--- a/src/HOL/Integ/int_arith1.ML Thu May 31 18:16:51 2007 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,610 +0,0 @@
-(* Title: HOL/Integ/int_arith1.ML
- ID: $Id$
- Authors: Larry Paulson and Tobias Nipkow
-
-Simprocs and decision procedure for linear arithmetic.
-*)
-
-(** Misc ML bindings **)
-
-val succ_Pls = thm "succ_Pls";
-val succ_Min = thm "succ_Min";
-val succ_1 = thm "succ_1";
-val succ_0 = thm "succ_0";
-
-val pred_Pls = thm "pred_Pls";
-val pred_Min = thm "pred_Min";
-val pred_1 = thm "pred_1";
-val pred_0 = thm "pred_0";
-
-val minus_Pls = thm "minus_Pls";
-val minus_Min = thm "minus_Min";
-val minus_1 = thm "minus_1";
-val minus_0 = thm "minus_0";
-
-val add_Pls = thm "add_Pls";
-val add_Min = thm "add_Min";
-val add_BIT_11 = thm "add_BIT_11";
-val add_BIT_10 = thm "add_BIT_10";
-val add_BIT_0 = thm "add_BIT_0";
-val add_Pls_right = thm "add_Pls_right";
-val add_Min_right = thm "add_Min_right";
-
-val mult_Pls = thm "mult_Pls";
-val mult_Min = thm "mult_Min";
-val mult_num1 = thm "mult_num1";
-val mult_num0 = thm "mult_num0";
-
-val neg_def = thm "neg_def";
-val iszero_def = thm "iszero_def";
-
-val number_of_succ = thm "number_of_succ";
-val number_of_pred = thm "number_of_pred";
-val number_of_minus = thm "number_of_minus";
-val number_of_add = thm "number_of_add";
-val diff_number_of_eq = thm "diff_number_of_eq";
-val number_of_mult = thm "number_of_mult";
-val double_number_of_BIT = thm "double_number_of_BIT";
-val numeral_0_eq_0 = thm "numeral_0_eq_0";
-val numeral_1_eq_1 = thm "numeral_1_eq_1";
-val numeral_m1_eq_minus_1 = thm "numeral_m1_eq_minus_1";
-val mult_minus1 = thm "mult_minus1";
-val mult_minus1_right = thm "mult_minus1_right";
-val minus_number_of_mult = thm "minus_number_of_mult";
-val zero_less_nat_eq = thm "zero_less_nat_eq";
-val eq_number_of_eq = thm "eq_number_of_eq";
-val iszero_number_of_Pls = thm "iszero_number_of_Pls";
-val nonzero_number_of_Min = thm "nonzero_number_of_Min";
-val iszero_number_of_BIT = thm "iszero_number_of_BIT";
-val iszero_number_of_0 = thm "iszero_number_of_0";
-val iszero_number_of_1 = thm "iszero_number_of_1";
-val less_number_of_eq_neg = thm "less_number_of_eq_neg";
-val le_number_of_eq = thm "le_number_of_eq";
-val not_neg_number_of_Pls = thm "not_neg_number_of_Pls";
-val neg_number_of_Min = thm "neg_number_of_Min";
-val neg_number_of_BIT = thm "neg_number_of_BIT";
-val le_number_of_eq_not_less = thm "le_number_of_eq_not_less";
-val abs_number_of = thm "abs_number_of";
-val number_of_reorient = thm "number_of_reorient";
-val add_number_of_left = thm "add_number_of_left";
-val mult_number_of_left = thm "mult_number_of_left";
-val add_number_of_diff1 = thm "add_number_of_diff1";
-val add_number_of_diff2 = thm "add_number_of_diff2";
-val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
-val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
-val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
-
-val arith_extra_simps = thms "arith_extra_simps";
-val arith_simps = thms "arith_simps";
-val rel_simps = thms "rel_simps";
-
-val zless_imp_add1_zle = thm "zless_imp_add1_zle";
-
-val combine_common_factor = thm "combine_common_factor";
-val eq_add_iff1 = thm "eq_add_iff1";
-val eq_add_iff2 = thm "eq_add_iff2";
-val less_add_iff1 = thm "less_add_iff1";
-val less_add_iff2 = thm "less_add_iff2";
-val le_add_iff1 = thm "le_add_iff1";
-val le_add_iff2 = thm "le_add_iff2";
-
-val arith_special = thms "arith_special";
-
-structure Int_Numeral_Base_Simprocs =
- struct
- fun prove_conv tacs ctxt (_: thm list) (t, u) =
- if t aconv u then NONE
- else
- let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
- in SOME (Goal.prove ctxt [] [] eq (K (EVERY tacs))) end
-
- fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];
-
- fun prep_simproc (name, pats, proc) =
- Simplifier.simproc (the_context()) name pats proc;
-
- fun is_numeral (Const(@{const_name Numeral.number_of}, _) $ w) = true
- | is_numeral _ = false
-
- fun simplify_meta_eq f_number_of_eq f_eq =
- mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
-
- (*reorientation simprules using ==, for the following simproc*)
- val meta_zero_reorient = zero_reorient RS eq_reflection
- val meta_one_reorient = one_reorient RS eq_reflection
