(* Title: HOL/int_arith1.ML ID: $Id: int_arith1.ML,v 1.9 2007/09/18 14:08:00 wenzelm Exp $ 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 = @{thm zero_reorient} RS eq_reflection val meta_one_reorient = @{thm 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 int_ord (abs i, abs j) of EQUAL => int_ord (Int.sign i, Int.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 one_of T = Const(@{const_name HOL.one},T); (* build product with trailing 1 rather than Numeral 1 in order to avoid the unnecessary restriction to type class number_ring which is not required for cancellation of common factors in divisions. *) fun mk_prod T = let val one = one_of T 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 [] = one_of T | 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) = 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, q1), (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', one_of T)) end; (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *) val add_0s = thms "add_0s"; val mult_1s = thms "mult_1s" @ [thm"mult_1_left", thm"mult_1_right", thm"divide_1"]; (*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 @ @{thms add_ac} val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms 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 HOL.less} val dest_bal = HOLogic.dest_bin @{const_name HOL.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 HOL.less_eq} val dest_bal = HOLogic.dest_bin @{const_name HOL.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 = int val iszero = (fn x => x = 0) val add = op + 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 @ @{thms add_ac} val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms 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 = int * int val iszero = (fn (p, 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 @ @{thms add_ac} val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms 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 @{thms 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 *) (* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<", and m and n are ground terms over rings (roughly speaking). That is, m and n consist only of 1s combined with "+", "-" and "*". *) local val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0}; val lhss0 = [@{cpat "0::?'a::ring"}]; fun proc0 phi ss ct = let val T = ctyp_of_term ct in if typ_of T = @{typ int} then NONE else SOME (instantiate' [SOME T] [] zeroth) end; val zero_to_of_int_zero_simproc = make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc", proc = proc0, identifier = []}; val oneth = (symmetric o mk_meta_eq) @{thm of_int_1}; val lhss1 = [@{cpat "1::?'a::ring_1"}]; fun proc1 phi ss ct = let val T = ctyp_of_term ct in if typ_of T = @{typ int} then NONE else SOME (instantiate' [SOME T] [] oneth) end; val one_to_of_int_one_simproc = make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc", proc = proc1, identifier = []}; val allowed_consts = [@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"}, @{const_name "HOL.minus"}, @{const_name "HOL.plus"}, @{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"}, @{const_name "HOL.less_eq"}]; fun check t = case t of Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int}) else s mem_string allowed_consts | a$b => check a andalso check b | _ => false; val conv = Simplifier.rewrite (HOL_basic_ss addsimps ((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult}, @{thm of_int_diff}, @{thm of_int_minus}])@ [@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}]) addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]); fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE val lhss' = [@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"}, @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"}, @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}] in val zero_one_idom_simproc = make_simproc {lhss = lhss' , name = "zero_one_idom_simproc", proc = sproc, identifier = []} end; 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}, @{thm of_nat_0}, @{thm of_nat_1}, @{thm of_nat_Suc}, @{thm of_nat_add}, @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, @{thm of_int_add}, @{thm of_int_mult}] val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2] val Int_Numeral_Base_Simprocs = assoc_fold_simproc :: zero_one_idom_simproc :: Int_Numeral_Simprocs.combine_numerals :: Int_Numeral_Simprocs.cancel_numerals; in val int_arith_setup = LinArith.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 (@{const_name of_nat}, 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"] (K LinArith.lin_arith_simproc); Addsimprocs [fast_int_arith_simproc];