author | paulson |
Tue, 16 Jan 2001 12:20:52 +0100 | |
changeset 10919 | 144ede948e58 |
parent 10784 | 27e4d90b35b5 |
child 11701 | 3d51fbf81c17 |
permissions | -rw-r--r-- |
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1 |
(* Title : RealPow.ML |
7219 | 2 |
ID : $Id$ |
7077
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3 |
Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
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parents:
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5 |
Description : Natural Powers of reals theory |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
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6 |
*) |
60b098bb8b8a
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parents:
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7 |
|
10690 | 8 |
bind_thm ("realpow_Suc", thm "realpow_Suc"); |
9 |
||
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
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10 |
Goal "(#0::real) ^ (Suc n) = #0"; |
10677 | 11 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
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12 |
qed "realpow_zero"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
13 |
Addsimps [realpow_zero]; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
14 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
15 |
Goal "r ~= (#0::real) --> r ^ n ~= #0"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
16 |
by (induct_tac "n" 1); |
9043
ca761fe227d8
First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
17 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
18 |
qed_spec_mp "realpow_not_zero"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
19 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
20 |
Goal "r ^ n = (#0::real) ==> r = #0"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
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|
21 |
by (rtac ccontr 1); |
9043
ca761fe227d8
First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
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|
22 |
by (auto_tac (claset() addDs [realpow_not_zero], simpset())); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
23 |
qed "realpow_zero_zero"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
24 |
|
10648 | 25 |
Goal "inverse ((r::real) ^ n) = (inverse r) ^ n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
26 |
by (induct_tac "n" 1); |
10648 | 27 |
by (auto_tac (claset(), simpset() addsimps [real_inverse_distrib])); |
28 |
qed "realpow_inverse"; |
|
7077
60b098bb8b8a
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paulson
parents:
diff
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|
29 |
|
8838 | 30 |
Goal "abs (r::real) ^ n = abs (r ^ n)"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
31 |
by (induct_tac "n" 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
32 |
by (auto_tac (claset(), simpset() addsimps [abs_mult])); |
8838 | 33 |
qed "realpow_abs"; |
7077
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parents:
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|
34 |
|
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
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|
35 |
Goal "(r::real) ^ (n + m) = (r ^ n) * (r ^ m)"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
36 |
by (induct_tac "n" 1); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
37 |
by (auto_tac (claset(),simpset() addsimps real_mult_ac)); |
60b098bb8b8a
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paulson
parents:
diff
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|
38 |
qed "realpow_add"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
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|
39 |
|
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
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|
40 |
Goal "(r::real) ^ 1 = r"; |
60b098bb8b8a
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paulson
parents:
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41 |
by (Simp_tac 1); |
60b098bb8b8a
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parents:
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42 |
qed "realpow_one"; |
60b098bb8b8a
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paulson
parents:
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|
43 |
Addsimps [realpow_one]; |
60b098bb8b8a
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parents:
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|
44 |
|
10712 | 45 |
Goal "(r::real)^2 = r * r"; |
7077
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parents:
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|
46 |
by (Simp_tac 1); |
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heavily revised by Jacques: coercions have alphabetic names;
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parents:
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47 |
qed "realpow_two"; |
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paulson
parents:
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48 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
49 |
Goal "(#0::real) < r --> #0 < r ^ n"; |
7077
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paulson
parents:
diff
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|
50 |
by (induct_tac "n" 1); |
9428 | 51 |
by (auto_tac (claset() addIs [rename_numerals real_mult_order], |
9070 | 52 |
simpset() addsimps [real_zero_less_one])); |
7077
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parents:
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53 |
qed_spec_mp "realpow_gt_zero"; |
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54 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
55 |
