_general_symmetry tool pol = let sumofsquares = sumofsquares_general_symmetry csdp;; -(* ------------------------------------------------------------------------- *) -(* Pure HOL SOS conversion. *) -(* ------------------------------------------------------------------------- *) -(* -let SOS_CONV = - let mk_square = - let pow_tm = `(pow)` and two_tm = `2` in - fun tm -> mk_comb(mk_comb(pow_tm,tm),two_tm) - and mk_prod = mk_binop `( * )` - and mk_sum = mk_binop `(+)` in - fun tm -> - let k,sos = sumofsquares(poly_of_term tm) in - let mk_sqtm(c,p) = - mk_prod (term_of_rat(k */ c)) (mk_square(term_of_poly p)) in - let tm' = end_itlist mk_sum (map mk_sqtm sos) in - let th = REAL_POLY_CONV tm and th' = REAL_POLY_CONV tm' in - TRANS th (SYM th');; -*) -(* ------------------------------------------------------------------------- *) -(* Attempt to prove &0 <= x by direct SOS decomposition. *) -(* ------------------------------------------------------------------------- *) -(* -let PURE_SOS_TAC = - let tac = - MATCH_ACCEPT_TAC(REWRITE_RULE[GSYM REAL_POW_2] REAL_LE_SQUARE) ORELSE - MATCH_ACCEPT_TAC REAL_LE_SQUARE ORELSE - (MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC) ORELSE - (MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) ORELSE - CONV_TAC(RAND_CONV REAL_RAT_REDUCE_CONV THENC REAL_RAT_LE_CONV) in - REPEAT GEN_TAC THEN REWRITE_TAC[real_ge] THEN - GEN_REWRITE_TAC I [GSYM REAL_SUB_LE] THEN - CONV_TAC(RAND_CONV SOS_CONV) THEN - REPEAT tac THEN NO_TAC;; - -let PURE_SOS tm = prove(tm,PURE_SOS_TAC);; -*) -(* ------------------------------------------------------------------------- *) -(* Examples. *) -(* ------------------------------------------------------------------------- *) - -(***** - -time REAL_SOS - `a1 >= &0 /\ a2 >= &0 /\ - (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + &2) /\ - (a1 * b1 + a2 * b2 = &0) - ==> a1 * a2 - b1 * b2 >= &0`;; - -time REAL_SOS `&3 * x + &7 * a < &4 /\ &3 < &2 * x ==> a < &0`;; - -time REAL_SOS - `b pow 2 < &4 * a * c ==> ~(a * x pow 2 + b * x + c = &0)`;; - -time REAL_SOS - `(a * x pow 2 + b * x + c = &0) ==> b pow 2 >= &4 * a * c`;; - -time REAL_SOS - `&0 <= x /\ x <= &1 /\ &0 <= y /\ y <= &1 - ==> x pow 2 + y pow 2 < &1 \/ - (x - &1) pow 2 + y pow 2 < &1 \/ - x pow 2 + (y - &1) pow 2 < &1 \/ - (x - &1) pow 2 + (y - &1) pow 2 < &1`;; - -time REAL_SOS - `&0 <= b /\ &0 <= c /\ &0 <= x /\ &0 <= y /\ - (x pow 2 = c) /\ (y pow 2 = a pow 2 * c + b) - ==> a * c <= y * x`;; - -time REAL_SOS - `&0 <= x /\ &0 <= y /\ &0 <= z /\ x + y + z <= &3 - ==> x * y + x * z + y * z >= &3 * x * y * z`;; - -time REAL_SOS - `(x pow 2 + y pow 2 + z pow 2 = &1) ==> (x + y + z) pow 2 <= &3`;; - -time REAL_SOS - `(w pow 2 + x pow 2 + y pow 2 + z pow 2 = &1) - ==> (w + x + y + z) pow 2 <= &4`;; - -time REAL_SOS - `x >= &1 /\ y >= &1 ==> x * y >= x + y - &1`;; - -time REAL_SOS - `x > &1 /\ y > &1 ==> x * y > x + y - &1`;; - -time REAL_SOS - `abs(x) <= &1 - ==> abs(&64 * x pow 7 - &112 * x pow 5 + &56 * x pow 3 - &7 * x) <= &1`;; - -time REAL_SOS - `abs(x - z) <= e /\ abs(y - z) <= e /\ &0 <= u /\ &0 <= v /\ (u + v = &1) - ==> abs((u * x + v * y) - z) <= e`;; - -(* ------------------------------------------------------------------------- *) -(* One component of denominator in dodecahedral example. *) -(* ------------------------------------------------------------------------- *) - -time REAL_SOS - `&2 <= x /\ x <= &125841 / &50000 /\ - &2 <= y /\ y <= &125841 / &50000 /\ - &2 <= z /\ z <= &125841 / &50000 - ==> &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= &0`;; - -(* ------------------------------------------------------------------------- *) -(* Over a larger but simpler interval. *) -(* ------------------------------------------------------------------------- *) - -time REAL_SOS - `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4 - ==> &0 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;; - -(* ------------------------------------------------------------------------- *) -(* We can do 12. I think 12 is a sharp bound; see PP's certificate. *) -(* ------------------------------------------------------------------------- *) - -time REAL_SOS - `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4 - ==> &12 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;; - -(* ------------------------------------------------------------------------- *) -(* Gloptipoly example. *) -(* ------------------------------------------------------------------------- *) - -(*** This works but normalization takes minutes - -time REAL_SOS - `(x - y - &2 * x pow 4 = &0) /\ &0 <= x /\ x <= &2 /\ &0 <= y /\ y <= &3 - ==> y pow 2 - &7 * y - &12 * x + &17 >= &0`;; - - ***) - -(* ------------------------------------------------------------------------- *) -(* Inequality from sci.math (see "Leon-Sotelo, por favor"). *) -(* ------------------------------------------------------------------------- *) - -time REAL_SOS - `&0 <= x /\ &0 <= y /\ (x * y = &1) - ==> x + y <= x pow 2 + y pow 2`;; - -time REAL_SOS - `&0 <= x /\ &0 <= y /\ (x * y = &1) - ==> x * y * (x + y) <= x pow 2 + y pow 2`;; - -time REAL_SOS - `&0 <= x /\ &0 <= y ==> x * y * (x + y) pow 2 <= (x pow 2 + y pow 2) pow 2`;; - -(* ------------------------------------------------------------------------- *) -(* Some examples over integers and natural numbers. *) -(* ------------------------------------------------------------------------- *) - -time SOS_RULE `!m n. 2 * m + n = (n + m) + m`;; -time SOS_RULE `!n. ~(n = 0) ==> (0 MOD n = 0)`;; -time SOS_RULE `!m n. m < n ==> (m DIV n = 0)`;; -time SOS_RULE `!n:num. n <= n * n`;; -time SOS_RULE `!m n. n * (m DIV n) <= m`;; -time SOS_RULE `!n. ~(n = 0) ==> (0 DIV n = 0)`;; -time SOS_RULE `!m n p. ~(p = 0) /\ m <= n ==> m DIV p <= n DIV p`;; -time SOS_RULE `!a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b)`;; - -(* ------------------------------------------------------------------------- *) -(* This is particularly gratifying --- cf hideous manual proof in arith.ml *) -(* ------------------------------------------------------------------------- *) - -(*** This doesn't now seem to work as well as it did; what changed? - -time SOS_RULE - `!a b c d. ~(b = 0) /\ b * c < (a + 1) * d ==> c DIV d <= a DIV b`;; - - ***) - -(* ------------------------------------------------------------------------- *) -(* Key lemma for injectivity of Cantor-type pairing functions. *) -(* ------------------------------------------------------------------------- *) - -time SOS_RULE - `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1) - ==> (x1 + y1 = x2 + y2)`;; - -time SOS_RULE - `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1) /\ - (x1 + y1 = x2 + y2) - ==> (x1 = x2) /\ (y1 = y2)`;; - -time SOS_RULE - `!x1 y1 x2 y2. - (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 = - ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2) - ==> (x1 + y1 = x2 + y2)`;; - -time SOS_RULE - `!x1 y1 x2 y2. - (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 = - ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2) /\ - (x1 + y1 = x2 + y2) - ==> (x1 = x2) /\ (y1 = y2)`;; - -(* ------------------------------------------------------------------------- *) -(* Reciprocal multiplication (actually just ARITH_RULE does these). *) -(* ------------------------------------------------------------------------- *) - -time SOS_RULE `x <= 127 ==> ((86 * x) DIV 256 = x DIV 3)`;; - -time SOS_RULE `x < 2 EXP 16 ==> ((104858 * x) DIV (2 EXP 20) = x DIV 10)`;; - -(* ------------------------------------------------------------------------- *) -(* This is more impressive since it's really nonlinear. See REMAINDER_DECODE *) -(* ------------------------------------------------------------------------- *) - -time SOS_RULE `0 < m /\ m < n ==> ((m * ((n * x) DIV m + 1)) DIV n = x)`;; - -(* ------------------------------------------------------------------------- *) -(* Some conversion examples. *) -(* ------------------------------------------------------------------------- *) - -time SOS_CONV - `&2 * x pow 4 + &2 * x pow 3 * y - x pow 2 * y pow 2 + &5 * y pow 4`;; - -time SOS_CONV - `x pow 4 - (&2 * y * z + &1) * x pow 2 + - (y pow 2 * z pow 2 + &2 * y * z + &2)`;; - -time SOS_CONV `&4 * x pow 4 + - &4 * x pow 3 * y - &7 * x pow 2 * y pow 2 - &2 * x * y pow 3 + - &10 * y pow 4`;; - -time SOS_CONV `&4 * x pow 4 * y pow 6 + x pow 2 - x * y pow 2 + y pow 2`;; - -time SOS_CONV - `&4096 * (x pow 4 + x pow 2 + z pow 6 - &3 * x pow 2 * z pow 2) + &729`;; - -time SOS_CONV - `&120 * x pow 2 - &63 * x pow 4 + &10 * x pow 6 + - &30 * x * y - &120 * y pow 2 + &120 * y pow 4 + &31`;; - -time SOS_CONV - `&9 * x pow 2 * y pow 4 + &9 * x pow 2 * z pow 4 + &36 * x pow 2 * y pow 3 + - &36 * x pow 2 * y pow 2 - &48 * x * y * z pow 2 + &4 * y pow 4 + - &4 * z pow 4 - &16 * y pow 3 + &16 * y pow 2`;; - -time SOS_CONV - `(x pow 2 + y pow 2 + z pow 2) * - (x pow 4 * y pow 2 + x pow 2 * y pow 4 + - z pow 6 - &3 * x pow 2 * y pow 2 * z pow 2)`;; - -time SOS_CONV - `x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3`;; - -(*** I think this will work, but normalization is slow - -time SOS_CONV - `&100 * (x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z) + &212`;; - - ***) - -time SOS_CONV - `&100 * ((&2 * x - &2) pow 2 + (x pow 3 - &8 * x - &2) pow 2) - &588`;; - -time SOS_CONV - `x pow 2 * (&120 - &63 * x pow 2 + &10 * x pow 4) + &30 * x * y + - &30 * y pow 2 * (&4 * y pow 2 - &4) + &31`;; - -(* ------------------------------------------------------------------------- *) -(* Example of basic rule. *) -(* ------------------------------------------------------------------------- *) - -time PURE_SOS - `!x. x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3 - >= &1 / &7`;; - -time PURE_SOS - `&0 <= &98 * x pow 12 + - -- &980 * x pow 10 + - &3038 * x pow 8 + - -- &2968 * x pow 6 + - &1022 * x pow 4 + - -- &84 * x pow 2 + - &2`;; - -time PURE_SOS - `!x. &0 <= &2 * x pow 14 + - -- &84 * x pow 12 + - &1022 * x pow 10 + - -- &2968 * x pow 8 + - &3038 * x pow 6 + - -- &980 * x pow 4 + - &98 * x pow 2`;; - -(* ------------------------------------------------------------------------- *) -(* From Zeng et al, JSC vol 37 (2004), p83-99. *) -(* All of them work nicely with pure SOS_CONV, except (maybe) the one noted. *) -(* ------------------------------------------------------------------------- *) - -PURE_SOS - `x pow 6 + y pow 6 + z pow 6 - &3 * x pow 2 * y pow 2 * z pow 2 >= &0`;; - -PURE_SOS `x pow 4 + y pow 4 + z pow 4 + &1 - &4*x*y*z >= &0`;; - -PURE_SOS `x pow 4 + &2*x pow 2*z + x pow 2 - &2*x*y*z + &2*y pow 2*z pow 2 + -&2*y*z pow 2 + &2*z pow 2 - &2*x + &2* y*z + &1 >= &0`;; - -(**** This is harder. Interestingly, this fails the pure SOS test, it seems. - Yet only on rounding(!?) Poor Newton polytope optimization or something? - But REAL_SOS does finally converge on the second run at level 12! - -REAL_SOS -`x pow 4*y pow 4 - &2*x pow 5*y pow 3*z pow 2 + x pow 6*y pow 2*z pow 4 + &2*x -pow 2*y pow 3*z - &4* x pow 3*y pow 2*z pow 3 + &2*x pow 4*y*z pow 5 + z pow -2*y pow 2 - &2*z pow 4*y*x + z pow 6*x pow 2 >= &0`;; - - ****) - -PURE_SOS -`x pow 4 + &4*x pow 2*y pow 2 + &2*x*y*z pow 2 + &2*x*y*w pow 2 + y pow 4 + z -pow 4 + w pow 4 + &2*z pow 2*w pow 2 + &2*x pow 2*w + &2*y pow 2*w + &2*x*y + -&3*w pow 2 + &2*z pow 2 + &1 >= &0`;; - -PURE_SOS -`w pow 6 + &2*z pow 2*w pow 3 + x pow 4 + y pow 4 + z pow 4 + &2*x pow 2*w + -&2*x pow 2*z + &3*x pow 2 + w pow 2 + &2*z*w + z pow 2 + &2*z + &2*w + &1 >= -&0`;; - -*****) -- cgit v1.2.3