1 (* Title: HOL/Real/real_arith.ML |
1 (* Title: HOL/Real/real_arith.ML |
2 ID: $Id$ |
2 ID: $Id$ |
3 Author: Tobias Nipkow, TU Muenchen |
3 Author: Tobias Nipkow, TU Muenchen |
4 Copyright 1999 TU Muenchen |
4 Copyright 1999 TU Muenchen |
5 |
5 |
6 Instantiation of the generic linear arithmetic package for type real. |
6 Augmentation of real linear arithmetic with rational coefficient handling |
7 *) |
7 *) |
8 |
8 |
9 local |
9 local |
10 |
10 |
11 (* reduce contradictory <= to False *) |
11 (* reduce contradictory <= to False *) |
12 val simps = [order_less_irrefl, zero_eq_numeral_0, one_eq_numeral_1, |
12 val simps = [True_implies_equals,inst "w" "number_of ?v" real_add_mult_distrib2, |
13 add_real_number_of, minus_real_number_of, diff_real_number_of, |
13 real_divide_1,real_times_divide1_eq,real_times_divide2_eq]; |
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]; |
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17 |
14 |
18 val add_rules = |
15 val simprocs = [hd(rev real_cancel_numeral_factors)]; |
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 |
16 |
50 fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
17 fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
51 |
18 |
52 val real_mult_mono_thms = |
19 val real_mult_mono_thms = |
53 [(rotate_prems 1 real_mult_less_mono2, |
20 [(rotate_prems 1 real_mult_less_mono2, |
55 (real_mult_le_mono2, |
22 (real_mult_le_mono2, |
56 cvar(real_mult_le_mono2, hd(tl(prems_of real_mult_le_mono2))))] |
23 cvar(real_mult_le_mono2, hd(tl(prems_of real_mult_le_mono2))))] |
57 |
24 |
58 in |
25 in |
59 |
26 |
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 = |
27 val real_arith_setup = |
64 [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} => |
28 [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} => |
65 {add_mono_thms = add_mono_thms @ add_mono_thms_real, |
29 {add_mono_thms = add_mono_thms, |
66 mult_mono_thms = mult_mono_thms @ real_mult_mono_thms, |
30 mult_mono_thms = mult_mono_thms @ real_mult_mono_thms, |
67 inj_thms = inj_thms, (*FIXME: add real*) |
31 inj_thms = inj_thms, |
68 lessD = lessD, (*We don't change LA_Data_Ref.lessD because the real ordering is dense!*) |
32 lessD = lessD, |
69 simpset = simpset addsimps (True_implies_equals :: add_rules @ simps) |
33 simpset = simpset addsimps simps addsimprocs simprocs})]; |
70 addsimprocs simprocs |
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71 addcongs [if_weak_cong]}), |
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72 arith_discrete ("RealDef.real",false), |
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73 Simplifier.change_simpset_of (op addsimprocs) [fast_real_arith_simproc]]; |
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74 |
34 |
75 end; |
35 end; |
76 |
36 |
77 |
37 (* |
78 (* Some test data [omitting examples that assume the ordering to be discrete!] |
38 Procedure "assoc_fold" needed? |
79 Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"; |
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80 by (fast_arith_tac 1); |
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81 qed ""; |
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82 |
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83 Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c"; |
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84 by (fast_arith_tac 1); |
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85 qed ""; |
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86 |
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87 Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j"; |
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88 by (fast_arith_tac 1); |
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89 qed ""; |
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90 |
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91 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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92 by (arith_tac 1); |
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93 qed ""; |
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94 |
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95 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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96 \ ==> a <= l"; |
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97 by (fast_arith_tac 1); |
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98 qed ""; |
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99 |
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100 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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101 \ ==> a+a+a+a <= l+l+l+l"; |
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102 by (fast_arith_tac 1); |
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103 qed ""; |
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104 |
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105 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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106 \ ==> a+a+a+a+a <= l+l+l+l+i"; |
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107 by (fast_arith_tac 1); |
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108 qed ""; |
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109 |
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110 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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111 \ ==> a+a+a+a+a+a <= l+l+l+l+i+l"; |
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112 by (fast_arith_tac 1); |
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113 qed ""; |
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114 |
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115 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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116 \ ==> #6*a <= #5*l+i"; |
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117 by (fast_arith_tac 1); |
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118 qed ""; |
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119 *) |
39 *) |