--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/lin_arith.ML Tue Jul 31 19:40:23 2007 +0200
@@ -0,0 +1,897 @@
+(* Title: HOL/Tools/lin_arith.ML
+ ID: $Id$
+ Author: Tjark Weber and Tobias Nipkow
+
+HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).
+*)
+
+signature BASIC_LIN_ARITH =
+sig
+ type arith_tactic
+ val mk_arith_tactic: string -> (Proof.context -> int -> tactic) -> arith_tactic
+ val eq_arith_tactic: arith_tactic * arith_tactic -> bool
+ val arith_split_add: attribute
+ val arith_discrete: string -> Context.generic -> Context.generic
+ val arith_inj_const: string * typ -> Context.generic -> Context.generic
+ val arith_tactic_add: arith_tactic -> Context.generic -> Context.generic
+ val fast_arith_split_limit: int ConfigOption.T
+ val fast_arith_neq_limit: int ConfigOption.T
+ val lin_arith_pre_tac: Proof.context -> int -> tactic
+ val fast_arith_tac: Proof.context -> int -> tactic
+ val fast_ex_arith_tac: Proof.context -> bool -> int -> tactic
+ val trace_arith: bool ref
+ val lin_arith_simproc: simpset -> term -> thm option
+ val fast_nat_arith_simproc: simproc
+ val simple_arith_tac: Proof.context -> int -> tactic
+ val arith_tac: Proof.context -> int -> tactic
+ val silent_arith_tac: Proof.context -> int -> tactic
+end;
+
+signature LIN_ARITH =
+sig
+ include BASIC_LIN_ARITH
+ val map_data:
+ ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
+ lessD: thm list, neqE: thm list, simpset: Simplifier.simpset} ->
+ {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
+ lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}) ->
+ Context.generic -> Context.generic
+ val setup: Context.generic -> Context.generic
+end;
+
+structure LinArith: LIN_ARITH =
+struct
+
+(* Parameters data for general linear arithmetic functor *)
+
+structure LA_Logic: LIN_ARITH_LOGIC =
+struct
+
+val ccontr = ccontr;
+val conjI = conjI;
+val notI = notI;
+val sym = sym;
+val not_lessD = @{thm linorder_not_less} RS iffD1;
+val not_leD = @{thm linorder_not_le} RS iffD1;
+val le0 = thm "le0";
+
+fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
+
+val mk_Trueprop = HOLogic.mk_Trueprop;
+
+fun atomize thm = case Thm.prop_of thm of
+ Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
+ atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
+ | _ => [thm];
+
+fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
+ | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
+ | neg_prop t = raise TERM ("neg_prop", [t]);
+
+fun is_False thm =
+ let val _ $ t = Thm.prop_of thm
+ in t = Const("False",HOLogic.boolT) end;
+
+fun is_nat(t) = fastype_of1 t = HOLogic.natT;
+
+fun mk_nat_thm sg t =
+ let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
+ in instantiate ([],[(cn,ct)]) le0 end;
+
+end;
+
+
+(* arith context data *)
+
+datatype arith_tactic =
+ ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
+
+fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
+
+fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
+
+structure ArithContextData = GenericDataFun
+(
+ type T = {splits: thm list,
+ inj_consts: (string * typ) list,
+ discrete: string list,
+ tactics: arith_tactic list};
+ val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
+ val extend = I;
+ fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
+ {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
+ {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
+ inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
+ discrete = Library.merge (op =) (discrete1, discrete2),
+ tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
+);
+
+val get_arith_data = ArithContextData.get o Context.Proof;
+
+val arith_split_add = Thm.declaration_attribute (fn thm =>
+ ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
+ {splits = insert Thm.eq_thm_prop thm splits,
+ inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
+
+fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
+ {splits = splits, inj_consts = inj_consts,
+ discrete = insert (op =) d discrete, tactics = tactics});
+
+fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
+ {splits = splits, inj_consts = insert (op =) c inj_consts,
+ discrete = discrete, tactics= tactics});
+
+fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
+ {splits = splits, inj_consts = inj_consts, discrete = discrete,
+ tactics = insert eq_arith_tactic tac tactics});
+
+
+val (fast_arith_split_limit, setup1) = ConfigOption.int "fast_arith_split_limit" 9;
+val (fast_arith_neq_limit, setup2) = ConfigOption.int "fast_arith_neq_limit" 9;
+val setup_options = setup1 #> setup2;
+
+
+structure LA_Data_Ref =
+struct
+
+val fast_arith_neq_limit = fast_arith_neq_limit;
+
+
+(* Decomposition of terms *)
+
+(*internal representation of linear (in-)equations*)
+type decompT = ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
+
+fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
+ | nT _ = false;
+
+fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
+ (term * Rat.rat) list * Rat.rat =
+ case AList.lookup (op =) p t of NONE => ((t, m) :: p, i)
+ | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
+
+exception Zero;
+
+fun rat_of_term (numt, dent) =
+ let
+ val num = HOLogic.dest_numeral numt
+ val den = HOLogic.dest_numeral dent
+ in
+ if den = 0 then raise Zero else Rat.rat_of_quotient (num, den)
+ end;
+
+(*Warning: in rare cases number_of encloses a non-numeral,
+ in which case dest_numeral raises TERM; hence all the handles below.
