--- a/src/HOL/Library/normarith.ML Sat May 15 17:59:06 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,415 +0,0 @@
-(* 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;