Goal "(#0::real) <= r --> #0 <= r ^ n"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
56 |
by (induct_tac "n" 1); |
10784 | 57 |
by (auto_tac (claset(), simpset() addsimps [real_0_le_mult_iff])); |
58 |
qed_spec_mp "realpow_ge_zero"; |
|
7077
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parents:
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59 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
60 |
Goal "(#0::real) <= x & x <= y --> x ^ n <= y ^ n"; |
7077
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parents:
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|
61 |
by (induct_tac "n" 1); |
10784 | 62 |
by (auto_tac (claset() addSIs [real_mult_le_mono], simpset())); |
63 |
by (asm_simp_tac (simpset() addsimps [realpow_ge_zero]) 1); |
|
64 |
qed_spec_mp "realpow_le"; |
|
7077
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parents:
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65 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
66 |
Goal "(#0::real) < x & x < y & 0 < n --> x ^ n < y ^ n"; |
7077
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paulson
parents:
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|
67 |
by (induct_tac "n" 1); |
9428 | 68 |
by (auto_tac (claset() addIs [rename_numerals real_mult_less_mono, gr0I] |
9070 | 69 |
addDs [realpow_gt_zero], |
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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70 |
simpset())); |
7077
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71 |
qed_spec_mp "realpow_less"; |
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parents:
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72 |
|
10752
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tidying, and separation of HOL-Hyperreal from HOL-Real
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73 |
Goal "#1 ^ n = (#1::real)"; |
7077
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paulson
parents:
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|
74 |
by (induct_tac "n" 1); |
10677 | 75 |
by Auto_tac; |
7077
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parents:
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|
76 |
qed "realpow_eq_one"; |
60b098bb8b8a
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parents:
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|
77 |
Addsimps [realpow_eq_one]; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
78 |
|
10677 | 79 |
Goal "abs((#-1) ^ n) = (#1::real)"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
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parents:
diff
changeset
|
80 |
by (induct_tac "n" 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
81 |
by (auto_tac (claset(), simpset() addsimps [abs_mult])); |
8838 | 82 |
qed "abs_realpow_minus_one"; |
83 |
Addsimps [abs_realpow_minus_one]; |
|
7077
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|
84 |
|
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
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parents:
diff
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|
85 |
Goal "((r::real) * s) ^ n = (r ^ n) * (s ^ n)"; |
60b098bb8b8a
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parents:
diff
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|
86 |
by (induct_tac "n" 1); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
87 |
by (auto_tac (claset(),simpset() addsimps real_mult_ac)); |
60b098bb8b8a
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paulson
parents:
diff
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|
88 |
qed "realpow_mult"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
89 |
|
10712 | 90 |
Goal "(#0::real) <= r^2"; |
9428 | 91 |
by (simp_tac (simpset() addsimps [rename_numerals real_le_square]) 1); |
7077
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92 |
qed "realpow_two_le"; |
60b098bb8b8a
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paulson
parents:
diff
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|
93 |
Addsimps [realpow_two_le]; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
94 |
|
10712 | 95 |
Goal "abs((x::real)^2) = x^2"; |
9070 | 96 |
by (simp_tac (simpset() addsimps [abs_eqI1, |
9428 | 97 |
rename_numerals real_le_square]) 1); |
8838 | 98 |
qed "abs_realpow_two"; |
99 |
Addsimps [abs_realpow_two]; |
|
7077
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paulson
parents:
diff
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100 |
|
10712 | 101 |
Goal "abs(x::real) ^ 2 = x^2"; |
10690 | 102 |
by (simp_tac (simpset() addsimps [realpow_abs,abs_eqI1] |
103 |
delsimps [realpow_Suc]) 1); |
|
8838 | 104 |
qed "realpow_two_abs"; |
105 |
Addsimps [realpow_two_abs]; |
|
7077
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paulson
parents:
diff
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|
106 |
|
10712 | 107 |
Goal "(#1::real) < r ==> #1 < r^2"; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
108 |
by Auto_tac; |
9428 | 109 |
by (cut_facts_tac [rename_numerals real_zero_less_one] 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
110 |
by (forw_inst_tac [("x","#0")] order_less_trans 1); |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
111 |
by (assume_tac 1); |
9070 | 112 |
by (dres_inst_tac [("z","r"),("x","#1")] |
9428 | 113 |
(rename_numerals real_mult_less_mono1) 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
114 |
by (auto_tac (claset() addIs [order_less_trans], simpset())); |
7077
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paulson
parents:
diff
changeset
|
115 |
qed "realpow_two_gt_one"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
116 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
117 |
Goal "(#1::real) < r --> #1 <= r ^ n"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