+ Same for Suc-terms that turn out not to be numerals -
+ although the simplifier should eliminate those anyway ...*)
+fun number_of_Sucs (Const ("Suc", _) $ n) : int =
+ number_of_Sucs n + 1
+ | number_of_Sucs t =
+ if HOLogic.is_zero t then 0 else raise TERM ("number_of_Sucs", []);
+
+(*decompose nested multiplications, bracketing them to the right and combining
+ all their coefficients*)
+fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
+let
+ fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) = (
+ (case s of
+ Const ("Numeral.number_class.number_of", _) $ n =>
+ demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
+ | Const (@{const_name HOL.uminus}, _) $ (Const ("Numeral.number_class.number_of", _) $ n) =>
+ demult (t, Rat.mult m (Rat.rat_of_int (~(HOLogic.dest_numeral n))))
+ | Const (@{const_name Suc}, _) $ _ =>
+ demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat s)))
+ | Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
+ demult (mC $ s1 $ (mC $ s2 $ t), m)
+ | Const (@{const_name HOL.divide}, _) $ numt $
+ (Const ("Numeral.number_class.number_of", _) $ dent) =>
+ let
+ val den = HOLogic.dest_numeral dent
+ in
+ if den = 0 then
+ raise Zero
+ else
+ demult (mC $ numt $ t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
+ end
+ | _ =>
+ atomult (mC, s, t, m)
+ ) handle TERM _ => atomult (mC, s, t, m)
+ )
+ | demult (atom as Const(@{const_name HOL.divide}, _) $ t $
+ (Const ("Numeral.number_class.number_of", _) $ dent), m) =
+ (let
+ val den = HOLogic.dest_numeral dent
+ in
+ if den = 0 then
+ raise Zero
+ else
+ demult (t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
+ end handle TERM _ => (SOME atom, m))
+ | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
+ | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
+ | demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) =
+ ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
+ handle TERM _ => (SOME t, m))
+ | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
+ | demult (t as Const f $ x, m) =
+ (if member (op =) inj_consts f then SOME x else SOME t, m)
+ | demult (atom, m) = (SOME atom, m)
+and
+ atomult (mC, atom, t, m) = (
+ case demult (t, m) of (NONE, m') => (SOME atom, m')
+ | (SOME t', m') => (SOME (mC $ atom $ t'), m')
+ )
+in demult end;
+
+fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
+ ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
+let
+ (* Turn term into list of summand * multiplicity plus a constant *)
+ fun poly (Const (@{const_name HOL.plus}, _) $ s $ t,
+ m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))
+ | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
+ if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
+ | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
+ if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
+ | poly (Const (@{const_name HOL.zero}, _), _, pi) =
+ pi
+ | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
+ (p, Rat.add i m)
+ | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
+ poly (t, m, (p, Rat.add i m))
+ | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
+ (case demult inj_consts (all, m) of
+ (NONE, m') => (p, Rat.add i m')
+ | (SOME u, m') => add_atom u m' pi)
+ | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
+ (case demult inj_consts (all, m) of
+ (NONE, m') => (p, Rat.add i m')
+ | (SOME u, m') => add_atom u m' pi)
+ | poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
+ (let val k = HOLogic.dest_numeral t
+ val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
+ in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
+ handle TERM _ => add_atom all m pi)
+ | poly (all as Const f $ x, m, pi) =
+ if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
+ | poly (all, m, pi) =
+ add_atom all m pi
+ val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
+ val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
+in
+ case rel of
+ @{const_name HOL.less} => SOME (p, i, "<", q, j)
+ | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
+ | "op =" => SOME (p, i, "=", q, j)
+ | _ => NONE
+end handle Zero => NONE;
+
+fun of_lin_arith_sort sg (U : typ) : bool =
+ Type.of_sort (Sign.tsig_of sg) (U, ["Ring_and_Field.ordered_idom"])
+
+fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
+ if of_lin_arith_sort sg U then
+ (true, D mem discrete)
+ else (* special cases *)
+ if D mem discrete then (true, true) else (false, false)
+ | allows_lin_arith sg discrete U =
+ (of_lin_arith_sort sg U, false);
+
+fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decompT option =
+ case T of
+ Type ("fun", [U, _]) =>
+ (case allows_lin_arith thy discrete U of
+ (true, d) =>
+ (case decomp0 inj_consts xxx of
+ NONE => NONE
+ | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
+ | (false, _) =>
+ NONE)
+ | _ => NONE;
+
+fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
+ | negate NONE = NONE;
+
+fun decomp_negation data
+ ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option =
+ decomp_typecheck data (T, (rel, lhs, rhs))
+ | decomp_negation data ((Const ("Trueprop", _)) $
+ (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
+ negate (decomp_typecheck data (T, (rel, lhs, rhs)))
+ | decomp_negation data _ =
+ NONE;
+
+fun decomp ctxt : term -> decompT option =
+ let
+ val thy = ProofContext.theory_of ctxt
+ val {discrete, inj_consts, ...} = get_arith_data ctxt
+ in decomp_negation (thy, discrete, inj_consts) end;
+
+fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T
+ | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
+ | domain_is_nat _ = false;
+
+fun number_of (n, T) = HOLogic.mk_number T n;
+
+(*---------------------------------------------------------------------------*)
+(* the following code performs splitting of certain constants (e.g. min, *)
+(* max) in a linear arithmetic problem; similar to what split_tac later does *)
+(* to the proof state *)
+(*---------------------------------------------------------------------------*)
+
+(* checks if splitting with 'thm' is implemented *)
+
+fun is_split_thm (thm : thm) : bool =
+ case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
+ (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
+ case head_of lhs of
+ Const (a, _) => member (op =) [@{const_name Orderings.max},
+ @{const_name Orderings.min},
+ @{const_name HOL.abs},
+ @{const_name HOL.minus},
+ "IntDef.nat",
+ "Divides.div_class.mod",
+ "Divides.div_class.div"] a
+ | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
+ Display.string_of_thm thm);
+ false))
+ | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
+ Display.string_of_thm thm);
+ false);
+
+(* substitute new for occurrences of old in a term, incrementing bound *)
+(* variables as needed when substituting inside an abstraction *)
+
+fun subst_term ([] : (term * term) list) (t : term) = t
+ | subst_term pairs t =
+ (case AList.lookup (op aconv) pairs t of
+ SOME new =>
+ new
+ | NONE =>
+ (case t of Abs (a, T, body) =>
+ let val pairs' = map (pairself (incr_boundvars 1)) pairs
+ in Abs (a, T, subst_term pairs' body) end
+ | t1 $ t2 =>
+ subst_term pairs t1 $ subst_term pairs t2
+ | _ => t));
+
+(* approximates the effect of one application of split_tac (followed by NNF *)
+(* normalization) on the subgoal represented by '(Ts, terms)'; returns a *)
+(* list of new subgoals (each again represented by a typ list for bound *)
+(* variables and a term list for premises), or NONE if split_tac would fail *)
+(* on the subgoal *)
+
+(* FIXME: currently only the effect of certain split theorems is reproduced *)
+(* (which is why we need 'is_split_thm'). A more canonical *)
+(* implementation should analyze the right-hand side of the split *)
+(* theorem that can be applied, and modify the subgoal accordingly. *)
+(* Or even better, the splitter should be extended to provide *)
+(* splitting on terms as well as splitting on theorems (where the *)
+(* former can have a faster implementation as it does not need to be *)
+(* proof-producing). *)
+
+fun split_once_items ctxt (Ts : typ list, terms : term list) :
+ (typ list * term list) list option =
+let
+ val thy = ProofContext.theory_of ctxt
+ (* takes a list [t1, ..., tn] to the term *)
+ (* tn' --> ... --> t1' --> False , *)
+ (* where ti' = HOLogic.dest_Trueprop ti *)
+ fun REPEAT_DETERM_etac_rev_mp terms' =
+ fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
+ val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
+ val cmap = Splitter.cmap_of_split_thms split_thms
+ val splits = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
+ val split_limit = ConfigOption.get ctxt fast_arith_split_limit
+in
+ if length splits > split_limit then
+ (tracing ("fast_arith_split_limit exceeded (current value is " ^
+ string_of_int split_limit ^ ")"); NONE)
+ else (
+ case splits of [] =>
+ (* split_tac would fail: no possible split *)
+ NONE
+ | ((_, _, _, split_type, split_term) :: _) => (
+ (* ignore all but the first possible split *)
+ case strip_comb split_term of
+ (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
+ (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
+ let
+ val rev_terms = rev terms
+ val terms1 = map (subst_term [(split_term, t1)]) rev_terms
+ val terms2 = map (subst_term [(split_term, t2)]) rev_terms
+ val t1_leq_t2 = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
+ val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (Ts, subgoal2)]
+ end
+ (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
+ | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
+ let
+ val rev_terms = rev terms
+ val terms1 = map (subst_term [(split_term, t1)]) rev_terms
+ val terms2 = map (subst_term [(split_term, t2)]) rev_terms
+ val t1_leq_t2 = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
+ val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (Ts, subgoal2)]
+ end
+ (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
+ | (Const (@{const_name HOL.abs}, _), [t1]) =>
+ let
+ val rev_terms = rev terms
+ val terms1 = map (subst_term [(split_term, t1)]) rev_terms
+ val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
+ split_type --> split_type) $ t1)]) rev_terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val zero_leq_t1 = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ zero $ t1
+ val t1_lt_zero = Const (@{const_name HOL.less},
+ split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (Ts, subgoal2)]
+ end
+ (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
+ | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
+ let
+ (* "d" in the above theorem becomes a new bound variable after NNF *)
+ (* transformation, therefore some adjustment of indices is necessary *)
+ val rev_terms = rev terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val d = Bound 0
+ val terms1 = map (subst_term [(split_term, zero)]) rev_terms
+ val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)])
+ (map (incr_boundvars 1) rev_terms)
+ val t1' = incr_boundvars 1 t1
+ val t2' = incr_boundvars 1 t2
+ val t1_lt_t2 = Const (@{const_name HOL.less},
+ split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
+ val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
+ (Const (@{const_name HOL.plus},
+ split_type --> split_type --> split_type) $ t2' $ d)
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
+ end
+ (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
+ | (Const ("IntDef.nat", _), [t1]) =>
+ let
+ val rev_terms = rev terms
+ val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT)
+ val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT)
+ val n = Bound 0
+ val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)])
+ (map (incr_boundvars 1) rev_terms)
+ val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms
+ val t1' = incr_boundvars 1 t1
+ val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
+ (Const ("Nat.of_nat", HOLogic.natT --> HOLogic.intT) $ n)
+ val t1_lt_zero = Const (@{const_name HOL.less},
+ HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
+ in
+ SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
+ end
+ (* "?P ((?n::nat) mod (number_of ?k)) =
+ ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
+ (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
+ | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
+ let
+ val rev_terms = rev terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val i = Bound 1
+ val j = Bound 0
+ val terms1 = map (subst_term [(split_term, t1)]) rev_terms
+ val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)])
+ (map (incr_boundvars 2) rev_terms)
+ val t1' = incr_boundvars 2 t1
+ val t2' = incr_boundvars 2 t2
+ val t2_eq_zero = Const ("op =",
+ split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
+ val t2_neq_zero = HOLogic.mk_not (Const ("op =",
+ split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
+ val j_lt_t2 = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ j $ t2'
+ val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
+ (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
+ (Const (@{const_name HOL.times},
+ split_type --> split_type --> split_type) $ t2' $ i) $ j)
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
+ val subgoal2 = (map HOLogic.mk_Trueprop
+ [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
+ @ terms2 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
+ end
+ (* "?P ((?n::nat) div (number_of ?k)) =
+ ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
+ (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
+ | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
+ let
+ val rev_terms = rev terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val i = Bound 1
+ val j = Bound 0
+ val terms1 = map (subst_term [(split_term, zero)]) rev_terms
+ val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)])
+ (map (incr_boundvars 2) rev_terms)
+ val t1' = incr_boundvars 2 t1
+ val t2' = incr_boundvars 2 t2
+ val t2_eq_zero = Const ("op =",
+ split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
+ val t2_neq_zero = HOLogic.mk_not (Const ("op =",
+ split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
+ val j_lt_t2 = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ j $ t2'
