1 (* WN.020812: theorems in the Reals,
2 necessary for special rule sets, in addition to Isabelle2002.
3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
4 !!! THIS IS THE _least_ NUMBER OF ADDITIONAL THEOREMS !!!
5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
6 xxxI contain ^^^ instead of ^ in the respective theorem xxx in 2002
7 changed by: Richard Lang 020912
11 use_thy"Knowledge/Poly";
13 use_thy_only"Knowledge/Poly";
16 use_thy"Knowledge/Isac";
25 (*-------------------- consts-----------------------------------------------*)
28 is'_expanded'_in :: "[real, real] => bool" ("_ is'_expanded'_in _")
29 is'_poly'_in :: "[real, real] => bool" ("_ is'_poly'_in _") (*RL DA *)
30 has'_degree'_in :: "[real, real] => real" ("_ has'_degree'_in _")(*RL DA *)
31 is'_polyrat'_in :: "[real, real] => bool" ("_ is'_polyrat'_in _")(*RL030626*)
33 is'_multUnordered :: "real => bool" ("_ is'_multUnordered")
34 is'_addUnordered :: "real => bool" ("_ is'_addUnordered") (*WN030618*)
35 is'_polyexp :: "real => bool" ("_ is'_polyexp")
40 ("((Script Expand'_binoms (_ =))// \
43 (*-------------------- rules------------------------------------------------*)
44 rules (*.not contained in Isabelle2002,
45 stated as axioms, TODO: prove as theorems;
46 theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*)
48 realpow_pow "(a ^^^ b) ^^^ c = a ^^^ (b * c)"
49 realpow_addI "r ^^^ (n + m) = r ^^^ n * r ^^^ m"
50 realpow_addI_assoc_l "r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s"
51 realpow_addI_assoc_r "s * r ^^^ n * r ^^^ m = s * r ^^^ (n + m)"
53 realpow_oneI "r ^^^ 1 = r"
54 realpow_zeroI "r ^^^ 0 = 1"
55 realpow_eq_oneI "1 ^^^ n = 1"
56 realpow_multI "(r * s) ^^^ n = r ^^^ n * s ^^^ n"
57 realpow_multI_poly "[| r is_polyexp; s is_polyexp |] ==> \
58 \(r * s) ^^^ n = r ^^^ n * s ^^^ n"
59 realpow_minus_oneI "-1 ^^^ (2 * n) = 1"
61 realpow_twoI "r ^^^ 2 = r * r"
62 realpow_twoI_assoc_l "r * (r * s) = r ^^^ 2 * s"
63 realpow_twoI_assoc_r "s * r * r = s * r ^^^ 2"
64 realpow_two_atom "r is_atom ==> r * r = r ^^^ 2"
65 realpow_plus_1 "r * r ^^^ n = r ^^^ (n + 1)"
66 realpow_plus_1_assoc_l "r * (r ^^^ m * s) = r ^^^ (1 + m) * s"
67 realpow_plus_1_assoc_l2 "r ^^^ m * (r * s) = r ^^^ (1 + m) * s"
68 realpow_plus_1_assoc_r "s * r * r ^^^ m = s * r ^^^ (1 + m)"
69 realpow_plus_1_atom "r is_atom ==> r * r ^^^ n = r ^^^ (1 + n)"
70 realpow_def_atom "[| Not (r is_atom); 1 < n |] \
71 \ ==> r ^^^ n = r * r ^^^ (n + -1)"
72 realpow_addI_atom "r is_atom ==> r ^^^ n * r ^^^ m = r ^^^ (n + m)"
75 realpow_minus_even "n is_even ==> (- r) ^^^ n = r ^^^ n"
76 realpow_minus_odd "Not (n is_even) ==> (- r) ^^^ n = -1 * r ^^^ n"
80 real_pp_binom_times "(a + b)*(c + d) = a*c + a*d + b*c + b*d"
81 real_pm_binom_times "(a + b)*(c - d) = a*c - a*d + b*c - b*d"
82 real_mp_binom_times "(a - b)*(c + d) = a*c + a*d - b*c - b*d"
83 real_mm_binom_times "(a - b)*(c - d) = a*c - a*d - b*c + b*d"
84 real_plus_binom_pow3 "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
85 real_plus_binom_pow3_poly "[| a is_polyexp; b is_polyexp |] ==> \
86 \(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
87 real_minus_binom_pow3 "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3"
88 real_minus_binom_pow3_p "(a + -1 * b)^^^3 = a^^^3 + -3*a^^^2*b + 3*a*b^^^2 + -1*b^^^3"
