author | huffman |
Tue, 08 May 2007 01:10:55 +0200 | |
changeset 22854 | 51087b1cc77d |
parent 22803 | 5129e02f4df2 |
child 23051 | e98ed26577a2 |
permissions | -rw-r--r-- |
7032 | 1 |
(* Title: HOL/NatBin.thy |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 |
Copyright 1999 University of Cambridge |
|
12838 | 5 |
*) |
7032 | 6 |
|
12838 | 7 |
header {* Binary arithmetic for the natural numbers *} |
7032 | 8 |
|
15131 | 9 |
theory NatBin |
15140 | 10 |
imports IntDiv |
15131 | 11 |
begin |
7032 | 12 |
|
12838 | 13 |
text {* |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
14 |
Arithmetic for naturals is reduced to that for the non-negative integers. |
12838 | 15 |
*} |
16 |
||
22319 | 17 |
instance nat :: number |
22803 | 18 |
nat_number_of_def [code inline]: "number_of v == nat (number_of (v\<Colon>int))" .. |
19380 | 19 |
|
20 |
abbreviation (xsymbols) |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
21 |
square :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where |
19380 | 22 |
"x\<twosuperior> == x^2" |
23 |
||
21210 | 24 |
notation (latex output) |
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19601
diff
changeset
|
25 |
square ("(_\<twosuperior>)" [1000] 999) |
19380 | 26 |
|
21210 | 27 |
notation (HTML output) |
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19601
diff
changeset
|
28 |
square ("(_\<twosuperior>)" [1000] 999) |
12838 | 29 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
30 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
31 |
subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
32 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
33 |
declare nat_0 [simp] nat_1 [simp] |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
34 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
35 |
lemma nat_number_of [simp]: "nat (number_of w) = number_of w" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
36 |
by (simp add: nat_number_of_def) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
37 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
38 |
lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
39 |
by (simp add: nat_number_of_def) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
40 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
41 |
lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
42 |
by (simp add: nat_1 nat_number_of_def) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
43 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
44 |
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
45 |
by (simp add: nat_numeral_1_eq_1) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
46 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
47 |
lemma numeral_2_eq_2: "2 = Suc (Suc 0)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
48 |
apply (unfold nat_number_of_def) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
49 |
apply (rule nat_2) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
50 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
51 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
52 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
53 |
text{*Distributive laws for type @{text nat}. The others are in theory |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
54 |
@{text IntArith}, but these require div and mod to be defined for type |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
55 |
"int". They also need some of the lemmas proved above.*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
56 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
57 |
lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
58 |
apply (case_tac "0 <= z'") |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
59 |
apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
60 |
apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
61 |
apply (auto elim!: nonneg_eq_int) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
62 |
apply (rename_tac m m') |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
63 |
apply (subgoal_tac "0 <= int m div int m'") |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
64 |
prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
65 |
apply (rule inj_int [THEN injD], simp) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
66 |
apply (rule_tac r = "int (m mod m') " in quorem_div) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
67 |
prefer 2 apply force |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
68 |
apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
69 |
zmult_int) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
70 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
71 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
72 |
(*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
73 |
lemma nat_mod_distrib: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
74 |
"[| (0::int) <= z; 0 <= z' |] ==> nat (z mod z') = nat z mod nat z'" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
75 |
apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
76 |
apply (auto elim!: nonneg_eq_int) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
77 |
apply (rename_tac m m') |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
78 |
apply (subgoal_tac "0 <= int m mod int m'") |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
79 |
prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
80 |
apply (rule inj_int [THEN injD], simp) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
81 |
apply (rule_tac q = "int (m div m') " in quorem_mod) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
82 |
prefer 2 apply force |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
83 |
apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int zmult_int) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
84 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
85 |
|
16413 | 86 |
text{*Suggested by Matthias Daum*} |
87 |
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)" |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20105
diff
changeset
|
88 |
apply (subgoal_tac "nat x div nat k < nat x") |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20105
diff
changeset
|
89 |
apply (simp (asm_lr) add: nat_div_distrib [symmetric]) |
16413 | 90 |
apply (rule Divides.div_less_dividend, simp_all) |
91 |
done |
|
92 |
||
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
93 |
subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
94 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
95 |
(*"neg" is used in rewrite rules for binary comparisons*) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
96 |
lemma int_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
97 |
"int (number_of v :: nat) = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
98 |
(if neg (number_of v :: int) then 0 |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
99 |
else (number_of v :: int))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
100 |
by (simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
101 |
add: neg_nat nat_number_of_def not_neg_nat add_assoc) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
102 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
103 |
|
14390 | 104 |
subsubsection{*Successor *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
105 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
106 |
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
107 |
apply (rule sym) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
108 |
apply (simp add: nat_eq_iff int_Suc) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
109 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
110 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
111 |
lemma Suc_nat_number_of_add: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
112 |
"Suc (number_of v + n) = |
20500 | 113 |
(if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
114 |
by (simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
115 |
add: nat_number_of_def neg_nat |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
116 |
Suc_nat_eq_nat_zadd1 number_of_succ) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