+ val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
+ (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
+ (Const (@{const_name HOL.times},
+ split_type --> split_type --> split_type) $ t2' $ i) $ j)
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
+ val subgoal2 = (map HOLogic.mk_Trueprop
+ [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
+ @ terms2 @ [not_false]
+ in
+ SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
+ end
+ (* "?P ((?n::int) mod (number_of ?k)) =
+ ((iszero (number_of ?k) --> ?P ?n) &
+ (neg (number_of (uminus ?k)) -->
+ (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
+ (neg (number_of ?k) -->
+ (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
+ | (Const ("Divides.div_class.mod",
+ Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
+ let
+ val rev_terms = rev terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val i = Bound 1
+ val j = Bound 0
+ val terms1 = map (subst_term [(split_term, t1)]) rev_terms
+ val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)])
+ (map (incr_boundvars 2) rev_terms)
+ val t1' = incr_boundvars 2 t1
+ val (t2' as (_ $ k')) = incr_boundvars 2 t2
+ val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
+ val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
+ (number_of $
+ (Const (@{const_name HOL.uminus},
+ HOLogic.intT --> HOLogic.intT) $ k'))
+ val zero_leq_j = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ zero $ j
+ val j_lt_t2 = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ j $ t2'
+ val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
+ (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
+ (Const (@{const_name HOL.times},
+ split_type --> split_type --> split_type) $ t2' $ i) $ j)
+ val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
+ val t2_lt_j = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ t2' $ j
+ val j_leq_zero = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ j $ zero
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
+ val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
+ @ hd terms2_3
+ :: (if tl terms2_3 = [] then [not_false] else [])
+ @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
+ @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
+ val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
+ @ hd terms2_3
+ :: (if tl terms2_3 = [] then [not_false] else [])
+ @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
+ @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
+ val Ts' = split_type :: split_type :: Ts
+ in
+ SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
+ end
+ (* "?P ((?n::int) div (number_of ?k)) =
+ ((iszero (number_of ?k) --> ?P 0) &
+ (neg (number_of (uminus ?k)) -->
+ (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
+ (neg (number_of ?k) -->
+ (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
+ | (Const ("Divides.div_class.div",
+ Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
+ let
+ val rev_terms = rev terms
+ val zero = Const (@{const_name HOL.zero}, split_type)
+ val i = Bound 1
+ val j = Bound 0
+ val terms1 = map (subst_term [(split_term, zero)]) rev_terms
+ val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)])
+ (map (incr_boundvars 2) rev_terms)
+ val t1' = incr_boundvars 2 t1
+ val (t2' as (_ $ k')) = incr_boundvars 2 t2
+ val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
+ val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
+ (number_of $
+ (Const (@{const_name HOL.uminus},
+ HOLogic.intT --> HOLogic.intT) $ k'))
+ val zero_leq_j = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ zero $ j
+ val j_lt_t2 = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ j $ t2'
+ val t1_eq_t2_times_i_plus_j = Const ("op =",
+ split_type --> split_type --> HOLogic.boolT) $ t1' $
+ (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
+ (Const (@{const_name HOL.times},
+ split_type --> split_type --> split_type) $ t2' $ i) $ j)
+ val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
+ val t2_lt_j = Const (@{const_name HOL.less},
+ split_type --> split_type--> HOLogic.boolT) $ t2' $ j
+ val j_leq_zero = Const (@{const_name HOL.less_eq},
+ split_type --> split_type --> HOLogic.boolT) $ j $ zero
+ val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
+ val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
+ val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k)
+ :: terms2_3
+ @ not_false
+ :: (map HOLogic.mk_Trueprop
+ [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
+ val subgoal3 = (HOLogic.mk_Trueprop neg_t2)
+ :: terms2_3
+ @ not_false
+ :: (map HOLogic.mk_Trueprop
+ [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
+ val Ts' = split_type :: split_type :: Ts
