(* Title: HOL/Tools/lin_arith.ML
Author: Tjark Weber and Tobias Nipkow, TU Muenchen
HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).
*)
signature LIN_ARITH =
sig
val pre_tac: Proof.context -> int -> tactic
val simple_tac: Proof.context -> int -> tactic
val tac: Proof.context -> int -> tactic
val simproc: simpset -> term -> thm option
val add_inj_thms: thm list -> Context.generic -> Context.generic
val add_lessD: thm -> Context.generic -> Context.generic
val add_simps: thm list -> Context.generic -> Context.generic
val add_simprocs: simproc list -> Context.generic -> Context.generic
val add_inj_const: string * typ -> Context.generic -> Context.generic
val add_discrete_type: string -> Context.generic -> Context.generic
val set_number_of: (theory -> typ -> int -> cterm) -> Context.generic ->
Context.generic
val setup: Context.generic -> Context.generic
val global_setup: theory -> theory
val split_limit: int Config.T
val neq_limit: int Config.T
val warning_count: int Unsynchronized.ref
val trace: bool Unsynchronized.ref
end;
structure Lin_Arith: 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 trueI = TrueI;
val not_lessD = @{thm linorder_not_less} RS iffD1;
val not_leD = @{thm linorder_not_le} RS iffD1;
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 (@{const_name "op &"}, _) $ _ $ _) =>
atomize (thm RS conjunct1) @ atomize (thm RS conjunct2)
| _ => [thm];
fun neg_prop ((TP as Const("Trueprop", _)) $ (Const (@{const_name 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 = HOLogic.false_const end;
fun is_nat t = (fastype_of1 t = HOLogic.natT);
fun mk_nat_thm thy t =
let
val cn = cterm_of thy (Var (("n", 0), HOLogic.natT))
and ct = cterm_of thy t
in instantiate ([], [(cn, ct)]) @{thm le0} end;
end; (* LA_Logic *)
(* arith context data *)
structure Lin_Arith_Data = Generic_Data
(
type T = {splits: thm list,
inj_consts: (string * typ) list,
discrete: string list};
val empty = {splits = [], inj_consts = [], discrete = []};
val extend = I;
fun merge
({splits= splits1, inj_consts= inj_consts1, discrete= discrete1},
{splits= splits2, inj_consts= inj_consts2, discrete= discrete2}) : T =
{splits = Thm.merge_thms (splits1, splits2),
inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
discrete = Library.merge (op =) (discrete1, discrete2)};
);
val get_arith_data = Lin_Arith_Data.get o Context.Proof;
fun add_split thm = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
{splits = update Thm.eq_thm_prop thm splits,
inj_consts = inj_consts, discrete = discrete});
fun add_discrete_type d = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
{splits = splits, inj_consts = inj_consts,
discrete = update (op =) d discrete});
fun add_inj_const c = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
{splits = splits, inj_consts = update (op =) c inj_consts,
discrete = discrete});
val (split_limit, setup_split_limit) = Attrib.config_int "linarith_split_limit" 9;
val (neq_limit, setup_neq_limit) = Attrib.config_int "linarith_neq_limit" 9;
structure LA_Data =
struct
val fast_arith_neq_limit = neq_limit;
(* Decomposition of terms *)
(*internal representation of linear (in-)equations*)
type decomp =
((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 Pattern.aeconv p t of
NONE => ((t, m) :: p, i)
| SOME n => (AList.update Pattern.aeconv (t, Rat.add n m) p, i);
(* decompose nested multiplications, bracketing them to the right and combining
all their coefficients
inj_consts: list of constants to be ignored when encountered
(e.g. arithmetic type conversions that preserve value)
m: multiplicity associated with the entire product
returns either (SOME term, associated multiplicity) or (NONE, constant)
*)
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 (@{const_name HOL.times}, _) $ s1 $ s2 =>
(* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)
demult (mC $ s1 $ (mC $ s2 $ t), m)
| _ =>
(* product 's * t', where either factor can be 'NONE' *)
(case demult (s, m) of
(SOME s', m') =>
(case demult (t, m') of
(SOME t', m'') => (SOME (mC $ s' $ t'), m'')
| (NONE, m'') => (SOME s', m''))
| (NONE, m') => demult (t, m')))
| demult ((mC as Const (@{const_name HOL.divide}, _)) $ s $ t, m) =
(* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could
become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ? Note that
if we choose to do so here, the simpset used by arith must be able to
perform the same simplifications. *)
(* FIXME: Currently we treat the numerator as atomic unless the
denominator can be reduced to a numeric constant. It might be better
to demult the numerator in any case, and invent a new term of the form
'1 / t' if the numerator can be reduced, but the denominator cannot. *)
(* FIXME: Currently we even treat the whole fraction as atomic unless the
denominator can be reduced to a numeric constant. It might be better
to use the partially reduced denominator (i.e. 's / (2*t)' could be
