--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/normarith.ML Sat May 15 18:11:00 2010 +0200
@@ -0,0 +1,415 @@
+(* Title: Library/normarith.ML
+ Author: Amine Chaieb, University of Cambridge
+ Description: A simple decision procedure for linear problems in euclidean space
+*)
+
+ (* Now the norm procedure for euclidean spaces *)
+
+
+signature NORM_ARITH =
+sig
+ val norm_arith : Proof.context -> conv
+ val norm_arith_tac : Proof.context -> int -> tactic
+end
+
+structure NormArith : NORM_ARITH =
+struct
+
+ open Conv;
+ val bool_eq = op = : bool *bool -> bool
+ fun dest_ratconst t = case term_of t of
+ Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+ | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
+ | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
+ fun is_ratconst t = can dest_ratconst t
+ fun augment_norm b t acc = case term_of t of
+ Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,Thm.dest_arg t) acc
+ | _ => acc
+ fun find_normedterms t acc = case term_of t of
+ @{term "op + :: real => _"}$_$_ =>
+ find_normedterms (Thm.dest_arg1 t) (find_normedterms (Thm.dest_arg t) acc)
+ | @{term "op * :: real => _"}$_$n =>
+ if not (is_ratconst (Thm.dest_arg1 t)) then acc else
+ augment_norm (dest_ratconst (Thm.dest_arg1 t) >=/ Rat.zero)
+ (Thm.dest_arg t) acc
+ | _ => augment_norm true t acc
+
+ val cterm_lincomb_neg = FuncUtil.Ctermfunc.map Rat.neg
+ fun cterm_lincomb_cmul c t =
+ if c =/ Rat.zero then FuncUtil.Ctermfunc.empty else FuncUtil.Ctermfunc.map (fn x => x */ c) t
+ fun cterm_lincomb_add l r = FuncUtil.Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
+ fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
+ fun cterm_lincomb_eq l r = FuncUtil.Ctermfunc.is_empty (cterm_lincomb_sub l r)
+
+ val int_lincomb_neg = FuncUtil.Intfunc.map Rat.neg
+ fun int_lincomb_cmul c t =
+ if c =/ Rat.zero then FuncUtil.Intfunc.empty else FuncUtil.Intfunc.map (fn x => x */ c) t
+ fun int_lincomb_add l r = FuncUtil.Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
+ fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
+ fun int_lincomb_eq l r = FuncUtil.Intfunc.is_empty (int_lincomb_sub l r)
+
+fun vector_lincomb t = case term_of t of
+ Const(@{const_name plus}, _) $ _ $ _ =>
+ cterm_lincomb_add (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name minus}, _) $ _ $ _ =>
+ cterm_lincomb_sub (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name scaleR}, _)$_$_ =>
+ cterm_lincomb_cmul (dest_ratconst (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
+ | Const(@{const_name uminus}, _)$_ =>
+ cterm_lincomb_neg (vector_lincomb (Thm.dest_arg t))
+(* FIXME: how should we handle numerals?
+ | Const(@ {const_name vec},_)$_ =>
+ let
+ val b = ((snd o HOLogic.dest_number o term_of o Thm.dest_arg) t = 0
+ handle TERM _=> false)
+ in if b then FuncUtil.Ctermfunc.onefunc (t,Rat.one)
+ else FuncUtil.Ctermfunc.empty
+ end
+*)
+ | _ => FuncUtil.Ctermfunc.onefunc (t,Rat.one)
+
+ fun vector_lincombs ts =
+ fold_rev
+ (fn t => fn fns => case AList.lookup (op aconvc) fns t of
+ NONE =>
+ let val f = vector_lincomb t
+ in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
+ SOME (_,f') => (t,f') :: fns
+ | NONE => (t,f) :: fns
+ end
+ | SOME _ => fns) ts []
+
+fun replacenegnorms cv t = case term_of t of
+ @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
+| @{term "op * :: real => _"}$_$_ =>
+ if dest_ratconst (Thm.dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t
+| _ => reflexive t
+fun flip v eq =
+ if FuncUtil.Ctermfunc.defined eq v
+ then FuncUtil.Ctermfunc.update (v, Rat.neg (FuncUtil.Ctermfunc.apply eq v)) eq else eq
+fun allsubsets s = case s of
+ [] => [[]]
+|(a::t) => let val res = allsubsets t in
+ map (cons a) res @ res end
+fun evaluate env lin =
+ FuncUtil.Intfunc.fold (fn (x,c) => fn s => s +/ c */ (FuncUtil.Intfunc.apply env x))
+ lin Rat.zero
+
+fun solve (vs,eqs) = case (vs,eqs) of