+ in
+ SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
+ end
+ (* this will only happen if a split theorem can be applied for which no *)
+ (* code exists above -- in which case either the split theorem should be *)
+ (* implemented above, or 'is_split_thm' should be modified to filter it *)
+ (* out *)
+ | (t, ts) => (
+ warning ("Lin. Arith.: split rule for " ^ ProofContext.string_of_term ctxt t ^
+ " (with " ^ string_of_int (length ts) ^
+ " argument(s)) not implemented; proof reconstruction is likely to fail");
+ NONE
+ ))
+ )
+end;
+
+(* remove terms that do not satisfy 'p'; change the order of the remaining *)
+(* terms in the same way as filter_prems_tac does *)
+
+fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
+let
+ fun filter_prems (t, (left, right)) =
+ if p t then (left, right @ [t]) else (left @ right, [])
+ val (left, right) = foldl filter_prems ([], []) terms
+in
+ right @ left
+end;
+
+(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a *)
+(* subgoal that has 'terms' as premises *)
+
+fun negated_term_occurs_positively (terms : term list) : bool =
+ List.exists
+ (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
+ | _ => false)
+ terms;
+
+fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
+let
+ (* repeatedly split (including newly emerging subgoals) until no further *)
+ (* splitting is possible *)
+ fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
+ | split_loop (subgoal::subgoals) = (
+ case split_once_items ctxt subgoal of
+ SOME new_subgoals => split_loop (new_subgoals @ subgoals)
+ | NONE => subgoal :: split_loop subgoals
+ )
+ fun is_relevant t = isSome (decomp ctxt t)
+ (* filter_prems_tac is_relevant: *)
+ val relevant_terms = filter_prems_tac_items is_relevant terms
+ (* split_tac, NNF normalization: *)
+ val split_goals = split_loop [(Ts, relevant_terms)]
+ (* necessary because split_once_tac may normalize terms: *)
+ val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
+ (* TRY (etac notE) THEN eq_assume_tac: *)
+ val result = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
+in
+ result
+end;
+
+(* takes the i-th subgoal [| A1; ...; An |] ==> B to *)
+(* An --> ... --> A1 --> B, performs splitting with the given 'split_thms' *)
+(* (resulting in a different subgoal P), takes P to ~P ==> False, *)
+(* performs NNF-normalization of ~P, and eliminates conjunctions, *)
+(* disjunctions and existential quantifiers from the premises, possibly (in *)
+(* the case of disjunctions) resulting in several new subgoals, each of the *)
+(* general form [| Q1; ...; Qm |] ==> False. Fails if more than *)
+(* !fast_arith_split_limit splits are possible. *)
+
+local
+ val nnf_simpset =
+ empty_ss setmkeqTrue mk_eq_True
+ setmksimps (mksimps mksimps_pairs)
+ addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
+ not_all, not_ex, not_not]
+ fun prem_nnf_tac i st =
+ full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
+in
+
+fun split_once_tac ctxt split_thms =
+ let
+ val thy = ProofContext.theory_of ctxt
+ val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
+ let
+ val Ts = rev (map snd (Logic.strip_params subgoal))
+ val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
+ val cmap = Splitter.cmap_of_split_thms split_thms
+ val splits = Splitter.split_posns cmap thy Ts concl
+ val split_limit = ConfigOption.get ctxt fast_arith_split_limit
+ in
+ if length splits > split_limit then no_tac
+ else split_tac split_thms i
+ end)
+ in
+ EVERY' [
+ REPEAT_DETERM o etac rev_mp,
+ cond_split_tac,
+ rtac ccontr,
+ prem_nnf_tac,
+ TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
+ ]
+ end;
+
+end; (* local *)
+
+(* remove irrelevant premises, then split the i-th subgoal (and all new *)
+(* subgoals) by using 'split_once_tac' repeatedly. Beta-eta-normalize new *)
+(* subgoals and finally attempt to solve them by finding an immediate *)
+(* contradiction (i.e. a term and its negation) in their premises. *)
+
+fun pre_tac ctxt i =
+let
+ val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
+ fun is_relevant t = isSome (decomp ctxt t)
+in
+ DETERM (
+ TRY (filter_prems_tac is_relevant i)
+ THEN (
+ (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
+ THEN_ALL_NEW
+ (CONVERSION Drule.beta_eta_conversion
+ THEN'
+ (TRY o (etac notE THEN' eq_assume_tac)))
+ ) i
+ )
+end;
+
+end; (* LA_Data_Ref *)
+
+
+val lin_arith_pre_tac = LA_Data_Ref.pre_tac;
+
+structure Fast_Arith =
+ Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
+
+val map_data = Fast_Arith.map_data;
+
+fun fast_arith_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
+val fast_ex_arith_tac = Fast_Arith.lin_arith_tac;
+val trace_arith = Fast_Arith.trace;
+
+(* reduce contradictory <= to False.
