diff -r eef345eff987 -r 69e55066dbca src/HOL/Integ/int_arith1.ML --- 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];