(* Title: HOL/Real/real_arith.ML ID: $Id: real_arith.ML,v 1.34 2005/07/14 15:16:52 wenzelm Exp $ Author: Tobias Nipkow, TU Muenchen Copyright 1999 TU Muenchen Simprocs for common factor cancellation & Rational coefficient handling Instantiation of the generic linear arithmetic package for type real. *) val real_le_def = thm "real_le_def"; val real_diff_def = thm "real_diff_def"; val real_divide_def = thm "real_divide_def"; val realrel_in_real = thm"realrel_in_real"; val real_add_commute = thm"real_add_commute"; val real_add_assoc = thm"real_add_assoc"; val real_add_zero_left = thm"real_add_zero_left"; val real_mult_commute = thm"real_mult_commute"; val real_mult_assoc = thm"real_mult_assoc"; val real_mult_1 = thm"real_mult_1"; val real_mult_1_right = thm"real_mult_1_right"; val preal_le_linear = thm"preal_le_linear"; val real_mult_inverse_left = thm"real_mult_inverse_left"; val real_not_refl2 = thm"real_not_refl2"; val real_of_preal_add = thm"real_of_preal_add"; val real_of_preal_mult = thm"real_of_preal_mult"; val real_of_preal_trichotomy = thm"real_of_preal_trichotomy"; val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero"; val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero"; val real_of_preal_zero_less = thm"real_of_preal_zero_less"; val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq"; val real_le_refl = thm"real_le_refl"; val real_le_linear = thm"real_le_linear"; val real_le_trans = thm"real_le_trans"; val real_less_le = thm"real_less_le"; val real_less_sum_gt_zero = thm"real_less_sum_gt_zero"; val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex"; val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex"; val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex"; val real_less_all_preal = thm "real_less_all_preal"; val real_less_all_real2 = thm "real_less_all_real2"; val real_of_preal_le_iff = thm "real_of_preal_le_iff"; val real_mult_order = thm "real_mult_order"; val real_add_less_le_mono = thm "real_add_less_le_mono"; val real_add_le_less_mono = thm "real_add_le_less_mono"; val real_add_order = thm "real_add_order"; val real_le_add_order = thm "real_le_add_order"; val real_le_square = thm "real_le_square"; val real_mult_less_mono2 = thm "real_mult_less_mono2"; val real_mult_less_iff1 = thm "real_mult_less_iff1"; val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1"; val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2"; val real_mult_less_mono = thm "real_mult_less_mono"; val real_mult_less_mono' = thm "real_mult_less_mono'"; val real_sum_squares_cancel = thm "real_sum_squares_cancel"; val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2"; val real_mult_left_cancel = thm"real_mult_left_cancel"; val real_mult_right_cancel = thm"real_mult_right_cancel"; val real_inverse_unique = thm "real_inverse_unique"; val real_inverse_gt_one = thm "real_inverse_gt_one"; val real_of_int_zero = thm"real_of_int_zero"; val real_of_one = thm"real_of_one"; val real_of_int_add = thm"real_of_int_add"; val real_of_int_minus = thm"real_of_int_minus"; val real_of_int_diff = thm"real_of_int_diff"; val real_of_int_mult = thm"real_of_int_mult"; val real_of_int_real_of_nat = thm"real_of_int_real_of_nat"; val real_of_int_inject = thm"real_of_int_inject"; val real_of_int_less_iff = thm"real_of_int_less_iff"; val real_of_int_le_iff = thm"real_of_int_le_iff"; val real_of_nat_zero = thm "real_of_nat_zero"; val real_of_nat_one = thm "real_of_nat_one"; val real_of_nat_add = thm "real_of_nat_add"; val real_of_nat_Suc = thm "real_of_nat_Suc"; val real_of_nat_less_iff = thm "real_of_nat_less_iff"; val real_of_nat_le_iff = thm "real_of_nat_le_iff"; val real_of_nat_ge_zero = thm "real_of_nat_ge_zero"; val real_of_nat_Suc_gt_zero = thm "real_of_nat_Suc_gt_zero"; val real_of_nat_mult = thm "real_of_nat_mult"; val real_of_nat_inject = thm "real_of_nat_inject"; val real_of_nat_diff = thm "real_of_nat_diff"; val real_of_nat_zero_iff = thm "real_of_nat_zero_iff"; val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff"; val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff"; val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero"; val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff"; val real_number_of = thm"real_number_of"; val real_of_nat_number_of = thm"real_of_nat_number_of"; val real_of_int_of_nat_eq = thm"real_of_int_of_nat_eq"; (****Instantiation of the generic linear arithmetic package****) local val simps = [real_of_nat_zero, real_of_nat_Suc, real_of_nat_add, real_of_nat_mult, real_of_int_zero, real_of_one, real_of_int_add, real_of_int_minus, real_of_int_diff, real_of_int_mult, real_of_int_of_nat_eq, real_of_nat_number_of, real_number_of]; val int_inj_thms = [real_of_int_le_iff RS iffD2, real_of_int_less_iff RS iffD2, real_of_int_inject RS iffD2]; val nat_inj_thms = [real_of_nat_le_iff RS iffD2, real_of_nat_less_iff RS iffD2, real_of_nat_inject RS iffD2]; in val fast_real_arith_simproc = Simplifier.simproc (Theory.sign_of (the_context ())) "fast_real_arith" ["(m::real) < n","(m::real) <= n", "(m::real) = n"] Fast_Arith.lin_arith_prover; val real_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 = mult_mono_thms, inj_thms = int_inj_thms @ nat_inj_thms @ inj_thms, lessD = lessD, (*Can't change LA_Data_Ref.lessD: the reals are dense!*) neqE = neqE, simpset = simpset addsimps simps}), arith_inj_const ("RealDef.real", HOLogic.natT --> HOLogic.realT), arith_inj_const ("RealDef.real", HOLogic.intT --> HOLogic.realT), Simplifier.change_simpset_of (op addsimprocs) [fast_real_arith_simproc]]; (* some thms for injection nat => real: real_of_nat_zero real_of_nat_add *) end; (* Some test data [omitting examples that assume the ordering to be discrete!] Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k"; by (arith_tac 1); qed ""; Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ \ ==> a <= l"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ \ ==> a+a+a+a <= l+l+l+l"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ \ ==> a+a+a+a+a <= l+l+l+l+i"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ \ ==> a+a+a+a+a+a <= l+l+l+l+i+l"; by (fast_arith_tac 1); qed ""; Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ \ ==> 6*a <= 5*l+i"; by (fast_arith_tac 1); qed ""; Goal "a<=b ==> a < b+(1::real)"; by (fast_arith_tac 1); qed ""; Goal "a<=b ==> a-(3::real) < b"; by (fast_arith_tac 1); qed ""; Goal "a<=b ==> a-(1::real) < b"; by (fast_arith_tac 1); qed ""; *)