1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/Tools/isac/Knowledge/Poly.thy Wed Aug 25 16:20:07 2010 +0200
1.3 @@ -0,0 +1,147 @@
1.4 +(* WN.020812: theorems in the Reals,
1.5 + necessary for special rule sets, in addition to Isabelle2002.
1.6 + !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1.7 + !!! THIS IS THE _least_ NUMBER OF ADDITIONAL THEOREMS !!!
1.8 + !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1.9 + xxxI contain ^^^ instead of ^ in the respective theorem xxx in 2002
1.10 + changed by: Richard Lang 020912
1.11 +*)
1.12 +
1.13 +(*
1.14 + use_thy"Knowledge/Poly";
1.15 + use_thy"Poly";
1.16 + use_thy_only"Knowledge/Poly";
1.17 +
1.18 + remove_thy"Poly";
1.19 + use_thy"Knowledge/Isac";
1.20 +
1.21 +
1.22 + use"ROOT.ML";
1.23 + cd"IsacKnowledge";
1.24 + *)
1.25 +
1.26 +Poly = Simplify +
1.27 +
1.28 +(*-------------------- consts-----------------------------------------------*)
1.29 +consts
1.30 +
1.31 + is'_expanded'_in :: "[real, real] => bool" ("_ is'_expanded'_in _")
1.32 + is'_poly'_in :: "[real, real] => bool" ("_ is'_poly'_in _") (*RL DA *)
1.33 + has'_degree'_in :: "[real, real] => real" ("_ has'_degree'_in _")(*RL DA *)
1.34 + is'_polyrat'_in :: "[real, real] => bool" ("_ is'_polyrat'_in _")(*RL030626*)
1.35 +
1.36 + is'_multUnordered :: "real => bool" ("_ is'_multUnordered")
1.37 + is'_addUnordered :: "real => bool" ("_ is'_addUnordered") (*WN030618*)
1.38 + is'_polyexp :: "real => bool" ("_ is'_polyexp")
1.39 +
1.40 + Expand'_binoms
1.41 + :: "['y, \
1.42 + \ 'y] => 'y"
1.43 + ("((Script Expand'_binoms (_ =))// \
1.44 + \ (_))" 9)
1.45 +
1.46 +(*-------------------- rules------------------------------------------------*)
1.47 +rules (*.not contained in Isabelle2002,
1.48 + stated as axioms, TODO: prove as theorems;
1.49 + theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*)
1.50 +
1.51 + realpow_pow "(a ^^^ b) ^^^ c = a ^^^ (b * c)"
1.52 + realpow_addI "r ^^^ (n + m) = r ^^^ n * r ^^^ m"
1.53 + realpow_addI_assoc_l "r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s"
1.54 + realpow_addI_assoc_r "s * r ^^^ n * r ^^^ m = s * r ^^^ (n + m)"
1.55 +
1.56 + realpow_oneI "r ^^^ 1 = r"
1.57 + realpow_zeroI "r ^^^ 0 = 1"
1.58 + realpow_eq_oneI "1 ^^^ n = 1"
1.59 + realpow_multI "(r * s) ^^^ n = r ^^^ n * s ^^^ n"
1.60 + realpow_multI_poly "[| r is_polyexp; s is_polyexp |] ==> \
1.61 + \(r * s) ^^^ n = r ^^^ n * s ^^^ n"
1.62 + realpow_minus_oneI "-1 ^^^ (2 * n) = 1"
1.63 +
1.64 + realpow_twoI "r ^^^ 2 = r * r"
1.65 + realpow_twoI_assoc_l "r * (r * s) = r ^^^ 2 * s"
1.66 + realpow_twoI_assoc_r "s * r * r = s * r ^^^ 2"
1.67 + realpow_two_atom "r is_atom ==> r * r = r ^^^ 2"
1.68 + realpow_plus_1 "r * r ^^^ n = r ^^^ (n + 1)"
1.69 + realpow_plus_1_assoc_l "r * (r ^^^ m * s) = r ^^^ (1 + m) * s"
1.70 + realpow_plus_1_assoc_l2 "r ^^^ m * (r * s) = r ^^^ (1 + m) * s"
1.71 + realpow_plus_1_assoc_r "s * r * r ^^^ m = s * r ^^^ (1 + m)"
1.72 + realpow_plus_1_atom "r is_atom ==> r * r ^^^ n = r ^^^ (1 + n)"
1.73 + realpow_def_atom "[| Not (r is_atom); 1 < n |] \
1.74 + \ ==> r ^^^ n = r * r ^^^ (n + -1)"
1.75 + realpow_addI_atom "r is_atom ==> r ^^^ n * r ^^^ m = r ^^^ (n + m)"
1.76 +
1.77 +
1.78 + realpow_minus_even "n is_even ==> (- r) ^^^ n = r ^^^ n"
1.79 + realpow_minus_odd "Not (n is_even) ==> (- r) ^^^ n = -1 * r ^^^ n"
1.80 +
1.81 +
1.82 +(* RL 020914 *)
1.83 + real_pp_binom_times "(a + b)*(c + d) = a*c + a*d + b*c + b*d"
1.84 + real_pm_binom_times "(a + b)*(c - d) = a*c - a*d + b*c - b*d"
1.85 + real_mp_binom_times "(a - b)*(c + d) = a*c + a*d - b*c - b*d"
1.86 + real_mm_binom_times "(a - b)*(c - d) = a*c - a*d - b*c + b*d"
1.87 + real_plus_binom_pow3 "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
1.88 + real_plus_binom_pow3_poly "[| a is_polyexp; b is_polyexp |] ==> \