117 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
118 |
lemma Suc_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
119 |
"Suc (number_of v) = |
20500 | 120 |
(if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
121 |
apply (cut_tac n = 0 in Suc_nat_number_of_add) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
122 |
apply (simp cong del: if_weak_cong) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
123 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
124 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
125 |
|
14390 | 126 |
subsubsection{*Addition *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
127 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
128 |
(*"neg" is used in rewrite rules for binary comparisons*) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
129 |
lemma add_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
130 |
"(number_of v :: nat) + number_of v' = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
131 |
(if neg (number_of v :: int) then number_of v' |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
132 |
else if neg (number_of v' :: int) then number_of v |
20485 | 133 |
else number_of (v + v'))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
134 |
by (force dest!: neg_nat |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
135 |
simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
136 |
simp add: nat_number_of_def nat_add_distrib [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
137 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
138 |
|
14390 | 139 |
subsubsection{*Subtraction *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
140 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
141 |
lemma diff_nat_eq_if: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
142 |
"nat z - nat z' = |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
143 |
(if neg z' then nat z |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
144 |
else let d = z-z' in |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
145 |
if neg d then 0 else nat d)" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
146 |
apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
147 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
148 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
149 |
lemma diff_nat_number_of [simp]: |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
150 |
"(number_of v :: nat) - number_of v' = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
151 |
(if neg (number_of v' :: int) then number_of v |
20485 | 152 |
else let d = number_of (v + uminus v') in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
153 |
if neg d then 0 else nat d)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
154 |
by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
155 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
156 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
157 |
|
14390 | 158 |
subsubsection{*Multiplication *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
159 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
160 |
lemma mult_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
161 |
"(number_of v :: nat) * number_of v' = |
20485 | 162 |
(if neg (number_of v :: int) then 0 else number_of (v * v'))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
163 |
by (force dest!: neg_nat |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
164 |
simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
165 |
simp add: nat_number_of_def nat_mult_distrib [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
166 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
167 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
168 |
|
14390 | 169 |
subsubsection{*Quotient *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
170 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
171 |
lemma div_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
172 |
"(number_of v :: nat) div number_of v' = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
173 |
(if neg (number_of v :: int) then 0 |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
174 |
else nat (number_of v div number_of v'))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
175 |
by (force dest!: neg_nat |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
176 |
simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
177 |
simp add: nat_number_of_def nat_div_distrib [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
178 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
179 |
lemma one_div_nat_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
180 |
"(Suc 0) div number_of v' = (nat (1 div number_of v'))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
181 |
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
182 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
183 |
|
14390 | 184 |
subsubsection{*Remainder *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
185 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
186 |
lemma mod_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
187 |
"(number_of v :: nat) mod number_of v' = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
188 |
(if neg (number_of v :: int) then 0 |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
189 |
else if neg (number_of v' :: int) then number_of v |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
190 |
else nat (number_of v mod number_of v'))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
191 |
by (force dest!: neg_nat |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
192 |
simp del: nat_number_of |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
193 |
simp add: nat_number_of_def nat_mod_distrib [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
194 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
195 |
lemma one_mod_nat_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
196 |
"(Suc 0) mod number_of v' = |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
197 |
(if neg (number_of v' :: int) then Suc 0 |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
198 |
else nat (1 mod number_of v'))" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
199 |
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
200 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
201 |
|
18978 | 202 |
subsubsection{* Divisibility *} |
203 |
||
18984 | 204 |
lemmas dvd_eq_mod_eq_0_number_of = |
205 |
dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard] |
|
206 |
||
207 |
declare dvd_eq_mod_eq_0_number_of [simp] |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
208 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
209 |
ML |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
210 |
{* |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
211 |
val nat_number_of_def = thm"nat_number_of_def"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
212 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
213 |
val nat_number_of = thm"nat_number_of"; |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
214 |
val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
215 |
val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1"; |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
216 |
val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
217 |
val numeral_2_eq_2 = thm"numeral_2_eq_2"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
218 |
val nat_div_distrib = thm"nat_div_distrib"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
219 |
val nat_mod_distrib = thm"nat_mod_distrib"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
220 |
val int_nat_number_of = thm"int_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
221 |
val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
222 |
val Suc_nat_number_of_add = thm"Suc_nat_number_of_add"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
223 |
val Suc_nat_number_of = thm"Suc_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
224 |
val add_nat_number_of = thm"add_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
225 |