118 |
by (induct_tac "n" 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
119 |
by Auto_tac; |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
120 |
by (subgoal_tac "#1*#1 <= r * r^n" 1); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
121 |
by (rtac real_mult_le_mono 2); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
122 |
by Auto_tac; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
123 |
qed_spec_mp "realpow_ge_one"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
124 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
125 |
Goal "(#1::real) <= r ==> #1 <= r ^ n"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
126 |
by (dtac order_le_imp_less_or_eq 1); |
7588
26384af93359
Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or
paulson
parents:
7292
diff
changeset
|
127 |
by (auto_tac (claset() addDs [realpow_ge_one], simpset())); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
128 |
qed "realpow_ge_one2"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
129 |
|
7588
26384af93359
Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or
paulson
parents:
7292
diff
changeset
|
130 |
Goal "(#1::real) <= #2 ^ n"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
131 |
by (res_inst_tac [("y","#1 ^ n")] order_trans 1); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
132 |
by (rtac realpow_le 2); |
10778
2c6605049646
more tidying, especially to remove real_of_posnat
paulson
parents:
10752
diff
changeset
|
133 |
by (auto_tac (claset() addIs [order_less_imp_le], simpset())); |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
134 |
qed "two_realpow_ge_one"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
135 |
|
10919
144ede948e58
renamings: real_of_nat, real_of_int -> (overloaded) real
paulson
parents:
10784
diff
changeset
|
136 |
Goal "real (n::nat) < #2 ^ n"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
137 |
by (induct_tac "n" 1); |
10778
2c6605049646
more tidying, especially to remove real_of_posnat
paulson
parents:
10752
diff
changeset
|
138 |
by (auto_tac (claset(), simpset() addsimps [real_of_nat_Suc])); |
9070 | 139 |
by (stac real_mult_2 1); |
140 |
by (rtac real_add_less_le_mono 1); |
|
10778
2c6605049646
more tidying, especially to remove real_of_posnat
paulson
parents:
10752
diff
changeset
|
141 |
by (auto_tac (claset(), simpset() addsimps [two_realpow_ge_one])); |
7077
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paulson
parents:
diff
changeset
|
142 |
qed "two_realpow_gt"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
143 |
Addsimps [two_realpow_gt,two_realpow_ge_one]; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
144 |
|
10677 | 145 |
Goal "(#-1) ^ (#2*n) = (#1::real)"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
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|
146 |
by (induct_tac "n" 1); |
10677 | 147 |
by Auto_tac; |
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148 |
qed "realpow_minus_one"; |
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paulson
parents:
diff
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|
149 |
Addsimps [realpow_minus_one]; |
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paulson
parents:
diff
changeset
|
150 |
|
10677 | 151 |
Goal "(#-1) ^ Suc (#2*n) = -(#1::real)"; |
152 |
by Auto_tac; |
|
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|
153 |
qed "realpow_minus_one_odd"; |
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parents:
diff
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|
154 |
Addsimps [realpow_minus_one_odd]; |
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paulson
parents:
diff
changeset
|
155 |
|
10677 | 156 |
Goal "(#-1) ^ Suc (Suc (#2*n)) = (#1::real)"; |
157 |
by Auto_tac; |
|
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parents:
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|
158 |
qed "realpow_minus_one_even"; |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
159 |
Addsimps [realpow_minus_one_even]; |
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parents:
diff
changeset
|
160 |
|
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|
161 |
Goal "(#0::real) < r & r < (#1::real) --> r ^ Suc n < r ^ n"; |
7077
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parents:
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|
162 |
by (induct_tac "n" 1); |
10677 | 163 |
by Auto_tac; |
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parents:
diff
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|
164 |
qed_spec_mp "realpow_Suc_less"; |
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parents:
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|
165 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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|
166 |
Goal "#0 <= r & r < (#1::real) --> r ^ Suc n <= r ^ n"; |
7077
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paulson
parents:
diff
changeset
|
167 |
by (induct_tac "n" 1); |
10752
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tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
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|
168 |
by (auto_tac (claset() addIs [order_less_imp_le] |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
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|
169 |
addSDs [order_le_imp_less_or_eq], |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
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|
170 |
simpset())); |
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parents:
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|
171 |
qed_spec_mp "realpow_Suc_le"; |
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parents:
diff
changeset
|
172 |
|