1.89 + \(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
1.90 + real_minus_binom_pow3 "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3"
1.91 + real_minus_binom_pow3_p "(a + -1 * b)^^^3 = a^^^3 + -3*a^^^2*b + 3*a*b^^^2 + -1*b^^^3"
1.92 +(* real_plus_binom_pow "[| n is_const; 3 < n |] ==> \
1.93 + \(a + b)^^^n = (a + b) * (a + b)^^^(n - 1)" *)
1.94 + real_plus_binom_pow4 "(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)"
1.95 + real_plus_binom_pow4_poly "[| a is_polyexp; b is_polyexp |] ==> \
1.96 + \(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)"
1.97 + 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)"
1.98 +
1.99 + real_plus_binom_pow5_poly "[| a is_polyexp; b is_polyexp |] ==> \
1.100 + \(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)"
1.101 +
1.102 + real_diff_plus "a - b = a + -b" (*17.3.03: do_NOT_use*)
1.103 + real_diff_minus "a - b = a + -1 * b"
1.104 + real_plus_binom_times "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2"
1.105 + real_minus_binom_times "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2"
1.106 + (*WN071229 changed for Schaerding -----vvv*)
1.107 + (*real_plus_binom_pow2 "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
1.108 + real_plus_binom_pow2 "(a + b)^^^2 = (a + b) * (a + b)"
1.109 + (*WN071229 changed for Schaerding -----^^^*)
1.110 + real_plus_binom_pow2_poly "[| a is_polyexp; b is_polyexp |] ==> \
1.111 + \(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"
1.112 + real_minus_binom_pow2 "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2"
1.113 + real_minus_binom_pow2_p "(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2"
1.114 + real_plus_minus_binom1 "(a + b)*(a - b) = a^^^2 - b^^^2"
1.115 + real_plus_minus_binom1_p "(a + b)*(a - b) = a^^^2 + -1*b^^^2"
1.116 + real_plus_minus_binom1_p_p "(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"
1.117 + real_plus_minus_binom2 "(a - b)*(a + b) = a^^^2 - b^^^2"
1.118 + real_plus_minus_binom2_p "(a - b)*(a + b) = a^^^2 + -1*b^^^2"
1.119 + real_plus_minus_binom2_p_p "(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"
1.120 + real_plus_binom_times1 "(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2"
1.121 + real_plus_binom_times2 "(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2"
1.122 +
1.123 + real_num_collect "[| l is_const; m is_const |] ==> \
1.124 + \l * n + m * n = (l + m) * n"
1.125 +(* FIXME.MG.0401: replace 'real_num_collect_assoc'
1.126 + by 'real_num_collect_assoc_l' ... are equal, introduced by MG ! *)
1.127 + real_num_collect_assoc "[| l is_const; m is_const |] ==> \
1.128 + \l * n + (m * n + k) = (l + m) * n + k"
1.129 + real_num_collect_assoc_l "[| l is_const; m is_const |] ==> \
1.130 + \l * n + (m * n + k) = (l + m)
1.131 + * n + k"
1.132 + real_num_collect_assoc_r "[| l is_const; m is_const |] ==> \
1.133 + \(k + m * n) + l * n = k + (l + m) * n"
1.134 + real_one_collect "m is_const ==> n + m * n = (1 + m) * n"
1.135 +(* FIXME.MG.0401: replace 'real_one_collect_assoc'
1.136 + by 'real_one_collect_assoc_l' ... are equal, introduced by MG ! *)
1.137 + real_one_collect_assoc "m is_const ==> n + (m * n + k) = (1 + m)* n + k"
1.138 +
1.139 + real_one_collect_assoc_l "m is_const ==> n + (m * n + k) = (1 + m) * n + k"
1.140 + real_one_collect_assoc_r "m is_const ==>(k + n) + m * n = k + (1 + m) * n"
1.141 +
1.142 +(* FIXME.MG.0401: replace 'real_mult_2_assoc'
1.143 + by 'real_mult_2_assoc_l' ... are equal, introduced by MG ! *)
1.144 + real_mult_2_assoc "z1 + (z1 + k) = 2 * z1 + k"
1.145 + real_mult_2_assoc_l "z1 + (z1 + k) = 2 * z1 + k"
1.146 + real_mult_2_assoc_r "(k + z1) + z1 = k + 2 * z1"
1.147 +
1.148 + real_add_mult_distrib_poly "w is_polyexp ==> (z1 + z2) * w = z1 * w + z2 * w"
1.149 + real_add_mult_distrib2_poly "w is_polyexp ==> w * (z1 + z2) = w * z1 + w * z2"
1.150 +end