val diff_nat_eq_if = thm"diff_nat_eq_if"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
226 |
val diff_nat_number_of = thm"diff_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
227 |
val mult_nat_number_of = thm"mult_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
228 |
val div_nat_number_of = thm"div_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
229 |
val mod_nat_number_of = thm"mod_nat_number_of"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
230 |
*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
231 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
232 |
|
14390 | 233 |
subsection{*Comparisons*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
234 |
|
14390 | 235 |
subsubsection{*Equals (=) *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
236 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
237 |
lemma eq_nat_nat_iff: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
238 |
"[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
239 |
by (auto elim!: nonneg_eq_int) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
240 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
241 |
(*"neg" is used in rewrite rules for binary comparisons*) |
14390 | 242 |
lemma eq_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
243 |
"((number_of v :: nat) = number_of v') = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
244 |
(if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int)) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
245 |
else if neg (number_of v' :: int) then iszero (number_of v :: int) |
20485 | 246 |
else iszero (number_of (v + uminus v') :: int))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
247 |
apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
248 |
eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
249 |
split add: split_if cong add: imp_cong) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
250 |
apply (simp only: nat_eq_iff nat_eq_iff2) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
251 |
apply (simp add: not_neg_eq_ge_0 [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
252 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
253 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
254 |
|
14390 | 255 |
subsubsection{*Less-than (<) *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
256 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
257 |
(*"neg" is used in rewrite rules for binary comparisons*) |
14390 | 258 |
lemma less_nat_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
259 |
"((number_of v :: nat) < number_of v') = |
20485 | 260 |
(if neg (number_of v :: int) then neg (number_of (uminus v') :: int) |
261 |
else neg (number_of (v + uminus v') :: int))" |
|
14390 | 262 |
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
263 |
nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless |
20485 | 264 |
cong add: imp_cong, simp add: Pls_def) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
265 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
266 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
267 |
(*Maps #n to n for n = 0, 1, 2*) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
268 |
lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2 |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
269 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
270 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
271 |
subsection{*Powers with Numeric Exponents*} |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
272 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
273 |
text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}. |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
274 |
We cannot prove general results about the numeral @{term "-1"}, so we have to |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
275 |
use @{term "- 1"} instead.*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
276 |
|
15003 | 277 |
lemma power2_eq_square: "(a::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = a * a" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
278 |
by (simp add: numeral_2_eq_2 Power.power_Suc) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
279 |
|
17724 | 280 |
lemma zero_power2 [simp]: "(0::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 0" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
281 |
by (simp add: power2_eq_square) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
282 |
|
17724 | 283 |
lemma one_power2 [simp]: "(1::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 1" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
284 |
by (simp add: power2_eq_square) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
285 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
286 |
lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
287 |
apply (subgoal_tac "3 = Suc (Suc (Suc 0))") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
288 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
289 |
apply (simp add: power_Suc mult_ac) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
290 |
apply (unfold nat_number_of_def) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
291 |
apply (subst nat_eq_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
292 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
293 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
294 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
295 |
text{*Squares of literal numerals will be evaluated.*} |
17085 | 296 |
lemmas power2_eq_square_number_of = |
297 |
power2_eq_square [of "number_of w", standard] |
|
298 |
declare power2_eq_square_number_of [simp] |
|
299 |
||
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
300 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
301 |
lemma zero_le_power2: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
302 |
by (simp add: power2_eq_square zero_le_square) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
303 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
304 |
lemma zero_less_power2: |
15003 | 305 |
"(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
306 |
by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
307 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
308 |
lemma power2_less_0: |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
309 |
fixes a :: "'a::{ordered_idom,recpower}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
310 |
shows "~ (a\<twosuperior> < 0)" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
311 |
by (force simp add: power2_eq_square mult_less_0_iff) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
312 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
313 |
lemma zero_eq_power2: |
15003 | 314 |
"(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
315 |
by (force simp add: power2_eq_square mult_eq_0_iff) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
316 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
317 |
lemma abs_power2: |
15003 | 318 |
"abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
319 |
by (simp add: power2_eq_square abs_mult abs_mult_self) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
320 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
321 |
lemma power2_abs: |
15003 | 322 |
"(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
323 |
by (simp add: power2_eq_square abs_mult_self) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
324 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
325 |
lemma power2_minus: |
15003 | 326 |
"(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
327 |
by (simp add: power2_eq_square) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
328 |
|
22854
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
329 |
lemma power2_le_imp_le: |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
330 |
fixes x y :: "'a::{ordered_semidom,recpower}" |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
331 |
shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y" |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
332 |
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base) |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