89 (* real_plus_binom_pow "[| n is_const; 3 < n |] ==> \
90 \(a + b)^^^n = (a + b) * (a + b)^^^(n - 1)" *)
91 real_plus_binom_pow4 "(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)"
92 real_plus_binom_pow4_poly "[| a is_polyexp; b is_polyexp |] ==> \
93 \(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)"
94 real_plus_binom_pow5 "(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)"
96 real_plus_binom_pow5_poly "[| a is_polyexp; b is_polyexp |] ==> \
97 \(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)"
99 real_diff_plus "a - b = a + -b" (*17.3.03: do_NOT_use*)
100 real_diff_minus "a - b = a + -1 * b"
101 real_plus_binom_times "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2"
102 real_minus_binom_times "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2"
103 (*WN071229 changed for Schaerding -----vvv*)
104 (*real_plus_binom_pow2 "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
105 real_plus_binom_pow2 "(a + b)^^^2 = (a + b) * (a + b)"
106 (*WN071229 changed for Schaerding -----^^^*)
107 real_plus_binom_pow2_poly "[| a is_polyexp; b is_polyexp |] ==> \
108 \(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"
109 real_minus_binom_pow2 "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2"
110 real_minus_binom_pow2_p "(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2"
111 real_plus_minus_binom1 "(a + b)*(a - b) = a^^^2 - b^^^2"
112 real_plus_minus_binom1_p "(a + b)*(a - b) = a^^^2 + -1*b^^^2"
113 real_plus_minus_binom1_p_p "(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"
114 real_plus_minus_binom2 "(a - b)*(a + b) = a^^^2 - b^^^2"
115 real_plus_minus_binom2_p "(a - b)*(a + b) = a^^^2 + -1*b^^^2"
116 real_plus_minus_binom2_p_p "(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"
117 real_plus_binom_times1 "(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2"
118 real_plus_binom_times2 "(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2"
120 real_num_collect "[| l is_const; m is_const |] ==> \
121 \l * n + m * n = (l + m) * n"
122 (* FIXME.MG.0401: replace 'real_num_collect_assoc'
123 by 'real_num_collect_assoc_l' ... are equal, introduced by MG ! *)
124 real_num_collect_assoc "[| l is_const; m is_const |] ==> \
125 \l * n + (m * n + k) = (l + m) * n + k"
126 real_num_collect_assoc_l "[| l is_const; m is_const |] ==> \
127 \l * n + (m * n + k) = (l + m)
129 real_num_collect_assoc_r "[| l is_const; m is_const |] ==> \
130 \(k + m * n) + l * n = k + (l + m) * n"
131 real_one_collect "m is_const ==> n + m * n = (1 + m) * n"
132 (* FIXME.MG.0401: replace 'real_one_collect_assoc'
133 by 'real_one_collect_assoc_l' ... are equal, introduced by MG ! *)
134 real_one_collect_assoc "m is_const ==> n + (m * n + k) = (1 + m)* n + k"
136 real_one_collect_assoc_l "m is_const ==> n + (m * n + k) = (1 + m) * n + k"
137 real_one_collect_assoc_r "m is_const ==>(k + n) + m * n = k + (1 + m) * n"
139 (* FIXME.MG.0401: replace 'real_mult_2_assoc'
140 by 'real_mult_2_assoc_l' ... are equal, introduced by MG ! *)
141 real_mult_2_assoc "z1 + (z1 + k) = 2 * z1 + k"
142 real_mult_2_assoc_l "z1 + (z1 + k) = 2 * z1 + k"
143 real_mult_2_assoc_r "(k + z1) + z1 = k + 2 * z1"
145 real_add_mult_distrib_poly "w is_polyexp ==> (z1 + z2) * w = z1 * w + z2 * w"
146 real_add_mult_distrib2_poly "w is_polyexp ==> w * (z1 + z2) = w * z1 + w * z2"