isabelle: src/HOL/Real/RealPow.ML@5c44de6aadf4 (annotated)

7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	1	(* Title : RealPow.ML
7219 4e3f386c2e37 inserted Id: lines paulson parents: 7077 diff changeset	2	ID : $Id$
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	3	Author : Jacques D. Fleuriot
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	4	Copyright : 1998 University of Cambridge
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	5	Description : Natural Powers of reals theory
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	6	*)
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	7
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	8	Goal "(#0::real) ^ (Suc n) = #0";
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	9	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	10	qed "realpow_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	11	Addsimps [realpow_zero];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	12
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	13	Goal "r ~= (#0::real) --> r ^ n ~= #0";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	14	by (induct_tac "n" 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	15	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	16	qed_spec_mp "realpow_not_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	17
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	18	Goal "r ^ n = (#0::real) ==> r = #0";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	19	by (rtac ccontr 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	20	by (auto_tac (claset() addDs [realpow_not_zero], simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	21	qed "realpow_zero_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	22
10648 a8c647cfa31f first stage in tidying up Real and Hyperreal. paulson parents: 10606 diff changeset	23	Goal "inverse ((r::real) ^ n) = (inverse r) ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	24	by (induct_tac "n" 1);
10648 a8c647cfa31f first stage in tidying up Real and Hyperreal. paulson parents: 10606 diff changeset	25	by (auto_tac (claset(), simpset() addsimps [real_inverse_distrib]));
a8c647cfa31f first stage in tidying up Real and Hyperreal. paulson parents: 10606 diff changeset	26	qed "realpow_inverse";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	27
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	28	Goal "abs (r::real) ^ n = abs (r ^ n)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	29	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	30	by (auto_tac (claset(),simpset() addsimps
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	31	[abs_mult,abs_one]));
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	32	qed "realpow_abs";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	33
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	34	Goal "(r::real) ^ (n + m) = (r ^ n) * (r ^ m)";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	35	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	36	by (auto_tac (claset(),simpset() addsimps real_mult_ac));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	37	qed "realpow_add";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	38
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	39	Goal "(r::real) ^ 1 = r";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	40	by (Simp_tac 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	41	qed "realpow_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	42	Addsimps [realpow_one];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	43
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	44	Goal "(r::real) ^ 2 = r * r";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	45	by (Simp_tac 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	46	qed "realpow_two";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	47
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	48	Goal "(#0::real) < r --> #0 <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	49	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	50	by (auto_tac (claset() addDs [real_less_imp_le]
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	51	addIs [rename_numerals real_le_mult_order],
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	52	simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	53	qed_spec_mp "realpow_ge_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	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 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	56	by (induct_tac "n" 1);
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	57	by (auto_tac (claset() addIs [rename_numerals real_mult_order],
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	58	simpset() addsimps [real_zero_less_one]));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	59	qed_spec_mp "realpow_gt_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	60
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	61	Goal "(#0::real) <= r --> #0 <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	62	by (induct_tac "n" 1);
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	63	by (auto_tac (claset() addIs [rename_numerals real_le_mult_order],
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	64	simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	65	qed_spec_mp "realpow_ge_zero2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	66
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	67	Goal "(#0::real) < x & x <= y --> x ^ n <= y ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	68	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	69	by (auto_tac (claset() addSIs [real_mult_le_mono],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	70	simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	71	by (asm_simp_tac (simpset() addsimps [realpow_ge_zero]) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	72	qed_spec_mp "realpow_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	73
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	74	Goal "(#0::real) <= x & x <= y --> x ^ n <= y ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	75	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	76	by (auto_tac (claset() addSIs [real_mult_le_mono4],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	77	simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	78	by (asm_simp_tac (simpset() addsimps [realpow_ge_zero2]) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	79	qed_spec_mp "realpow_le2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	80
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	81	Goal "(#0::real) < x & x < y & 0 < n --> x ^ n < y ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	82	by (induct_tac "n" 1);
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	83	by (auto_tac (claset() addIs [rename_numerals real_mult_less_mono, gr0I]
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	84	addDs [realpow_gt_zero],
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	85	simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	86	qed_spec_mp "realpow_less";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	87