333 |
|
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
334 |
lemma power2_less_imp_less: |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
335 |
fixes x y :: "'a::{ordered_semidom,recpower}" |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
336 |
shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y" |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
337 |
by (rule power_less_imp_less_base) |
51087b1cc77d
add lemmas power2_le_imp_le and power2_less_imp_less
huffman
parents:
22803
diff
changeset
|
338 |
|
15003 | 339 |
lemma power_minus1_even: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})" |
15251 | 340 |
apply (induct "n") |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
341 |
apply (auto simp add: power_Suc power_add power2_minus) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
342 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
343 |
|
15003 | 344 |
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2" |
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
20835
diff
changeset
|
345 |
by (subst mult_commute) (simp add: power_mult) |
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
346 |
|
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
347 |
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2" |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
348 |
by (simp add: power_even_eq) |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
349 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
350 |
lemma power_minus_even [simp]: |
15003 | 351 |
"(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
352 |
by (simp add: power_minus1_even power_minus [of a]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
353 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
354 |
lemma zero_le_even_power': |
15003 | 355 |
"0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
356 |
proof (induct "n") |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
357 |
case 0 |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
358 |
show ?case by (simp add: zero_le_one) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
359 |
next |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
360 |
case (Suc n) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
361 |
have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
362 |
by (simp add: mult_ac power_add power2_eq_square) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
363 |
thus ?case |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
364 |
by (simp add: prems zero_le_square zero_le_mult_iff) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
365 |
qed |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
366 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
367 |
lemma odd_power_less_zero: |
15003 | 368 |
"(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
369 |
proof (induct "n") |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
370 |
case 0 |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
371 |
show ?case by (simp add: Power.power_Suc) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
372 |
next |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
373 |
case (Suc n) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
374 |
have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
375 |
by (simp add: mult_ac power_add power2_eq_square Power.power_Suc) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
376 |
thus ?case |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16642
diff
changeset
|
377 |
by (simp add: prems mult_less_0_iff mult_neg_neg) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
378 |
qed |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
379 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
380 |
lemma odd_0_le_power_imp_0_le: |
15003 | 381 |
"0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})" |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
382 |
apply (insert odd_power_less_zero [of a n]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
383 |
apply (force simp add: linorder_not_less [symmetric]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
384 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
385 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
386 |
text{*Simprules for comparisons where common factors can be cancelled.*} |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
387 |
lemmas zero_compare_simps = |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
388 |
add_strict_increasing add_strict_increasing2 add_increasing |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
389 |
zero_le_mult_iff zero_le_divide_iff |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
390 |
zero_less_mult_iff zero_less_divide_iff |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
391 |
mult_le_0_iff divide_le_0_iff |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
392 |
mult_less_0_iff divide_less_0_iff |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15140
diff
changeset
|
393 |
zero_le_power2 power2_less_0 |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
394 |
|
14390 | 395 |
subsubsection{*Nat *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
396 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
397 |
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
398 |
by (simp add: numerals) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
399 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
400 |
(*Expresses a natural number constant as the Suc of another one. |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
401 |
NOT suitable for rewriting because n recurs in the condition.*) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
402 |
lemmas expand_Suc = Suc_pred' [of "number_of v", standard] |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
403 |
|
14390 | 404 |
subsubsection{*Arith *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
405 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
406 |
lemma Suc_eq_add_numeral_1: "Suc n = n + 1" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
407 |
by (simp add: numerals) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
408 |
|
14467 | 409 |
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n" |
410 |
by (simp add: numerals) |
|
411 |
||
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
412 |
(* These two can be useful when m = number_of... *) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
413 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
414 |
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
415 |
apply (case_tac "m") |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
416 |
apply (simp_all add: numerals) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
417 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
418 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
419 |
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
420 |
apply (case_tac "m") |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
421 |
apply (simp_all add: numerals) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
422 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
423 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
424 |
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
425 |
apply (case_tac "m") |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
426 |
apply (simp_all add: numerals) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
427 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
428 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
429 |
|
14390 | 430 |
subsection{*Comparisons involving (0::nat) *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
431 |
|
14390 | 432 |
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*} |
433 |
||
434 |
lemma eq_number_of_0 [simp]: |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
435 |
"(number_of v = (0::nat)) = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
436 |
(if neg (number_of v :: int) then True else iszero (number_of v :: int))" |
14390 | 437 |
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
438 |
|
14390 | 439 |