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|
173 |
Goal "(#0::real) <= #0 ^ n"; |
8442
96023903c2df
case_tac now subsumes both boolean and datatype cases;
wenzelm
parents:
8423
diff
changeset
|
174 |
by (case_tac "n" 1); |
10677 | 175 |
by Auto_tac; |
7077
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parents:
diff
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|
176 |
qed "realpow_zero_le"; |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
177 |
Addsimps [realpow_zero_le]; |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
178 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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changeset
|
179 |
Goal "#0 < r & r < (#1::real) --> r ^ Suc n <= r ^ n"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
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parents:
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|
180 |
by (blast_tac (claset() addSIs [order_less_imp_le, |
7077
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|
181 |
realpow_Suc_less]) 1); |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
182 |
qed_spec_mp "realpow_Suc_le2"; |
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paulson
parents:
diff
changeset
|
183 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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changeset
|
184 |
Goal "[| #0 <= r; r < (#1::real) |] ==> r ^ Suc n <= r ^ n"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
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parents:
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changeset
|
185 |
by (etac (order_le_imp_less_or_eq RS disjE) 1); |
7077
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paulson
parents:
diff
changeset
|
186 |
by (rtac realpow_Suc_le2 1); |
10677 | 187 |
by Auto_tac; |
7077
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paulson
parents:
diff
changeset
|
188 |
qed "realpow_Suc_le3"; |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
189 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
190 |
Goal "#0 <= r & r < (#1::real) & n < N --> r ^ N <= r ^ n"; |
7077
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heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
191 |
by (induct_tac "N" 1); |
10784 | 192 |
by (ALLGOALS Asm_simp_tac); |
193 |
by (Clarify_tac 1); |
|
194 |
by (subgoal_tac "r * r ^ na <= #1 * r ^ n" 1); |
|
195 |
by (Asm_full_simp_tac 1); |
|
196 |
by (rtac real_mult_le_mono 1); |
|
197 |
by (auto_tac (claset(), simpset() addsimps [realpow_ge_zero, less_Suc_eq])); |
|
7077
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parents:
diff
changeset
|
198 |
qed_spec_mp "realpow_less_le"; |
60b098bb8b8a
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paulson
parents:
diff
changeset
|
199 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
200 |
Goal "[| #0 <= r; r < (#1::real); n <= N |] ==> r ^ N <= r ^ n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
201 |
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
202 |
by (auto_tac (claset() addIs [realpow_less_le], |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
203 |
simpset())); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
204 |
qed "realpow_le_le"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
205 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
206 |
Goal "[| #0 < r; r < (#1::real) |] ==> r ^ Suc n <= r"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
207 |
by (dres_inst_tac [("n","1"),("N","Suc n")] |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
208 |
(order_less_imp_le RS realpow_le_le) 1); |
10677 | 209 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
210 |
qed "realpow_Suc_le_self"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
211 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
212 |
Goal "[| #0 < r; r < (#1::real) |] ==> r ^ Suc n < #1"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
213 |
by (blast_tac (claset() addIs [realpow_Suc_le_self, order_le_less_trans]) 1); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
214 |
qed "realpow_Suc_less_one"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
215 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
216 |
Goal "(#1::real) <= r --> r ^ n <= r ^ Suc n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
217 |
by (induct_tac "n" 1); |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
218 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
219 |
qed_spec_mp "realpow_le_Suc"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
220 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
221 |
Goal "(#1::real) < r --> r ^ n < r ^ Suc n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
222 |
by (induct_tac "n" 1); |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
223 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
224 |
qed_spec_mp "realpow_less_Suc"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
225 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
226 |
Goal "(#1::real) < r --> r ^ n <= r ^ Suc n"; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
227 |
by (blast_tac (claset() addSIs [order_less_imp_le, realpow_less_Suc]) 1); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
228 |
qed_spec_mp "realpow_le_Suc2"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
229 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
230 |
Goal "(#1::real) < r & n < N --> r ^ n <= r ^ N"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
231 |
by (induct_tac "N" 1); |
10677 | 232 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
233 |
by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one)); |
9428 | 234 |
by (ALLGOALS(dtac (rename_numerals real_mult_self_le))); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