demult'ed to 's / t' with multiplicity .5). This would require a
very simple change only below, but it breaks existing proofs. *)
(* quotient 's / t', where the denominator t can be NONE *)
(* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)
(case demult (t, Rat.one) of
(SOME _, _) => (SOME (mC $ s $ t), m)
| (NONE, m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))
(* terms that evaluate to numeric constants *)
| demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
| demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
| demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
(*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 ...*)
| demult (t as Const ("Int.number_class.number_of", _) $ n, m) =
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
handle TERM _ => (SOME t, m))
| demult (t as Const (@{const_name Suc}, _) $ _, m) =
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
handle TERM _ => (SOME t, m))
(* injection constants are ignored *)
| demult (t as Const f $ x, m) =
if member (op =) inj_consts f then demult (x, m) else (SOME t, m)
(* everything else is considered atomic *)
| demult (atom, m) = (SOME atom, 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
(* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of
summands and associated multiplicities, plus a constant 'i' (with implicit
multiplicity 1) *)
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 ("Int.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 Rat.DIVZERO => NONE;
fun of_lin_arith_sort thy U =
Sign.of_sort thy (U, @{sort Ring_and_Field.ordered_idom});
fun allows_lin_arith thy (discrete : string list) (U as Type (D, [])) : bool * bool =
if of_lin_arith_sort thy U then (true, member (op =) discrete D)
else if member (op =) discrete D 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) : decomp 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)) : decomp 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 -> decomp 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;
(*---------------------------------------------------------------------------*)
(* 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 =
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},
"Int.nat",
"Divides.div_class.mod",
"Divides.div_class.div"] a
| _ => (warning ("Lin. Arith.: wrong format for split rule " ^
Display.string_of_thm_without_context thm);
false))
| _ => (warning ("Lin. Arith.: wrong format for split rule " ^
Display.string_of_thm_without_context 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 Pattern.aeconv 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 tms =
fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop tms)
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 goal_tm = REPEAT_DETERM_etac_rev_mp terms
val splits = Splitter.split_posns cmap thy Ts goal_tm
val split_limit = Config.get ctxt split_limit
in
if length splits > split_limit then (
tracing ("linarith_split_limit exceeded (current value is " ^
string_of_int split_limit ^ ")");
NONE
) else case splits of
[] =>
(* split_tac would fail: no possible split *)
NONE
| (_, _::_, _, _, _) :: _ =>
(* disallow a split that involves non-locally bound variables (except *)
(* when bound by outermost meta-quantifiers) *)
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 ("Int.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 (@{const_name 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 ("Int.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 ("Int.iszero", split_type --> HOLogic.boolT) $ t2
val neg_minus_k = Const ("Int.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 ("Int.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 ("Int.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 ("Int.iszero", split_type --> HOLogic.boolT) $ t2
val neg_minus_k = Const ("Int.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 ("Int.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 " ^ Syntax.string_of_term ctxt t ^
" (with " ^ string_of_int (length ts) ^
" argument(s)) not implemented; proof reconstruction is likely to fail");
NONE
))
end; (* split_once_items *)
(* 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) = fold 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 Pattern.aeconv 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 = is_some (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 = filter_out (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 *)
(* !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
in
if null splits orelse length splits > Config.get ctxt split_limit then
no_tac
else if null (#2 (hd splits)) then
split_tac split_thms i
else
(* disallow a split that involves non-locally bound variables *)
(* (except when bound by outermost meta-quantifiers) *)
no_tac
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 = is_some (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 *)