lemma eq_0_number_of [simp]: |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
440 |
"((0::nat) = number_of v) = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
441 |
(if neg (number_of v :: int) then True else iszero (number_of v :: int))" |
14390 | 442 |
by (rule trans [OF eq_sym_conv eq_number_of_0]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
443 |
|
14390 | 444 |
lemma less_0_number_of [simp]: |
20485 | 445 |
"((0::nat) < number_of v) = neg (number_of (uminus v) :: int)" |
446 |
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def) |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
447 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
448 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
449 |
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
450 |
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
451 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
452 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
453 |
|
22190
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
454 |
subsection{*Comparisons involving @{term Suc} *} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
455 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
456 |
lemma eq_number_of_Suc [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
457 |
"(number_of v = Suc n) = |
20500 | 458 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
459 |
if neg pv then False else nat pv = n)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
460 |
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
461 |
number_of_pred nat_number_of_def |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
462 |
split add: split_if) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
463 |
apply (rule_tac x = "number_of v" in spec) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
464 |
apply (auto simp add: nat_eq_iff) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
465 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
466 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
467 |
lemma Suc_eq_number_of [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
468 |
"(Suc n = number_of v) = |
20500 | 469 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
470 |
if neg pv then False else nat pv = n)" |
14390 | 471 |
by (rule trans [OF eq_sym_conv eq_number_of_Suc]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
472 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
473 |
lemma less_number_of_Suc [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
474 |
"(number_of v < Suc n) = |
20500 | 475 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
476 |
if neg pv then True else nat pv < n)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
477 |
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
478 |
number_of_pred nat_number_of_def |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
479 |
split add: split_if) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
480 |
apply (rule_tac x = "number_of v" in spec) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
481 |
apply (auto simp add: nat_less_iff) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
482 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
483 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
484 |
lemma less_Suc_number_of [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
485 |
"(Suc n < number_of v) = |
20500 | 486 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
487 |
if neg pv then False else n < nat pv)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
488 |
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
489 |
number_of_pred nat_number_of_def |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
490 |
split add: split_if) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
491 |
apply (rule_tac x = "number_of v" in spec) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
492 |
apply (auto simp add: zless_nat_eq_int_zless) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
493 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
494 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
495 |
lemma le_number_of_Suc [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
496 |
"(number_of v <= Suc n) = |
20500 | 497 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
498 |
if neg pv then True else nat pv <= n)" |
14390 | 499 |
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
500 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
501 |
lemma le_Suc_number_of [simp]: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
502 |
"(Suc n <= number_of v) = |
20500 | 503 |
(let pv = number_of (Numeral.pred v) in |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
504 |
if neg pv then False else n <= nat pv)" |
14390 | 505 |
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric]) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
506 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
507 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
508 |
(* Push int(.) inwards: *) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
509 |
declare zadd_int [symmetric, simp] |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
510 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
511 |
lemma lemma1: "(m+m = n+n) = (m = (n::int))" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
512 |
by auto |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
513 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
514 |
lemma lemma2: "m+m ~= (1::int) + (n + n)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
515 |
apply auto |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
516 |
apply (drule_tac f = "%x. x mod 2" in arg_cong) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
517 |
apply (simp add: zmod_zadd1_eq) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
518 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
519 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
520 |
lemma eq_number_of_BIT_BIT: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
521 |
"((number_of (v BIT x) ::int) = number_of (w BIT y)) = |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
522 |
(x=y & (((number_of v) ::int) = number_of w))" |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
523 |
apply (simp only: number_of_BIT lemma1 lemma2 eq_commute |
14738 | 524 |
OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0 |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
525 |
split add: bit.split) |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
526 |
apply simp |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
527 |
done |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
528 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
529 |
lemma eq_number_of_BIT_Pls: |
15013 | 530 |
"((number_of (v BIT x) ::int) = Numeral0) = |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
531 |
(x=bit.B0 & (((number_of v) ::int) = Numeral0))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
532 |
apply (simp only: simp_thms add: number_of_BIT number_of_Pls eq_commute |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
533 |
split add: bit.split cong: imp_cong) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
534 |
apply (rule_tac x = "number_of v" in spec, safe) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
535 |
apply (simp_all (no_asm_use)) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
536 |
apply (drule_tac f = "%x. x mod 2" in arg_cong) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
537 |
apply (simp add: zmod_zadd1_eq) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
538 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
539 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
540 |
lemma eq_number_of_BIT_Min: |
15013 | 541 |
"((number_of (v BIT x) ::int) = number_of Numeral.Min) = |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
542 |
(x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
543 |
apply (simp only: simp_thms add: number_of_BIT number_of_Min eq_commute |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
544 |
split add: bit.split cong: imp_cong) |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