235 |
by (assume_tac 1); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
236 |
by (assume_tac 2); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
237 |
by (auto_tac (claset() addIs [order_trans], |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
238 |
simpset() addsimps [less_Suc_eq])); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
239 |
qed_spec_mp "realpow_gt_ge"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
240 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
241 |
Goal "(#1::real) <= r & n < N --> r ^ n <= r ^ N"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
242 |
by (induct_tac "N" 1); |
10677 | 243 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
244 |
by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one2)); |
9428 | 245 |
by (ALLGOALS(dtac (rename_numerals real_mult_self_le2))); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
246 |
by (assume_tac 1); |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
247 |
by (assume_tac 2); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
248 |
by (auto_tac (claset() addIs [order_trans], |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
249 |
simpset() addsimps [less_Suc_eq])); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
250 |
qed_spec_mp "realpow_gt_ge2"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
251 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
252 |
Goal "[| (#1::real) < r; n <= N |] ==> r ^ n <= r ^ N"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
253 |
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
254 |
by (auto_tac (claset() addIs [realpow_gt_ge], simpset())); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
255 |
qed "realpow_ge_ge"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
256 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
257 |
Goal "[| (#1::real) <= r; n <= N |] ==> r ^ n <= r ^ N"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
258 |
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1); |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
259 |
by (auto_tac (claset() addIs [realpow_gt_ge2], simpset())); |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
260 |
qed "realpow_ge_ge2"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
261 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
262 |
Goal "(#1::real) < r ==> r <= r ^ Suc n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
263 |
by (dres_inst_tac [("n","1"),("N","Suc n")] |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
264 |
realpow_ge_ge 1); |
10677 | 265 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
266 |
qed_spec_mp "realpow_Suc_ge_self"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
267 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
268 |
Goal "(#1::real) <= r ==> r <= r ^ Suc n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
269 |
by (dres_inst_tac [("n","1"),("N","Suc n")] |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
270 |
realpow_ge_ge2 1); |
10677 | 271 |
by Auto_tac; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
272 |
qed_spec_mp "realpow_Suc_ge_self2"; |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
273 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
274 |
Goal "[| (#1::real) < r; 0 < n |] ==> r <= r ^ n"; |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
275 |
by (dtac (less_not_refl2 RS not0_implies_Suc) 1); |
60b098bb8b8a
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276 |
by (auto_tac (claset() addSIs |
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277 |
[realpow_Suc_ge_self],simpset())); |
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278 |
qed "realpow_ge_self"; |
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|
279 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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280 |
Goal "[| (#1::real) <= r; 0 < n |] ==> r <= r ^ n"; |
7077
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|
281 |
by (dtac (less_not_refl2 RS not0_implies_Suc) 1); |
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parents:
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|
282 |
by (auto_tac (claset() addSIs [realpow_Suc_ge_self2],simpset())); |
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|
283 |
qed "realpow_ge_self2"; |
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parents:
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|
284 |
|
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|
285 |
Goal "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"; |
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parents:
diff
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|
286 |
by (induct_tac "n" 1); |
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parents:
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|
287 |
by (auto_tac (claset(),simpset() |
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|
288 |
addsimps [real_mult_commute])); |
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|
289 |
qed_spec_mp "realpow_minus_mult"; |
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|
290 |
Addsimps [realpow_minus_mult]; |
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parents:
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|
291 |
|
10606 | 292 |
Goal "r ~= #0 ==> r * inverse r ^ 2 = inverse (r::real)"; |
7077
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parents:
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|
293 |
by (asm_simp_tac (simpset() addsimps [realpow_two, |
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294 |
real_mult_assoc RS sym]) 1); |
10606 | 295 |
qed "realpow_two_mult_inverse"; |
296 |
Addsimps [realpow_two_mult_inverse]; |
|