val pre_tac = LA_Data.pre_tac;
structure Fast_Arith = Fast_Lin_Arith(structure LA_Logic = LA_Logic and LA_Data = LA_Data);
fun map_inj_thms f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = f inj_thms,
lessD = lessD, neqE = neqE, simpset = simpset, number_of = number_of};
fun map_lessD f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
lessD = f lessD, neqE = neqE, simpset = simpset, number_of = number_of};
fun map_simpset f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
lessD = lessD, neqE = neqE, simpset = f simpset, number_of = number_of};
fun map_number_of f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
lessD = lessD, neqE = neqE, simpset = simpset, number_of = f number_of};
fun add_inj_thms thms = Fast_Arith.map_data (map_inj_thms (append thms));
fun add_lessD thm = Fast_Arith.map_data (map_lessD (fn thms => thms @ [thm]));
fun add_simps thms = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimps thms));
fun add_simprocs procs = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimprocs procs));
fun set_number_of f = Fast_Arith.map_data (map_number_of (K (serial (), f)))
fun simple_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
val lin_arith_tac = Fast_Arith.lin_arith_tac;
val trace = Fast_Arith.trace;
val warning_count = Fast_Arith.warning_count;
(* 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, number_of, ...} =>
{add_mono_thms = @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field} @ add_mono_thms,
mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} ::
@{lemma "a = b ==> c*a = c*b" by (rule arg_cong)} :: 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 @{thms ring_distribs}
addsimps [@{thm if_True}, @{thm if_False}]
addsimps
[@{thm "monoid_add_class.add_0_left"},
@{thm "monoid_add_class.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_Arith.nat_cancel_sums_add
addcongs [if_weak_cong],
number_of = number_of}) #>
add_discrete_type @{type_name nat};
fun add_arith_facts ss =
add_prems (Arith_Data.get_arith_facts (MetaSimplifier.the_context ss)) ss;
val simproc = add_arith_facts #> Fast_Arith.lin_arith_simproc;
(* generic refutation procedure *)
(* parameters:
test: term -> bool
tests if a term is at all relevant to the refutation proof;
if not, then it can be discarded. Can improve performance,
esp. if disjunctions can be discarded (no case distinction needed!).
prep_tac: int -> tactic
A preparation tactic to be applied to the goal once all relevant premises
have been moved to the conclusion.
ref_tac: int -> tactic
the actual refutation tactic. Should be able to deal with goals
[| A1; ...; An |] ==> False
where the Ai are atomic, i.e. no top-level &, | or EX
*)
local
val nnf_simpset =
empty_ss setmkeqTrue mk_eq_True
setmksimps (mksimps mksimps_pairs)
addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
@{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm 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 refute_tac test prep_tac ref_tac =
let val refute_prems_tac =
REPEAT_DETERM
(eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
filter_prems_tac test 1 ORELSE
etac @{thm disjE} 1) THEN
(DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
ref_tac 1);
in EVERY'[TRY o filter_prems_tac test,
REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
end;
end;
(* arith proof method *)
local
fun raw_tac ctxt ex =
(* FIXME: K true should be replaced by a sensible test (perhaps "is_some 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 simple_tac, but not completely -- *)
(* split_tac may use split theorems that have not been implemented in *)
(* simple_tac (cf. pre_decomp and split_once_items above), and *)
(* split_limit may trigger. *)
(* Therefore splitting outside of simple_tac may allow us to prove *)
(* some goals that simple_tac alone would fail on. *)
(REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
(lin_arith_tac ctxt ex);
in
fun gen_tac ex ctxt = FIRST' [simple_tac ctxt,
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_tac ctxt ex];
val tac = gen_tac true;
end;
(* context setup *)
val setup =
init_arith_data #>
Simplifier.map_ss (fn ss => ss addsimprocs [Simplifier.simproc (@{theory}) "fast_nat_arith"
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K 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. *)
addSolver (mk_solver' "lin_arith"
(add_arith_facts #> Fast_Arith.cut_lin_arith_tac)))
val global_setup =
setup_split_limit #> setup_neq_limit #>
Attrib.setup @{binding arith_split} (Scan.succeed (Thm.declaration_attribute add_split))
"declaration of split rules for arithmetic procedure" #>
Method.setup @{binding linarith}
(Args.bang_facts >> (fn prems => fn ctxt =>
METHOD (fn facts =>
HEADGOAL (Method.insert_tac (prems @ Arith_Data.get_arith_facts ctxt @ facts)
THEN' tac ctxt)))) "linear arithmetic" #>
Arith_Data.add_tactic "linear arithmetic" gen_tac;
end;