- val meta_number_of_reorient = number_of_reorient RS eq_reflection
-
- (*reorientation simplification procedure: reorients (polymorphic)
- 0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
- fun reorient_proc sg _ (_ $ t $ u) =
- case u of
- Const(@{const_name HOL.zero}, _) => NONE
- | Const(@{const_name HOL.one}, _) => NONE
- | Const(@{const_name Numeral.number_of}, _) $ _ => NONE
- | _ => SOME (case t of
- Const(@{const_name HOL.zero}, _) => meta_zero_reorient
- | Const(@{const_name HOL.one}, _) => meta_one_reorient
- | Const(@{const_name Numeral.number_of}, _) $ _ => meta_number_of_reorient)
-
- val reorient_simproc =
- prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)
-
- end;
-
-
-Addsimprocs [Int_Numeral_Base_Simprocs.reorient_simproc];
-
-
-structure Int_Numeral_Simprocs =
-struct
-
-(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in Int_Numeral_Base_Simprocs
- isn't complicated by the abstract 0 and 1.*)
-val numeral_syms = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];
-
-(** New term ordering so that AC-rewriting brings numerals to the front **)
-
-(*Order integers by absolute value and then by sign. The standard integer
- ordering is not well-founded.*)
-fun num_ord (i,j) =
- (case IntInf.compare (IntInf.abs i, IntInf.abs j) of
- EQUAL => int_ord (IntInf.sign i, IntInf.sign j)
- | ord => ord);
-
-(*This resembles Term.term_ord, but it puts binary numerals before other
- non-atomic terms.*)
-local open Term
-in
-fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
- (case numterm_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
- | numterm_ord
- (Const(@{const_name Numeral.number_of}, _) $ v, Const(@{const_name Numeral.number_of}, _) $ w) =
- num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
- | numterm_ord (Const(@{const_name Numeral.number_of}, _) $ _, _) = LESS
- | numterm_ord (_, Const(@{const_name Numeral.number_of}, _) $ _) = GREATER
- | numterm_ord (t, u) =
- (case int_ord (size_of_term t, size_of_term u) of
- EQUAL =>
- let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
- (case hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
- end
- | ord => ord)
-and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
-end;
-
-fun numtermless tu = (numterm_ord tu = LESS);
-
-(*Defined in this file, but perhaps needed only for Int_Numeral_Base_Simprocs of type nat.*)
-val num_ss = HOL_ss settermless numtermless;
-
-
-(** Utilities **)
-
-fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
-
-fun find_first_numeral past (t::terms) =
- ((snd (HOLogic.dest_number t), rev past @ terms)
- handle TERM _ => find_first_numeral (t::past) terms)
- | find_first_numeral past [] = raise TERM("find_first_numeral", []);
-
-val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
-
-fun mk_minus t =
- let val T = Term.fastype_of t
- in Const (@{const_name HOL.uminus}, T --> T) $ t
- end;
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum T [] = mk_number T 0
- | mk_sum T [t,u] = mk_plus (t, u)
- | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum T [] = mk_number T 0
- | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
-
-(*decompose additions AND subtractions as a sum*)
-fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
- dest_summing (pos, t, dest_summing (pos, u, ts))
- | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
- dest_summing (pos, t, dest_summing (not pos, u, ts))
- | dest_summing (pos, t, ts) =
- if pos then t::ts else mk_minus t :: ts;
-
-fun dest_sum t = dest_summing (true, t, []);
-
-val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
-val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
-
-val mk_times = HOLogic.mk_binop @{const_name HOL.times};
-
-fun mk_prod T =
- let val one = mk_number T 1
- fun mk [] = one
- | mk [t] = t
- | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
- in mk end;
-
-(*This version ALWAYS includes a trailing one*)
-fun long_mk_prod T [] = mk_number T 1
- | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
-
-val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
-
-fun dest_prod t =
- let val (t,u) = dest_times t
- in dest_prod t @ dest_prod u end
- handle TERM _ => [t];
-
-(*DON'T do the obvious simplifications; that would create special cases*)
-fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
-
-(*Express t as a product of (possibly) a numeral with other sorted terms*)
-fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t
- | dest_coeff sign t =
- let val ts = sort Term.term_ord (dest_prod t)
- val (n, ts') = find_first_numeral [] ts
- handle TERM _ => (1, ts)
- in (sign*n, mk_prod (Term.fastype_of t) ts') end;
-
-(*Find first coefficient-term THAT MATCHES u*)
-fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
- | find_first_coeff past u (t::terms) =
- let val (n,u') = dest_coeff 1 t
- in if u aconv u' then (n, rev past @ terms)
- else find_first_coeff (t::past) u terms
- end
- handle TERM _ => find_first_coeff (t::past) u terms;
-
-(*Fractions as pairs of ints. Can't use Rat.rat because the representation
- needs to preserve negative values in the denominator.*)
-fun mk_frac (p, q : IntInf.int) = if q = 0 then raise Div else (p, q);
-
-(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
- Fractions are reduced later by the cancel_numeral_factor simproc.*)
-fun add_frac ((p1 : IntInf.int, q1 : IntInf.int), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
-
-val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
-
-(*Build term (p / q) * t*)
-fun mk_fcoeff ((p, q), t) =
- let val T = Term.fastype_of t
- in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
-
-(*Express t as a product of a fraction with other sorted terms*)
-fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t
- | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) =
- let val (p, t') = dest_coeff sign t
- val (q, u') = dest_coeff 1 u
- in (mk_frac (p, q), mk_divide (t', u')) end
- | dest_fcoeff sign t =
- let val (p, t') = dest_coeff sign t
- val T = Term.fastype_of t
- in (mk_frac (p, 1), mk_divide (t', mk_number T 1)) end;
-
-
-(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
-val add_0s = thms "add_0s";
-val mult_1s = thms "mult_1s";
-
-(*Simplify inverse Numeral1, a/Numeral1*)
-val inverse_1s = [@{thm inverse_numeral_1}];
-val divide_1s = [@{thm divide_numeral_1}];
-
-(*To perform binary arithmetic. The "left" rewriting handles patterns
- created by the Int_Numeral_Base_Simprocs, such as 3 * (5 * x). *)
-val simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
- add_number_of_left, mult_number_of_left] @
- arith_simps @ rel_simps;
-
-(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
- during re-arrangement*)
-val non_add_simps =
- subtract Thm.eq_thm [add_number_of_left, number_of_add RS sym] simps;
-
-(*To evaluate binary negations of coefficients*)
-val minus_simps = [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
- minus_1, minus_0, minus_Pls, minus_Min,
- pred_1, pred_0, pred_Pls, pred_Min];
-
-(*To let us treat subtraction as addition*)
-val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
-
-(*To let us treat division as multiplication*)
-val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
-
-(*push the unary minus down: - x * y = x * - y *)
-val minus_mult_eq_1_to_2 =
- [@{thm minus_mult_left} RS sym, @{thm minus_mult_right}] MRS trans |> standard;
-
-(*to extract again any uncancelled minuses*)
-val minus_from_mult_simps =
- [@{thm minus_minus}, @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym];
-
-(*combine unary minus with numeric literals, however nested within a product*)
-val mult_minus_simps =
- [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
-
-(*Apply the given rewrite (if present) just once*)
-fun trans_tac NONE = all_tac
- | trans_tac (SOME th) = ALLGOALS (rtac (th RS trans));
-
-fun simplify_meta_eq rules =
- let val ss0 = HOL_basic_ss addeqcongs [eq_cong2] addsimps rules
- in fn ss => simplify (Simplifier.inherit_context ss ss0) o mk_meta_eq end
-
-structure CancelNumeralsCommon =
- struct
- val mk_sum = mk_sum
- val dest_sum = dest_sum
- val mk_coeff = mk_coeff
- val dest_coeff = dest_coeff 1
- val find_first_coeff = find_first_coeff []
- val trans_tac = fn _ => trans_tac
-
- val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
- diff_simps @ minus_simps @ add_ac
- val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
- val norm_ss3 = num_ss addsimps minus_from_mult_simps @ add_ac @ mult_ac
- fun norm_tac ss =
- ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
-
- val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
- val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
- end;
-
-
-structure EqCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
- val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
- val mk_bal = HOLogic.mk_eq
- val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
- val bal_add1 = eq_add_iff1 RS trans
- val bal_add2 = eq_add_iff2 RS trans
-);
-
-structure LessCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
- val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
- val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
- val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
- val bal_add1 = less_add_iff1 RS trans
- val bal_add2 = less_add_iff2 RS trans
-);
-
-structure LeCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
- val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
- val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
- val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
- val bal_add1 = le_add_iff1 RS trans
- val bal_add2 = le_add_iff2 RS trans
-);
-
-val cancel_numerals =
- map Int_Numeral_Base_Simprocs.prep_simproc
- [("inteq_cancel_numerals",
- ["(l::'a::number_ring) + m = n",
- "(l::'a::number_ring) = m + n",
- "(l::'a::number_ring) - m = n",
- "(l::'a::number_ring) = m - n",
- "(l::'a::number_ring) * m = n",
- "(l::'a::number_ring) = m * n"],
- K EqCancelNumerals.proc),
- ("intless_cancel_numerals",
- ["(l::'a::{ordered_idom,number_ring}) + m < n",
- "(l::'a::{ordered_idom,number_ring}) < m + n",
- "(l::'a::{ordered_idom,number_ring}) - m < n",
- "(l::'a::{ordered_idom,number_ring}) < m - n",
- "(l::'a::{ordered_idom,number_ring}) * m < n",
- "(l::'a::{ordered_idom,number_ring}) < m * n"],
- K LessCancelNumerals.proc),
- ("intle_cancel_numerals",
- ["(l::'a::{ordered_idom,number_ring}) + m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m + n",
- "(l::'a::{ordered_idom,number_ring}) - m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m - n",
- "(l::'a::{ordered_idom,number_ring}) * m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m * n"],
- K LeCancelNumerals.proc)];
-
-
-structure CombineNumeralsData =
- struct
- type coeff = IntInf.int
- val iszero = (fn x : IntInf.int => x = 0)
- val add = IntInf.+
- val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *)
- val dest_sum = dest_sum
- val mk_coeff = mk_coeff
- val dest_coeff = dest_coeff 1
- val left_distrib = combine_common_factor RS trans
- val prove_conv = Int_Numeral_Base_Simprocs.prove_conv_nohyps
- val trans_tac = fn _ => trans_tac
-
- val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
- diff_simps @ minus_simps @ add_ac
- val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
- val norm_ss3 = num_ss addsimps minus_from_mult_simps @ add_ac @ mult_ac
- fun norm_tac ss =
- ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
-
- val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
- val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
- end;
-
-structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
-
-(*Version for fields, where coefficients can be fractions*)
-structure FieldCombineNumeralsData =
- struct
- type coeff = IntInf.int * IntInf.int
- val iszero = (fn (p : IntInf.int, q) => p = 0)
- val add = add_frac
- val mk_sum = long_mk_sum
- val dest_sum = dest_sum
- val mk_coeff = mk_fcoeff
- val dest_coeff = dest_fcoeff 1
- val left_distrib = combine_common_factor RS trans
- val prove_conv = Int_Numeral_Base_Simprocs.prove_conv_nohyps
- val trans_tac = fn _ => trans_tac
-
- val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
- inverse_1s @ divide_simps @ diff_simps @ minus_simps @ add_ac
- val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
- val norm_ss3 = num_ss addsimps minus_from_mult_simps @ add_ac @ mult_ac
- fun norm_tac ss =
- ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
- THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
-
- val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
- val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
- end;
-
-structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
-
-val combine_numerals =
- Int_Numeral_Base_Simprocs.prep_simproc