+ Most of the work is done by the cancel tactics. *)
+
+val init_arith_data =
+ Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
+ {add_mono_thms = add_mono_thms @
+ @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
+ mult_mono_thms = mult_mono_thms,
+ inj_thms = inj_thms,
+ lessD = lessD @ [thm "Suc_leI"],
+ neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
+ simpset = HOL_basic_ss
+ addsimps
+ [@{thm "monoid_add_class.zero_plus.add_0_left"},
+ @{thm "monoid_add_class.zero_plus.add_0_right"},
+ @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
+ @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
+ @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
+ @{thm "not_one_less_zero"}]
+ addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
+ (*abel_cancel helps it work in abstract algebraic domains*)
+ addsimprocs nat_cancel_sums_add}) #>
+ arith_discrete "nat";
+
+val lin_arith_simproc = Fast_Arith.lin_arith_simproc;
+
+val fast_nat_arith_simproc =
+ Simplifier.simproc (the_context ()) "fast_nat_arith"
+ ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
+
+(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
+useful to detect inconsistencies among the premises for subgoals which are
+*not* themselves (in)equalities, because the latter activate
+fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
+solver all the time rather than add the additional check. *)
+
+
+(* arith proof method *)
+
+local
+
+fun raw_arith_tac ctxt ex =
+ (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
+ decomp sg"? -- but note that the test is applied to terms already before
+ they are split/normalized) to speed things up in case there are lots of
+ irrelevant terms involved; elimination of min/max can be optimized:
+ (max m n + k <= r) = (m+k <= r & n+k <= r)
+ (l <= min m n + k) = (l <= m+k & l <= n+k)
+ *)
+ refute_tac (K true)
+ (* Splitting is also done inside fast_arith_tac, but not completely -- *)
+ (* split_tac may use split theorems that have not been implemented in *)
+ (* fast_arith_tac (cf. pre_decomp and split_once_items above), and *)
+ (* fast_arith_split_limit may trigger. *)
+ (* Therefore splitting outside of fast_arith_tac may allow us to prove *)
+ (* some goals that fast_arith_tac alone would fail on. *)
+ (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
+ (fast_ex_arith_tac ctxt ex);
+
+fun more_arith_tacs ctxt =
+ let val tactics = #tactics (get_arith_data ctxt)
+ in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
+
+in
+
+fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
+ ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
+
+fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
+ ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
+ more_arith_tacs ctxt];
+
+fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
+ ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
+ more_arith_tacs ctxt];
+
+fun arith_method src =
+ Method.syntax Args.bang_facts src
+ #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
+ HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
+
+end;
+
+
+(* context setup *)
+
+val setup =
+ init_arith_data #>
+ Simplifier.map_ss (fn ss => ss addsimprocs [fast_nat_arith_simproc]
+ addSolver (mk_solver' "lin_arith" Fast_Arith.cut_lin_arith_tac)) #>
+ Context.mapping
+ (setup_options #>
+ Method.add_methods
+ [("arith", arith_method, "decide linear arithmethic")] #>
+ Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
+ "declaration of split rules for arithmetic procedure")]) I;
+
+end;
+
+structure BasicLinArith: BASIC_LIN_ARITH = LinArith;
+open BasicLinArith;