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	88	Goal "#1 ^ n = (#1::real)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	89	by (induct_tac "n" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	90	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	91	qed "realpow_eq_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	92	Addsimps [realpow_eq_one];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	93
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	94	Goal "abs(-(#1 ^ n)) = (#1::real)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	95	by (simp_tac (simpset() addsimps
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	96	[abs_minus_cancel,abs_one]) 1);
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	97	qed "abs_minus_realpow_one";
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	98	Addsimps [abs_minus_realpow_one];
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	99
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	100	Goal "abs((#-1) ^ n) = (#1::real)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	101	by (induct_tac "n" 1);
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	102	by (auto_tac (claset(),simpset() addsimps [abs_mult,
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	103	abs_minus_cancel,abs_one]));
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	104	qed "abs_realpow_minus_one";
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	105	Addsimps [abs_realpow_minus_one];
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	106
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	107	Goal "((r::real) * s) ^ n = (r ^ n) * (s ^ n)";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	108	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	109	by (auto_tac (claset(),simpset() addsimps real_mult_ac));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	110	qed "realpow_mult";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	111
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	112	Goal "(#0::real) <= r ^ 2";
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	113	by (simp_tac (simpset() addsimps [rename_numerals real_le_square]) 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	114	qed "realpow_two_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	115	Addsimps [realpow_two_le];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	116
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	117	Goal "abs((x::real) ^ 2) = x ^ 2";
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	118	by (simp_tac (simpset() addsimps [abs_eqI1,
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	119	rename_numerals real_le_square]) 1);
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	120	qed "abs_realpow_two";
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	121	Addsimps [abs_realpow_two];
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	122
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	123	Goal "abs(x::real) ^ 2 = x ^ 2";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	124	by (simp_tac (simpset() addsimps
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	125	[realpow_abs,abs_eqI1] delsimps [thm "realpow_Suc"]) 1);
8838 4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	126	qed "realpow_two_abs";
4eaa99f0d223 replaced rabs by overloaded abs; wenzelm parents: 8442 diff changeset	127	Addsimps [realpow_two_abs];
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	128
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	129	Goal "(#1::real) < r ==> #1 < r ^ 2";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	130	by Auto_tac;
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	131	by (cut_facts_tac [rename_numerals real_zero_less_one] 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	132	by (forw_inst_tac [("R1.0","#0")] real_less_trans 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	133	by (assume_tac 1);
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	134	by (dres_inst_tac [("z","r"),("x","#1")]
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	135	(rename_numerals real_mult_less_mono1) 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	136	by (auto_tac (claset() addIs [real_less_trans],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	137	qed "realpow_two_gt_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	138
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	139	Goal "(#1::real) < r --> #1 <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	140	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	141	by (auto_tac (claset() addSDs [real_le_imp_less_or_eq],
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	142	simpset()));
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	143	by (dtac (rename_numerals (real_zero_less_one RS real_mult_less_mono)) 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	144	by (dtac sym 4);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	145	by (auto_tac (claset() addSIs [real_less_imp_le],
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	146	simpset() addsimps [real_zero_less_one]));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	147	qed_spec_mp "realpow_ge_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	148
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	149	Goal "(#1::real) < r ==> #1 < r ^ (Suc n)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	150	by (forw_inst_tac [("n","n")] realpow_ge_one 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	151	by (dtac real_le_imp_less_or_eq 1 THEN Step_tac 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	152	by (dtac sym 2);
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	153	by (ftac (rename_numerals (real_zero_less_one RS real_less_trans)) 1);
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	154	by (dres_inst_tac [("y","r ^ n")]
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	155	(rename_numerals real_mult_less_mono2) 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	156	by (assume_tac 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	157	by (auto_tac (claset() addDs [real_less_trans],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	158	simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	159	qed "realpow_Suc_gt_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	160
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	161	Goal "(#1::real) <= r ==> #1 <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	162	by (dtac real_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	163	by (auto_tac (claset() addDs [realpow_ge_one], simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	164	qed "realpow_ge_one2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	165
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	166	Goal "(#0::real) < r ==> #0 < r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	167	by (forw_inst_tac [("n","n")] realpow_ge_zero 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	168	by (forw_inst_tac [("n1","n")]
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	169	((real_not_refl2 RS not_sym) RS realpow_not_zero RS not_sym) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	170	by (auto_tac (claset() addSDs [real_le_imp_less_or_eq]
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	171	addIs [rename_numerals real_mult_order],