- ("int_combine_numerals",
- ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
- K CombineNumerals.proc);
-
-val field_combine_numerals =
- Int_Numeral_Base_Simprocs.prep_simproc
- ("field_combine_numerals",
- ["(i::'a::{number_ring,field,division_by_zero}) + j",
- "(i::'a::{number_ring,field,division_by_zero}) - j"],
- K FieldCombineNumerals.proc);
-
-end;
-
-Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
-Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
-Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
-
-(*examples:
-print_depth 22;
-set timing;
-set trace_simp;
-fun test s = (Goal s, by (Simp_tac 1));
-
-test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
-
-test "2*u = (u::int)";
-test "(i + j + 12 + (k::int)) - 15 = y";
-test "(i + j + 12 + (k::int)) - 5 = y";
-
-test "y - b < (b::int)";
-test "y - (3*b + c) < (b::int) - 2*c";
-
-test "(2*x - (u*v) + y) - v*3*u = (w::int)";
-test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
-test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
-test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
-
-test "(i + j + 12 + (k::int)) = u + 15 + y";
-test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
-
-test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";
-
-test "a + -(b+c) + b = (d::int)";
-test "a + -(b+c) - b = (d::int)";
-
-(*negative numerals*)
-test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
-test "(i + j + -3 + (k::int)) < u + 5 + y";
-test "(i + j + 3 + (k::int)) < u + -6 + y";
-test "(i + j + -12 + (k::int)) - 15 = y";
-test "(i + j + 12 + (k::int)) - -15 = y";
-test "(i + j + -12 + (k::int)) - -15 = y";
-*)
-
-
-(** Constant folding for multiplication in semirings **)
-
-(*We do not need folding for addition: combine_numerals does the same thing*)
-
-structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
-struct
- val assoc_ss = HOL_ss addsimps mult_ac
- val eq_reflection = eq_reflection
-end;
-
-structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
-
-val assoc_fold_simproc =
- Int_Numeral_Base_Simprocs.prep_simproc
- ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
- K Semiring_Times_Assoc.proc);
-
-Addsimprocs [assoc_fold_simproc];
-
-
-
-
-(*** decision procedure for linear arithmetic ***)
-
-(*---------------------------------------------------------------------------*)
-(* Linear arithmetic *)
-(*---------------------------------------------------------------------------*)
-
-(*
-Instantiation of the generic linear arithmetic package for int.
-*)
-
-(* Update parameters of arithmetic prover *)
-local
-
-(* reduce contradictory <= to False *)
-val add_rules =
- simp_thms @ arith_simps @ rel_simps @ arith_special @
- [@{thm neg_le_iff_le}, @{thm numeral_0_eq_0}, @{thm numeral_1_eq_1},
- @{thm minus_zero}, @{thm diff_minus}, @{thm left_minus}, @{thm right_minus},
- @{thm mult_zero_left}, @{thm mult_zero_right}, @{thm mult_num1}, @{thm mult_1_right},
- @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym,
- @{thm minus_add_distrib}, @{thm minus_minus}, @{thm mult_assoc},
- of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,
- of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat]
-
-val nat_inj_thms = [zle_int RS iffD2, int_int_eq RS iffD2]
-
-val Int_Numeral_Base_Simprocs = assoc_fold_simproc
- :: Int_Numeral_Simprocs.combine_numerals
- :: Int_Numeral_Simprocs.cancel_numerals;
-
-in
-
-val int_arith_setup =
- Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
- {add_mono_thms = add_mono_thms,
- mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
- inj_thms = nat_inj_thms @ inj_thms,
- lessD = lessD @ [zless_imp_add1_zle],
- neqE = neqE,
- simpset = simpset addsimps add_rules
- addsimprocs Int_Numeral_Base_Simprocs
- addcongs [if_weak_cong]}) #>
- arith_inj_const ("Nat.of_nat", HOLogic.natT --> HOLogic.intT) #>
- arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT) #>
- arith_discrete "IntDef.int"
-
-end;
-
-val fast_int_arith_simproc =
- Simplifier.simproc @{theory}
- "fast_int_arith"
- ["(m::'a::{ordered_idom,number_ring}) < n",
- "(m::'a::{ordered_idom,number_ring}) <= n",
- "(m::'a::{ordered_idom,number_ring}) = n"] Fast_Arith.lin_arith_prover;
-
-Addsimprocs [fast_int_arith_simproc];