+ ([],[]) => SOME (FuncUtil.Intfunc.onefunc (0,Rat.one))
+ |(_,eq::oeqs) =>
+ (case filter (member (op =) vs) (FuncUtil.Intfunc.dom eq) of (*FIXME use find_first here*)
+ [] => NONE
+ | v::_ =>
+ if FuncUtil.Intfunc.defined eq v
+ then
+ let
+ val c = FuncUtil.Intfunc.apply eq v
+ val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq
+ fun eliminate eqn = if not (FuncUtil.Intfunc.defined eqn v) then eqn
+ else int_lincomb_add (int_lincomb_cmul (FuncUtil.Intfunc.apply eqn v) vdef) eqn
+ in (case solve (remove (op =) v vs, map eliminate oeqs) of
+ NONE => NONE
+ | SOME soln => SOME (FuncUtil.Intfunc.update (v, evaluate soln (FuncUtil.Intfunc.delete_safe v vdef)) soln))
+ end
+ else NONE)
+
+fun combinations k l = if k = 0 then [[]] else
+ case l of
+ [] => []
+| h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
+
+
+fun forall2 p l1 l2 = case (l1,l2) of
+ ([],[]) => true
+ | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
+ | _ => false;
+
+
+fun vertices vs eqs =
+ let
+ fun vertex cmb = case solve(vs,cmb) of
+ NONE => NONE
+ | SOME soln => SOME (map (fn v => FuncUtil.Intfunc.tryapplyd soln v Rat.zero) vs)
+ val rawvs = map_filter vertex (combinations (length vs) eqs)
+ val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs
+ in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset []
+ end
+
+fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m
+
+fun subsume todo dun = case todo of
+ [] => dun
+|v::ovs =>
+ let val dun' = if exists (fn w => subsumes w v) dun then dun
+ else v::(filter (fn w => not(subsumes v w)) dun)
+ in subsume ovs dun'
+ end;
+
+fun match_mp PQ P = P RS PQ;
+
+fun cterm_of_rat x =
+let val (a, b) = Rat.quotient_of_rat x
+in
+ if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
+ else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
+ (Numeral.mk_cnumber @{ctyp "real"} a))
+ (Numeral.mk_cnumber @{ctyp "real"} b)
+end;
+
+fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
+
+fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm};
+
+ (* I think here the static context should be sufficient!! *)
+fun inequality_canon_rule ctxt =
+ let
+ (* FIXME : Should be computed statically!! *)
+ val real_poly_conv =
+ Semiring_Normalizer.semiring_normalize_wrapper ctxt
+ (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+ in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Numeral_Simprocs.field_comp_conv then_conv real_poly_conv)))
+end;
+
+ fun absc cv ct = case term_of ct of
+ Abs (v,_, _) =>
+ let val (x,t) = Thm.dest_abs (SOME v) ct
+ in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
+ end
+ | _ => all_conv ct;
+
+fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct;
+fun botc1 conv ct =
+ ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct;
+
+ val apply_pth1 = rewr_conv @{thm pth_1};
+ val apply_pth2 = rewr_conv @{thm pth_2};
+ val apply_pth3 = rewr_conv @{thm pth_3};
+ val apply_pth4 = rewrs_conv @{thms pth_4};
+ val apply_pth5 = rewr_conv @{thm pth_5};
+ val apply_pth6 = rewr_conv @{thm pth_6};
+ val apply_pth7 = rewrs_conv @{thms pth_7};
+ val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Numeral_Simprocs.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
+ val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Numeral_Simprocs.field_comp_conv);
+ val apply_ptha = rewr_conv @{thm pth_a};
+ val apply_pthb = rewrs_conv @{thms pth_b};
+ val apply_pthc = rewrs_conv @{thms pth_c};
+ val apply_pthd = try_conv (rewr_conv @{thm pth_d});
+
+fun headvector t = case t of
+ Const(@{const_name plus}, _)$
+ (Const(@{const_name scaleR}, _)$l$v)$r => v
+ | Const(@{const_name scaleR}, _)$l$v => v
+ | _ => error "headvector: non-canonical term"
+
+fun vector_cmul_conv ct =
+ ((apply_pth5 then_conv arg1_conv Numeral_Simprocs.field_comp_conv) else_conv
+ (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
+
+fun vector_add_conv ct = apply_pth7 ct
+ handle CTERM _ =>
+ (apply_pth8 ct
+ handle CTERM _ =>