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	172	simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	173	qed "realpow_Suc_gt_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	174
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	175	Goal "(#0::real) <= r ==> #0 <= r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	176	by (etac (real_le_imp_less_or_eq RS disjE) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	177	by (etac (realpow_ge_zero) 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	178	by (dtac sym 1);
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	179	by (Asm_simp_tac 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	180	qed "realpow_Suc_ge_zero";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	181
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	182	Goal "(#1::real) <= #2 ^ n";
26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	183	by (res_inst_tac [("j","#1 ^ n")] real_le_trans 1);
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	184	by (rtac realpow_le 2);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	185	by (auto_tac (claset() addIs [real_less_imp_le],simpset()));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	186	qed "two_realpow_ge_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	187
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	188	Goal "real_of_nat n < #2 ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	189	by (induct_tac "n" 1);
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	190	by Auto_tac;
99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	191	by (stac real_mult_2 1);
99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	192	by (rtac real_add_less_le_mono 1);
7292 dff3470c5c62 real literals using binary arithmetic paulson parents: 7219 diff changeset	193	by (auto_tac (claset(),
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	194	simpset() addsimps [two_realpow_ge_one]));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	195	qed "two_realpow_gt";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	196	Addsimps [two_realpow_gt,two_realpow_ge_one];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	197
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	198	Goal "(#-1) ^ (#2*n) = (#1::real)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	199	by (induct_tac "n" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	200	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	201	qed "realpow_minus_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	202	Addsimps [realpow_minus_one];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	203
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	204	Goal "(#-1) ^ Suc (#2*n) = -(#1::real)";
36625483213f further round of tidying paulson parents: 10648 diff changeset	205	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	206	qed "realpow_minus_one_odd";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	207	Addsimps [realpow_minus_one_odd];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	208
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	209	Goal "(#-1) ^ Suc (Suc (#2*n)) = (#1::real)";
36625483213f further round of tidying paulson parents: 10648 diff changeset	210	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	211	qed "realpow_minus_one_even";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	212	Addsimps [realpow_minus_one_even];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	213
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	214	Goal "(#0::real) < r & r < (#1::real) --> r ^ Suc n < r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	215	by (induct_tac "n" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	216	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	217	qed_spec_mp "realpow_Suc_less";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	218
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	219	Goal "#0 <= r & r < (#1::real) --> r ^ Suc n <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	220	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	221	by (auto_tac (claset() addIs [real_less_imp_le] addSDs
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	222	[real_le_imp_less_or_eq],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	223	qed_spec_mp "realpow_Suc_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	224
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	225	Goal "(#0::real) <= #0 ^ n";
8442 96023903c2df case_tac now subsumes both boolean and datatype cases; wenzelm parents: 8423 diff changeset	226	by (case_tac "n" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	227	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	228	qed "realpow_zero_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	229	Addsimps [realpow_zero_le];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	230
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	231	Goal "#0 < r & r < (#1::real) --> r ^ Suc n <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	232	by (blast_tac (claset() addSIs [real_less_imp_le,
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	233	realpow_Suc_less]) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	234	qed_spec_mp "realpow_Suc_le2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	235
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	236	Goal "[\| #0 <= r; r < (#1::real) \|] ==> r ^ Suc n <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	237	by (etac (real_le_imp_less_or_eq RS disjE) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	238	by (rtac realpow_Suc_le2 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	239	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	240	qed "realpow_Suc_le3";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	241
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	242	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	243	by (induct_tac "N" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	244	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	245	by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_zero2));
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	246	by (ALLGOALS(dtac (rename_numerals real_mult_le_mono3)));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	247	by (REPEAT(assume_tac 1));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	248	by (REPEAT(assume_tac 3));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	249	by (auto_tac (claset(),simpset() addsimps
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	250	[less_Suc_eq]));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	251	qed_spec_mp "realpow_less_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	252
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	253	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	254	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	255	by (auto_tac (claset() addIs [realpow_less_le],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	256	simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	257	qed "realpow_le_le";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	258
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	259	Goal "[\| #0 < r; r < (#1::real) \|] ==> r ^ Suc n <= r";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	260	by (dres_inst_tac [("n","1"),("N","Suc n")]