545 |
apply (rule_tac x = "number_of v" in spec, auto) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
546 |
apply (drule_tac f = "%x. x mod 2" in arg_cong, auto) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
547 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
548 |
|
15013 | 549 |
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min" |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
550 |
by auto |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
551 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
552 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
553 |
|
22190
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
554 |
subsection{*Max and Min Combined with @{term Suc} *} |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
555 |
|
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
556 |
lemma max_number_of_Suc [simp]: |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
557 |
"max (Suc n) (number_of v) = |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
558 |
(let pv = number_of (Numeral.pred v) in |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
559 |
if neg pv then Suc n else Suc(max n (nat pv)))" |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
560 |
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
561 |
split add: split_if nat.split) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
562 |
apply (rule_tac x = "number_of v" in spec) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
563 |
apply auto |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
564 |
done |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
565 |
|
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
566 |
lemma max_Suc_number_of [simp]: |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
567 |
"max (number_of v) (Suc n) = |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
568 |
(let pv = number_of (Numeral.pred v) in |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
569 |
if neg pv then Suc n else Suc(max (nat pv) n))" |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
570 |
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
571 |
split add: split_if nat.split) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
572 |
apply (rule_tac x = "number_of v" in spec) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
573 |
apply auto |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
574 |
done |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
575 |
|
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
576 |
lemma min_number_of_Suc [simp]: |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
577 |
"min (Suc n) (number_of v) = |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
578 |
(let pv = number_of (Numeral.pred v) in |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
579 |
if neg pv then 0 else Suc(min n (nat pv)))" |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
580 |
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
581 |
split add: split_if nat.split) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
582 |
apply (rule_tac x = "number_of v" in spec) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
583 |
apply auto |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
584 |
done |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
585 |
|
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
586 |
lemma min_Suc_number_of [simp]: |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
587 |
"min (number_of v) (Suc n) = |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
588 |
(let pv = number_of (Numeral.pred v) in |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
589 |
if neg pv then 0 else Suc(min (nat pv) n))" |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
590 |
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
591 |
split add: split_if nat.split) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
592 |
apply (rule_tac x = "number_of v" in spec) |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
593 |
apply auto |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
594 |
done |
d31dec6397be
simplification of Suc/numeral combinations with min, max
paulson
parents:
22046
diff
changeset
|
595 |
|
14390 | 596 |
subsection{*Literal arithmetic involving powers*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
597 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
598 |
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n" |
15251 | 599 |
apply (induct "n") |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
600 |
apply (simp_all (no_asm_simp) add: nat_mult_distrib) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
601 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
602 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
603 |
lemma power_nat_number_of: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
604 |
"(number_of v :: nat) ^ n = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
605 |
(if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))" |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
606 |
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
607 |
split add: split_if cong: imp_cong) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
608 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
609 |
|
17085 | 610 |
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard] |
611 |
declare power_nat_number_of_number_of [simp] |
|
612 |
||
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
613 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
614 |
|
14390 | 615 |
text{*For the integers*} |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
616 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
617 |
lemma zpower_number_of_even: |
20485 | 618 |
"(z::int) ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)" |
619 |
unfolding Let_def nat_number_of_def number_of_BIT bit.cases |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
620 |
apply (rule_tac x = "number_of w" in spec, clarify) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
621 |
apply (case_tac " (0::int) <= x") |
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
622 |
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
623 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
624 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
625 |
lemma zpower_number_of_odd: |
20485 | 626 |
"(z::int) ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w |
627 |
then (let w = z ^ (number_of w) in z * w * w) else 1)" |
|
628 |
unfolding Let_def nat_number_of_def number_of_BIT bit.cases |
|
629 |
apply (rule_tac x = "number_of w" in spec, auto) |
|
630 |
apply (simp only: nat_add_distrib nat_mult_distrib) |
|
631 |
apply simp |
|
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
632 |
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
633 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
634 |
|
17085 | 635 |
lemmas zpower_number_of_even_number_of = |
636 |
zpower_number_of_even [of "number_of v", standard] |
|
637 |
declare zpower_number_of_even_number_of [simp] |
|
638 |
||
639 |
lemmas zpower_number_of_odd_number_of = |
|
640 |
zpower_number_of_odd [of "number_of v", standard] |
|
641 |
declare zpower_number_of_odd_number_of [simp] |
|
642 |
||
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
643 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
644 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
645 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
646 |
ML |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
647 |
{* |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
648 |
val numerals = thms"numerals"; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
649 |
val numeral_ss = simpset() addsimps numerals; |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
650 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
651 |
val nat_bin_arith_setup = |
18708 | 652 |
Fast_Arith.map_data |
15921 | 653 |
(fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} => |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
654 |
