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1 (* Title: HOL/Real/real_arith.ML |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow, TU Muenchen |
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4 Copyright 1999 TU Muenchen |
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5 |
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6 Instantiation of the generic linear arithmetic package for type real. |
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7 *) |
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8 |
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9 local |
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10 |
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11 (* reduce contradictory <= to False *) |
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12 val simps = [order_less_irrefl, zero_eq_numeral_0, one_eq_numeral_1, |
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13 add_real_number_of, minus_real_number_of, diff_real_number_of, |
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14 mult_real_number_of, eq_real_number_of, less_real_number_of, |
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15 le_real_number_of_eq_not_less, real_diff_def, |
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16 real_minus_add_distrib, real_minus_minus, real_mult_assoc]; |
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17 |
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18 val add_rules = |
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19 map rename_numerals |
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20 [real_add_zero_left, real_add_zero_right, |
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21 real_add_minus, real_add_minus_left, |
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22 real_mult_0, real_mult_0_right, |
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23 real_mult_1, real_mult_1_right, |
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24 real_mult_minus_1, real_mult_minus_1_right]; |
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25 |
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26 val simprocs = [Real_Times_Assoc.conv, Real_Numeral_Simprocs.combine_numerals]@ |
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27 Real_Numeral_Simprocs.cancel_numerals(* @ real_cancel_numeral_factors*); |
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28 |
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29 val mono_ss = simpset() addsimps |
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30 [real_add_le_mono,real_add_less_mono, |
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31 real_add_less_le_mono,real_add_le_less_mono]; |
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32 |
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33 val add_mono_thms_real = |
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34 map (fn s => prove_goal (the_context ()) s |
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35 (fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1])) |
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36 ["(i <= j) & (k <= l) ==> i + k <= j + (l::real)", |
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37 "(i = j) & (k <= l) ==> i + k <= j + (l::real)", |
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38 "(i <= j) & (k = l) ==> i + k <= j + (l::real)", |
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39 "(i = j) & (k = l) ==> i + k = j + (l::real)", |
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40 "(i < j) & (k = l) ==> i + k < j + (l::real)", |
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41 "(i = j) & (k < l) ==> i + k < j + (l::real)", |
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42 "(i < j) & (k <= l) ==> i + k < j + (l::real)", |
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43 "(i <= j) & (k < l) ==> i + k < j + (l::real)", |
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44 "(i < j) & (k < l) ==> i + k < j + (l::real)"]; |
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45 |
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46 val real_arith_simproc_pats = |
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47 map (fn s => Thm.read_cterm (Theory.sign_of (the_context ())) (s, HOLogic.boolT)) |
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48 ["(m::real) < n","(m::real) <= n", "(m::real) = n"]; |
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49 |
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50 fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
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51 |
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52 val real_mult_mono_thms = |
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53 [(rotate_prems 1 real_mult_less_mono2, |
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54 cvar(real_mult_less_mono2, hd(prems_of real_mult_less_mono2))), |
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55 (real_mult_le_mono2, |
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56 cvar(real_mult_le_mono2, hd(tl(prems_of real_mult_le_mono2))))] |
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57 |
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58 in |
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59 |
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60 val fast_real_arith_simproc = mk_simproc |
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61 "fast_real_arith" real_arith_simproc_pats Fast_Arith.lin_arith_prover; |
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62 |
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63 val real_arith_setup = |
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64 [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} => |
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65 {add_mono_thms = add_mono_thms @ add_mono_thms_real, |
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66 mult_mono_thms = mult_mono_thms @ real_mult_mono_thms, |
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67 inj_thms = inj_thms, (*FIXME: add real*) |
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68 lessD = lessD, (*We don't change LA_Data_Ref.lessD because the real ordering is dense!*) |
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69 simpset = simpset addsimps (add_rules @ simps) |
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70 addsimprocs simprocs}), |
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71 arith_discrete ("RealDef.real",false), |
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72 Simplifier.change_simpset_of (op addsimprocs) [fast_real_arith_simproc]]; |
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73 |
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74 end; |
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75 |
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76 |
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77 (* Some test data [omitting examples that assume the ordering to be discrete!] |
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78 Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"; |
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79 by (fast_arith_tac 1); |
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80 qed ""; |
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81 |
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82 Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c"; |
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83 by (fast_arith_tac 1); |
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84 qed ""; |
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85 |
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86 Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j"; |
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87 by (fast_arith_tac 1); |
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88 qed ""; |
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89 |
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90 Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k"; |
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91 by (arith_tac 1); |
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92 qed ""; |
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93 |
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94 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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95 \ ==> a <= l"; |
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96 by (fast_arith_tac 1); |
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97 qed ""; |
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98 |
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99 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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100 \ ==> a+a+a+a <= l+l+l+l"; |
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101 by (fast_arith_tac 1); |
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102 qed ""; |
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103 |
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104 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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105 \ ==> a+a+a+a+a <= l+l+l+l+i"; |
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106 by (fast_arith_tac 1); |
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107 qed ""; |
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108 |
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109 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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110 \ ==> a+a+a+a+a+a <= l+l+l+l+i+l"; |
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111 by (fast_arith_tac 1); |
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112 qed ""; |
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113 |
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114 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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115 \ ==> #6*a <= #5*l+i"; |
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116 by (fast_arith_tac 1); |
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117 qed ""; |
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118 *) |