+ (case term_of ct of
+ Const(@{const_name plus},_)$lt$rt =>
+ let
+ val l = headvector lt
+ val r = headvector rt
+ in (case Term_Ord.fast_term_ord (l,r) of
+ LESS => (apply_pthb then_conv arg_conv vector_add_conv
+ then_conv apply_pthd) ct
+ | GREATER => (apply_pthc then_conv arg_conv vector_add_conv
+ then_conv apply_pthd) ct
+ | EQUAL => (apply_pth9 then_conv
+ ((apply_ptha then_conv vector_add_conv) else_conv
+ arg_conv vector_add_conv then_conv apply_pthd)) ct)
+ end
+ | _ => reflexive ct))
+
+fun vector_canon_conv ct = case term_of ct of
+ Const(@{const_name plus},_)$_$_ =>
+ let
+ val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
+ val lth = vector_canon_conv l
+ val rth = vector_canon_conv r
+ val th = Drule.binop_cong_rule p lth rth
+ in fconv_rule (arg_conv vector_add_conv) th end
+
+| Const(@{const_name scaleR}, _)$_$_ =>
+ let
+ val (p,r) = Thm.dest_comb ct
+ val rth = Drule.arg_cong_rule p (vector_canon_conv r)
+ in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth
+ end
+
+| Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct
+
+| Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct
+
+(* FIXME
+| Const(@{const_name vec},_)$n =>
+ let val n = Thm.dest_arg ct
+ in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero)
+ then reflexive ct else apply_pth1 ct
+ end
+*)
+| _ => apply_pth1 ct
+
+fun norm_canon_conv ct = case term_of ct of
+ Const(@{const_name norm},_)$_ => arg_conv vector_canon_conv ct
+ | _ => raise CTERM ("norm_canon_conv", [ct])
+
+fun fold_rev2 f [] [] z = z
+ | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z)
+ | fold_rev2 f _ _ _ = raise UnequalLengths;
+
+fun int_flip v eq =
+ if FuncUtil.Intfunc.defined eq v
+ then FuncUtil.Intfunc.update (v, Rat.neg (FuncUtil.Intfunc.apply eq v)) eq else eq;
+
+local
+ val pth_zero = @{thm norm_zero}
+ val tv_n = (ctyp_of_term o Thm.dest_arg o Thm.dest_arg1 o Thm.dest_arg o cprop_of)
+ pth_zero
+ val concl = Thm.dest_arg o cprop_of
+ fun real_vector_combo_prover ctxt translator (nubs,ges,gts) =
+ let
+ (* FIXME: Should be computed statically!!*)
+ val real_poly_conv =
+ Semiring_Normalizer.semiring_normalize_wrapper ctxt
+ (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+ val sources = map (Thm.dest_arg o Thm.dest_arg1 o concl) nubs
+ val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) []
+ val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
+ else ()
+ val dests = distinct (op aconvc) (map snd rawdests)
+ val srcfuns = map vector_lincomb sources
+ val destfuns = map vector_lincomb dests
+ val vvs = fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) (srcfuns @ destfuns) []
+ val n = length srcfuns
+ val nvs = 1 upto n
+ val srccombs = srcfuns ~~ nvs
+ fun consider d =
+ let
+ fun coefficients x =
+ let
+ val inp = if FuncUtil.Ctermfunc.defined d x then FuncUtil.Intfunc.onefunc (0, Rat.neg(FuncUtil.Ctermfunc.apply d x))
+ else FuncUtil.Intfunc.empty
+ in fold_rev (fn (f,v) => fn g => if FuncUtil.Ctermfunc.defined f x then FuncUtil.Intfunc.update (v, FuncUtil.Ctermfunc.apply f x) g else g) srccombs inp
+ end
+ val equations = map coefficients vvs
+ val inequalities = map (fn n => FuncUtil.Intfunc.onefunc (n,Rat.one)) nvs
+ fun plausiblevertices f =
+ let
+ val flippedequations = map (fold_rev int_flip f) equations
+ val constraints = flippedequations @ inequalities
+ val rawverts = vertices nvs constraints
+ fun check_solution v =
+ let
+ val f = fold_rev2 (curry FuncUtil.Intfunc.update) nvs v (FuncUtil.Intfunc.onefunc (0, Rat.one))
+ in forall (fn e => evaluate f e =/ Rat.zero) flippedequations
+ end
+ val goodverts = filter check_solution rawverts
+ val signfixups = map (fn n => if member (op =) f n then ~1 else 1) nvs
+ in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts
+ end
+ val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) []
+ in subsume allverts []
+ end
+ fun compute_ineq v =
+ let