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
655 |
inj_thms = inj_thms, |
15921 | 656 |
lessD = lessD, neqE = neqE, |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
657 |
simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of, |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
658 |
not_neg_number_of_Pls, |
18708 | 659 |
neg_number_of_Min,neg_number_of_BIT]}) |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
660 |
*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14194
diff
changeset
|
661 |
|
12838 | 662 |
setup nat_bin_arith_setup |
663 |
||
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13154
diff
changeset
|
664 |
(* Enable arith to deal with div/mod k where k is a numeral: *) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13154
diff
changeset
|
665 |
declare split_div[of _ _ "number_of k", standard, arith_split] |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13154
diff
changeset
|
666 |
declare split_mod[of _ _ "number_of k", standard, arith_split] |
13154 | 667 |
|
15013 | 668 |
lemma nat_number_of_Pls: "Numeral0 = (0::nat)" |
12838 | 669 |
by (simp add: number_of_Pls nat_number_of_def) |
670 |
||
15013 | 671 |
lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)" |
12838 | 672 |
apply (simp only: number_of_Min nat_number_of_def nat_zminus_int) |
673 |
done |
|
7032 | 674 |
|
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
675 |
lemma nat_number_of_BIT_1: |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
676 |
"number_of (w BIT bit.B1) = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
677 |
(if neg (number_of w :: int) then 0 |
12838 | 678 |
else let n = number_of w in Suc (n + n))" |
679 |
apply (simp only: nat_number_of_def Let_def split: split_if) |
|
680 |
apply (intro conjI impI) |
|
681 |
apply (simp add: neg_nat neg_number_of_BIT) |
|
682 |
apply (rule int_int_eq [THEN iffD1]) |
|
683 |
apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms) |
|
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
684 |
apply (simp only: number_of_BIT zadd_assoc split: bit.split) |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
685 |
apply simp |
12838 | 686 |
done |
7032 | 687 |
|
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
688 |
lemma nat_number_of_BIT_0: |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
689 |
"number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)" |
12838 | 690 |
apply (simp only: nat_number_of_def Let_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
691 |
apply (cases "neg (number_of w :: int)") |
12838 | 692 |
apply (simp add: neg_nat neg_number_of_BIT) |
693 |
apply (rule int_int_eq [THEN iffD1]) |
|
694 |
apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms) |
|
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
695 |
apply (simp only: number_of_BIT zadd_assoc) |
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
696 |
apply simp |
12838 | 697 |
done |
698 |
||
13043
ad1828b479b7
renamed nat_number_of to nat_number (avoid clash with separate theorem);
wenzelm
parents:
12933
diff
changeset
|
699 |
lemmas nat_number = |
12838 | 700 |
nat_number_of_Pls nat_number_of_Min |
15620
8ccdc8bc66a2
replaced bool by a new datatype "bit" for binary numerals
paulson
parents:
15531
diff
changeset
|
701 |
nat_number_of_BIT_1 nat_number_of_BIT_0 |
12838 | 702 |
|
703 |
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)" |
|
704 |
by (simp add: Let_def) |
|
10574
8f98f0301d67
Linear arithmetic now copes with mixed nat/int formulae.
nipkow
parents:
9509
diff
changeset
|
705 |
|
15003 | 706 |
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})" |
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
707 |
by (simp add: power_mult); |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
708 |
|
15003 | 709 |
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})" |
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
710 |
by (simp add: power_mult power_Suc); |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14430
diff
changeset
|
711 |
|
12440 | 712 |
|
14390 | 713 |
subsection{*Literal arithmetic and @{term of_nat}*} |
714 |
||
715 |
lemma of_nat_double: |
|
716 |
"0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)" |
|
717 |
by (simp only: mult_2 nat_add_distrib of_nat_add) |
|
718 |
||
719 |
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)" |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20105
diff
changeset
|
720 |
by (simp only: nat_number_of_def) |
14390 | 721 |
|
722 |
lemma of_nat_number_of_lemma: |
|
723 |
"of_nat (number_of v :: nat) = |
|
724 |
(if 0 \<le> (number_of v :: int) |
|
725 |
then (number_of v :: 'a :: number_ring) |
|
726 |
else 0)" |
|
15013 | 727 |
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat); |
14390 | 728 |
|
729 |
lemma of_nat_number_of_eq [simp]: |
|
730 |
"of_nat (number_of v :: nat) = |
|
731 |
(if neg (number_of v :: int) then 0 |
|
732 |
else (number_of v :: 'a :: number_ring))" |
|
733 |
by (simp only: of_nat_number_of_lemma neg_def, simp) |
|
734 |
||
735 |
||
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
736 |
subsection {*Lemmas for the Combination and Cancellation Simprocs*} |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
737 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
738 |
lemma nat_number_of_add_left: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
739 |
"number_of v + (number_of v' + (k::nat)) = |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
740 |
(if neg (number_of v :: int) then number_of v' + k |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
741 |
else if neg (number_of v' :: int) then number_of v + k |
20485 | 742 |
else number_of (v + v') + k)" |
14390 | 743 |
by simp |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
744 |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
745 |
lemma nat_number_of_mult_left: |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
746 |
"number_of v * (number_of v' * (k::nat)) = |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
747 |
(if neg (number_of v :: int) then 0 |
20485 | 748 |
else number_of (v * v') * k)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
749 |
by simp |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
750 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
751 |
|
14390 | 752 |
subsubsection{*For @{text combine_numerals}*} |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
753 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
754 |
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
755 |
by (simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
756 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
757 |
|
14390 | 758 |
subsubsection{*For @{text cancel_numerals}*} |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
759 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
760 |
lemma nat_diff_add_eq1: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
761 |
"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
762 |
by (simp split add: nat_diff_split add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
763 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
764 |
lemma nat_diff_add_eq2: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
765 |
"i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
766 |
by (simp split add: nat_diff_split add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
767 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
768 |
lemma nat_eq_add_iff1: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
769 |
"j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
770 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
771 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
772 |
lemma nat_eq_add_iff2: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
773 |
"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
774 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
775 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
776 |
lemma nat_less_add_iff1: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
777 |
"j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