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	261	(real_less_imp_le RS realpow_le_le) 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	262	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	263	qed "realpow_Suc_le_self";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	264
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	265	Goal "[\| #0 < r; r < (#1::real) \|] ==> r ^ Suc n < #1";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	266	by (blast_tac (claset() addIs [realpow_Suc_le_self,
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	267	real_le_less_trans]) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	268	qed "realpow_Suc_less_one";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	269
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	270	Goal "(#1::real) <= r --> r ^ n <= r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	271	by (induct_tac "n" 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	272	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	273	qed_spec_mp "realpow_le_Suc";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	274
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	275	Goal "(#1::real) < r --> r ^ n < r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	276	by (induct_tac "n" 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	277	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	278	qed_spec_mp "realpow_less_Suc";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	279
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	280	Goal "(#1::real) < r --> r ^ n <= r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	281	by (blast_tac (claset() addSIs [real_less_imp_le,
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	282	realpow_less_Suc]) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	283	qed_spec_mp "realpow_le_Suc2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	284
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	285	Goal "(#1::real) < r & n < N --> r ^ n <= r ^ N";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	286	by (induct_tac "N" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	287	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	288	by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one));
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	289	by (ALLGOALS(dtac (rename_numerals real_mult_self_le)));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	290	by (assume_tac 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	291	by (assume_tac 2);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	292	by (auto_tac (claset() addIs [real_le_trans],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	293	simpset() addsimps [less_Suc_eq]));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	294	qed_spec_mp "realpow_gt_ge";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	295
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	296	Goal "(#1::real) <= r & n < N --> r ^ n <= r ^ N";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	297	by (induct_tac "N" 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	298	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	299	by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one2));
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	300	by (ALLGOALS(dtac (rename_numerals real_mult_self_le2)));
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	301	by (assume_tac 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	302	by (assume_tac 2);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	303	by (auto_tac (claset() addIs [real_le_trans],
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	304	simpset() addsimps [less_Suc_eq]));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	305	qed_spec_mp "realpow_gt_ge2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	306
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	307	Goal "[\| (#1::real) < r; n <= N \|] ==> r ^ n <= r ^ N";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	308	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	309	by (auto_tac (claset() addIs [realpow_gt_ge],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	310	qed "realpow_ge_ge";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	311
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	312	Goal "[\| (#1::real) <= r; n <= N \|] ==> r ^ n <= r ^ N";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	313	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	314	by (auto_tac (claset() addIs [realpow_gt_ge2],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	315	qed "realpow_ge_ge2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	316
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	317	Goal "(#1::real) < r ==> r <= r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	318	by (dres_inst_tac [("n","1"),("N","Suc n")]
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	319	realpow_ge_ge 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	320	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	321	qed_spec_mp "realpow_Suc_ge_self";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	322
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	323	Goal "(#1::real) <= r ==> r <= r ^ Suc n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	324	by (dres_inst_tac [("n","1"),("N","Suc n")]
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	325	realpow_ge_ge2 1);
10677 36625483213f further round of tidying paulson parents: 10648 diff changeset	326	by Auto_tac;
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	327	qed_spec_mp "realpow_Suc_ge_self2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	328
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	329	Goal "[\| (#1::real) < r; 0 < n \|] ==> r <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	330	by (dtac (less_not_refl2 RS not0_implies_Suc) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	331	by (auto_tac (claset() addSIs
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	332	[realpow_Suc_ge_self],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	333	qed "realpow_ge_self";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	334
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	335	Goal "[\| (#1::real) <= r; 0 < n \|] ==> r <= r ^ n";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	336	by (dtac (less_not_refl2 RS not0_implies_Suc) 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	337	by (auto_tac (claset() addSIs [realpow_Suc_ge_self2],simpset()));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	338	qed "realpow_ge_self2";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	339
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	340	Goal "0 < n --> (x::real) ^ (n - 1) * x = x ^ n";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	341	by (induct_tac "n" 1);
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	342	by (auto_tac (claset(),simpset()
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	343	addsimps [real_mult_commute]));
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	344	qed_spec_mp "realpow_minus_mult";
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	345	Addsimps [realpow_minus_mult];
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	346