+ val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE
+ else SOME(norm_cmul_rule v t))
+ (v ~~ nubs)
+ fun end_itlist f xs = split_last xs |> uncurry (fold_rev f)
+ in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
+ end
+ val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
+ map (inequality_canon_rule ctxt) nubs @ ges
+ val zerodests = filter
+ (fn t => null (FuncUtil.Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)
+
+ in fst (RealArith.real_linear_prover translator
+ (map (fn t => instantiate ([(tv_n, ctyp_of_term t)],[]) pth_zero)
+ zerodests,
+ map (fconv_rule (try_conv (Conv.top_sweep_conv (K norm_canon_conv) ctxt) then_conv
+ arg_conv (arg_conv real_poly_conv))) ges',
+ map (fconv_rule (try_conv (Conv.top_sweep_conv (K norm_canon_conv) ctxt) then_conv
+ arg_conv (arg_conv real_poly_conv))) gts))
+ end
+in val real_vector_combo_prover = real_vector_combo_prover
+end;
+
+local
+ val pth = @{thm norm_imp_pos_and_ge}
+ val norm_mp = match_mp pth
+ val concl = Thm.dest_arg o cprop_of
+ fun conjunct1 th = th RS @{thm conjunct1}
+ fun conjunct2 th = th RS @{thm conjunct2}
+fun real_vector_ineq_prover ctxt translator (ges,gts) =
+ let
+(* val _ = error "real_vector_ineq_prover: pause" *)
+ val ntms = fold_rev find_normedterms (map (Thm.dest_arg o concl) (ges @ gts)) []
+ val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
+ val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
+ fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
+ fun mk_norm t = Thm.capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: real_normed_vector) => real"}) t
+ fun mk_equals l r = Thm.capply (Thm.capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r
+ val asl = map2 (fn (t,_) => fn n => assume (mk_equals (mk_norm t) (cterm_of (ProofContext.theory_of ctxt') (Free(n,@{typ real}))))) lctab fxns
+ val replace_conv = try_conv (rewrs_conv asl)
+ val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
+ val ges' =
+ fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
+ asl (map replace_rule ges)
+ val gts' = map replace_rule gts
+ val nubs = map (conjunct2 o norm_mp) asl
+ val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
+ val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1))
+ val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1])
+ val cps = map (swap o Thm.dest_equals) (cprems_of th11)
+ val th12 = instantiate ([], cps) th11
+ val th13 = fold Thm.elim_implies (map (reflexive o snd) cps) th12;
+ in hd (Variable.export ctxt' ctxt [th13])
+ end
+in val real_vector_ineq_prover = real_vector_ineq_prover
+end;
+
+local
+ val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
+ fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
+ fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS;
+ (* FIXME: Lookup in the context every time!!! Fix this !!!*)
+ fun splitequation ctxt th acc =
+ let
+ val real_poly_neg_conv = #neg
+ (Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+ (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
+ val (th1,th2) = conj_pair(rawrule th)
+ in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc
+ end
+in fun real_vector_prover ctxt _ translator (eqs,ges,gts) =
+ (real_vector_ineq_prover ctxt translator
+ (fold_rev (splitequation ctxt) eqs ges,gts), RealArith.Trivial)
+end;
+
+ fun init_conv ctxt =
+ Simplifier.rewrite (Simplifier.context ctxt
+ (HOL_basic_ss addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
+ then_conv Numeral_Simprocs.field_comp_conv
+ then_conv nnf_conv
+
+ fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
+ fun norm_arith ctxt ct =
+ let
+ val ctxt' = Variable.declare_term (term_of ct) ctxt
+ val th = init_conv ctxt' ct
+ in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th))
+ (pure ctxt' (Thm.rhs_of th))
+ end
+
+ fun norm_arith_tac ctxt =
+ clarify_tac HOL_cs THEN'
+ Object_Logic.full_atomize_tac THEN'
+ CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i);
+
+end;