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|
297 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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298 |
(* 05/00 *) |
10712 | 299 |
Goal "(-x)^2 = (x::real) ^ 2"; |
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Updated files to remove 0r and 1r from theorems in descendant theories
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|
300 |
by (Simp_tac 1); |
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Updated files to remove 0r and 1r from theorems in descendant theories
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301 |
qed "realpow_two_minus"; |
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|
302 |
Addsimps [realpow_two_minus]; |
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Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or
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303 |
|
10712 | 304 |
Goalw [real_diff_def] "(x::real)^2 - y^2 = (x - y) * (x + y)"; |
305 |
by (simp_tac (simpset() addsimps |
|
306 |
[real_add_mult_distrib2, real_add_mult_distrib, |
|
307 |
real_minus_mult_eq2 RS sym] @ real_mult_ac) 1); |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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changeset
|
308 |
qed "realpow_two_diff"; |
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Updated files to remove 0r and 1r from theorems in descendant theories
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changeset
|
309 |
|
10712 | 310 |
Goalw [real_diff_def] "((x::real)^2 = y^2) = (x = y | x = -y)"; |
311 |
by (cut_inst_tac [("x","x"),("y","y")] realpow_two_diff 1); |
|
312 |
by (auto_tac (claset(), simpset() delsimps [realpow_Suc])); |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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changeset
|
313 |
qed "realpow_two_disj"; |
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Updated files to remove 0r and 1r from theorems in descendant theories
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changeset
|
314 |
|
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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changeset
|
315 |
(* used in Transc *) |
10606 | 316 |
Goal "[|(x::real) ~= #0; m <= n |] ==> x ^ (n - m) = x ^ n * inverse (x ^ m)"; |
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parents:
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|
317 |
by (auto_tac (claset(), |
10712 | 318 |
simpset() addsimps [le_eq_less_or_eq, less_iff_Suc_add, realpow_add, |
319 |
realpow_not_zero] @ real_mult_ac)); |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
8838
diff
changeset
|
320 |
qed "realpow_diff"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
321 |
|
10919
144ede948e58
renamings: real_of_nat, real_of_int -> (overloaded) real
paulson
parents:
10784
diff
changeset
|
322 |
Goal "real (m::nat) ^ n = real (m ^ n)"; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
323 |
by (induct_tac "n" 1); |
9043
ca761fe227d8
First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
324 |
by (auto_tac (claset(), |
ca761fe227d8
First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
325 |
simpset() addsimps [real_of_nat_one, real_of_nat_mult])); |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
326 |
qed "realpow_real_of_nat"; |
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parents:
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changeset
|
327 |
|
10919
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renamings: real_of_nat, real_of_int -> (overloaded) real
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parents:
10784
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changeset
|
328 |
Goal "#0 < real (2 ^ n)"; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
329 |
by (induct_tac "n" 1); |
9070 | 330 |
by (auto_tac (claset(), |
331 |
simpset() addsimps [real_of_nat_mult, real_zero_less_mult_iff])); |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
332 |
qed "realpow_real_of_nat_two_pos"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
333 |
Addsimps [realpow_real_of_nat_two_pos]; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
334 |
|
9043
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First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
335 |
|
9070 | 336 |
Goal "(#0::real) <= x --> #0 <= y --> x ^ Suc n <= y ^ Suc n --> x <= y"; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
337 |
by (induct_tac "n" 1); |
9043
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First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
338 |
by Auto_tac; |
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
339 |
by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
340 |
by (swap_res_tac [real_mult_less_mono'] 1); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
341 |
by Auto_tac; |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
342 |
by (auto_tac (claset(), simpset() addsimps [real_0_le_mult_iff])); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
343 |
by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym])); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
344 |
by (dres_inst_tac [("n","n")] realpow_gt_zero 1); |
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10715
diff
changeset
|
345 |
by Auto_tac; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
346 |
qed_spec_mp "realpow_increasing"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
347 |
|
9070 | 348 |
Goal "[| (#0::real) <= x; #0 <= y; x ^ Suc n = y ^ Suc n |] ==> x = y"; |
349 |
by (blast_tac (claset() addIs [realpow_increasing, order_antisym, |
|
350 |
order_eq_refl, sym]) 1); |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
351 |
qed_spec_mp "realpow_Suc_cancel_eq"; |