--- a/src/HOL/arith_data.ML Tue Jul 31 19:40:25 2007 +0200
+++ b/src/HOL/arith_data.ML Tue Jul 31 19:40:26 2007 +0200
@@ -1,13 +1,11 @@
(* Title: HOL/arith_data.ML
ID: $Id$
- Author: Markus Wenzel, Stefan Berghofer and Tobias Nipkow
+ Author: Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
-Various arithmetic proof procedures.
+Basic arithmetic proof tools.
*)
-(*---------------------------------------------------------------------------*)
-(* 1. Cancellation of common terms *)
-(*---------------------------------------------------------------------------*)
+(*** cancellation of common terms ***)
structure NatArithUtils =
struct
@@ -28,6 +26,7 @@
funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
end;
+
(* dest_sum *)
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
@@ -42,6 +41,8 @@
SOME (t, u) => dest_sum t @ dest_sum u
| NONE => [tm]));
+
+
(** generic proof tools **)
(* prove conversions *)
@@ -61,21 +62,25 @@
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (the_context ()) name pats proc;
-end; (* NatArithUtils *)
+end;
+
+(** ArithData **)
+
signature ARITH_DATA =
sig
val nat_cancel_sums_add: simproc list
val nat_cancel_sums: simproc list
+ val arith_data_setup: Context.generic -> Context.generic
end;
-
structure ArithData: ARITH_DATA =
struct
open NatArithUtils;
+
(** cancel common summands **)
structure Sum =
@@ -92,6 +97,7 @@
fun gen_uncancel_tac rule ct =
rtac (instantiate' [] [NONE, SOME ct] (rule RS @{thm subst_equals})) 1;
+
(* nat eq *)
structure EqCancelSums = CancelSumsFun
@@ -102,6 +108,7 @@
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
end);
+
(* nat less *)
structure LessCancelSums = CancelSumsFun
@@ -112,6 +119,7 @@
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
end);
+
(* nat le *)
structure LeCancelSums = CancelSumsFun
@@ -122,6 +130,7 @@
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
end);
+
(* nat diff *)
structure DiffCancelSums = CancelSumsFun
@@ -132,7 +141,8 @@
val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
end);
-(** prepare nat_cancel simprocs **)
+
+(* prepare nat_cancel simprocs *)
val nat_cancel_sums_add = map prep_simproc
[("nateq_cancel_sums",
@@ -150,848 +160,11 @@
["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"],
K DiffCancelSums.proc)];
-end; (* ArithData *)
-
-open ArithData;
-
-
-(*---------------------------------------------------------------------------*)
-(* 2. Linear arithmetic *)
-(*---------------------------------------------------------------------------*)
-
-(* 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; (* LA_Logic *)
-
-
-(* arith theory data *)
-
-datatype arithtactic = ArithTactic of {name: string, tactic: 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: arithtactic 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});
-
-
-signature HOL_LIN_ARITH_DATA =
-sig
- include LIN_ARITH_DATA
- val fast_arith_split_limit: int ConfigOption.T
- val setup_options: theory -> theory
-end;
-
-structure LA_Data_Ref: HOL_LIN_ARITH_DATA =
-struct
-
-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;
+val arith_data_setup =
+ Simplifier.map_ss (fn ss => ss addsimprocs nat_cancel_sums);
-(* internal representation of linear (in-)equations *)
-type decompT = ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
-
-(* Decomposition of terms *)
-
-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 *)
-
-
-structure Fast_Arith =
- Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
-
-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 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) 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;
-
+(* FIXME dead code *)
(* antisymmetry:
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y
@@ -1036,17 +209,6 @@
end;
*)
-(* theory setup *)
+end;
-val arith_setup =
- init_arith_data #>
- Simplifier.map_ss (fn ss => ss
- addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc])
- addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)) #>
- Context.mapping
- (LA_Data_Ref.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;
+open ArithData;