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	347	Goal "r ~= #0 ==> r * inverse r ^ 2 = inverse (r::real)";
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	348	by (asm_simp_tac (simpset() addsimps [realpow_two,
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	349	real_mult_assoc RS sym]) 1);
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	350	qed "realpow_two_mult_inverse";
e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	351	Addsimps [realpow_two_mult_inverse];
7077 60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names; paulson parents: diff changeset	352
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	353	(* 05/00 *)
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	354	Goal "(-x) ^ 2 = (x::real) ^ 2";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	355	by (Simp_tac 1);
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	356	qed "realpow_two_minus";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	357	Addsimps [realpow_two_minus];
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	358
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	359	Goal "(x::real) ^ 2 + - (y ^ 2) = (x + -y) * (x + y)";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	360	by (simp_tac (simpset() addsimps [real_add_mult_distrib2,
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	361	real_add_mult_distrib, real_minus_mult_eq2 RS sym]
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	362	@ real_mult_ac) 1);
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	363	qed "realpow_two_add_minus";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	364
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	365	Goalw [real_diff_def]
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	366	"(x::real) ^ 2 - y ^ 2 = (x - y) * (x + y)";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	367	by (simp_tac (simpset() addsimps [realpow_two_add_minus]
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	368	delsimps [thm "realpow_Suc"]) 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	369	qed "realpow_two_diff";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	370
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	371	Goalw [real_diff_def]
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	372	"((x::real) ^ 2 = y ^ 2) = (x = y \| x = -y)";
9428 c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	373	by (auto_tac (claset(),simpset() delsimps [thm "realpow_Suc"]));
c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	374	by (dtac (rename_numerals (real_eq_minus_iff RS iffD1 RS sym)) 1);
c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	375	by (auto_tac (claset() addDs [rename_numerals real_add_minus_eq_minus],
c8eb573114de rename_numerals: use implicit theory context; wenzelm parents: 9070 diff changeset	376	simpset() addsimps [realpow_two_add_minus] delsimps [thm "realpow_Suc"]));
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	377	qed "realpow_two_disj";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	378
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	379	(* used in Transc *)
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	380	Goal "[\|(x::real) ~= #0; m <= n \|] ==> x ^ (n - m) = x ^ n * inverse (x ^ m)";
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	381	by (auto_tac (claset(),
ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	382	simpset() addsimps [le_eq_less_or_eq,
ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	383	less_iff_Suc_add,realpow_add,
ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	384	realpow_not_zero] @ real_mult_ac));
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	385	qed "realpow_diff";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	386
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	387	Goal "real_of_nat (m) ^ n = real_of_nat (m ^ n)";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	388	by (induct_tac "n" 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	389	by (auto_tac (claset(),
ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	390	simpset() addsimps [real_of_nat_one, real_of_nat_mult]));
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	391	qed "realpow_real_of_nat";
7588 26384af93359 Tidying to exploit the new arith_tac. RealBin no longer imports RealPow or paulson parents: 7292 diff changeset	392
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	393	Goal "#0 < real_of_nat (2 ^ n)";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	394	by (induct_tac "n" 1);
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	395	by (auto_tac (claset(),
99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	396	simpset() addsimps [real_of_nat_mult, real_zero_less_mult_iff]));
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	397	qed "realpow_real_of_nat_two_pos";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	398	Addsimps [realpow_real_of_nat_two_pos];
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	399
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	400
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	401	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	402	by (induct_tac "n" 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	403	by Auto_tac;
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	404	by (rtac real_leI 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	405	by (auto_tac (claset(),
ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	406	simpset() addsimps [inst "x" "#0" order_le_less,
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	407	real_mult_le_0_iff]));
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	408	by (subgoal_tac "inverse x * (x * (x * x ^ n)) <= inverse y * (y * (y * y ^ n))" 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	409	by (rtac real_mult_le_mono 2);
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	410	by (asm_full_simp_tac (simpset() addsimps [realpow_ge_zero, real_0_le_mult_iff]) 4);
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	411	by (asm_full_simp_tac (simpset() addsimps [real_inverse_le_iff]) 3);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	412	by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 1);
10606 e3229a37d53f converted rinv to inverse; bauerg parents: 10043 diff changeset	413	by (rtac real_inverse_gt_zero 1);
9043 ca761fe227d8 First round of changes, towards installation of simprocs paulson parents: 9013 diff changeset	414	by Auto_tac;
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	415	qed_spec_mp "realpow_increasing";
9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	416
9070 99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	417	Goal "[\| (#0::real) <= x; #0 <= y; x ^ Suc n = y ^ Suc n \|] ==> x = y";
99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	418	by (blast_tac (claset() addIs [realpow_increasing, order_antisym,
99d93349914b full_rename_numerals -> rename_numerals; tidied paulson parents: 9043 diff changeset	419	order_eq_refl, sym]) 1);
9013 9dd0274f76af Updated files to remove 0r and 1r from theorems in descendant theories fleuriot parents: 8838 diff changeset	420	qed_spec_mp "realpow_Suc_cancel_eq";

author	paulson
	Mon, 18 Dec 2000 12:21:54 +0100
changeset 10689	5c44de6aadf4
parent 10677	36625483213f
child 10690	cd80241125b0
permissions	-rw-r--r--