778 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
779 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
780 |
lemma nat_less_add_iff2: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
781 |
"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
782 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
783 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
784 |
lemma nat_le_add_iff1: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
785 |
"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
786 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
787 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
788 |
lemma nat_le_add_iff2: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
789 |
"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
790 |
by (auto split add: nat_diff_split simp add: add_mult_distrib) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
791 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
792 |
|
14390 | 793 |
subsubsection{*For @{text cancel_numeral_factors} *} |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
794 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
795 |
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
796 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
797 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
798 |
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
799 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
800 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
801 |
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
802 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
803 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
804 |
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
805 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
806 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
807 |
|
14390 | 808 |
subsubsection{*For @{text cancel_factor} *} |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
809 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
810 |
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
811 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
812 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
813 |
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
814 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
815 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
816 |
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
817 |
by auto |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
818 |
|
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
819 |
lemma nat_mult_div_cancel_disj: |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
820 |
"(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)" |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
821 |
by (simp add: nat_mult_div_cancel1) |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
822 |
|
20355 | 823 |
|
19601 | 824 |
subsection {* legacy ML bindings *} |
825 |
||
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
826 |
ML |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
827 |
{* |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
828 |
val eq_nat_nat_iff = thm"eq_nat_nat_iff"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
829 |
val eq_nat_number_of = thm"eq_nat_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
830 |
val less_nat_number_of = thm"less_nat_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
831 |
val power2_eq_square = thm "power2_eq_square"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
832 |
val zero_le_power2 = thm "zero_le_power2"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
833 |
val zero_less_power2 = thm "zero_less_power2"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
834 |
val zero_eq_power2 = thm "zero_eq_power2"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
835 |
val abs_power2 = thm "abs_power2"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
836 |
val power2_abs = thm "power2_abs"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
837 |
val power2_minus = thm "power2_minus"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
838 |
val power_minus1_even = thm "power_minus1_even"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
839 |
val power_minus_even = thm "power_minus_even"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
840 |
val odd_power_less_zero = thm "odd_power_less_zero"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
841 |
val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
842 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
843 |
val Suc_pred' = thm"Suc_pred'"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
844 |
val expand_Suc = thm"expand_Suc"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
845 |
val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1"; |
14467 | 846 |
val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left"; |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
847 |
val add_eq_if = thm"add_eq_if"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
848 |
val mult_eq_if = thm"mult_eq_if"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
849 |
val power_eq_if = thm"power_eq_if"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
850 |
val eq_number_of_0 = thm"eq_number_of_0"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
851 |
val eq_0_number_of = thm"eq_0_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
852 |
val less_0_number_of = thm"less_0_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
853 |
val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
854 |
val eq_number_of_Suc = thm"eq_number_of_Suc"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
855 |
val Suc_eq_number_of = thm"Suc_eq_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
856 |
val less_number_of_Suc = thm"less_number_of_Suc"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
857 |
val less_Suc_number_of = thm"less_Suc_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
858 |
val le_number_of_Suc = thm"le_number_of_Suc"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
859 |
val le_Suc_number_of = thm"le_Suc_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
860 |
val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
861 |
val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
862 |
val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
863 |
val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min"; |
14390 | 864 |
val of_nat_number_of_eq = thm"of_nat_number_of_eq"; |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
865 |
val nat_power_eq = thm"nat_power_eq"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
866 |
val power_nat_number_of = thm"power_nat_number_of"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
867 |
val zpower_number_of_even = thm"zpower_number_of_even"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
868 |
val zpower_number_of_odd = thm"zpower_number_of_odd"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
869 |
val nat_number_of_Pls = thm"nat_number_of_Pls"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
870 |
val nat_number_of_Min = thm"nat_number_of_Min"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
871 |
val Let_Suc = thm"Let_Suc"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
872 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
873 |
val nat_number = thms"nat_number"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
874 |
|
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
875 |
val nat_number_of_add_left = thm"nat_number_of_add_left"; |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14417
diff
changeset
|
876 |
val nat_number_of_mult_left = thm"nat_number_of_mult_left"; |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
877 |
val left_add_mult_distrib = thm"left_add_mult_distrib"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
878 |
val nat_diff_add_eq1 = thm"nat_diff_add_eq1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
879 |
val nat_diff_add_eq2 = thm"nat_diff_add_eq2"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
880 |
val nat_eq_add_iff1 = thm"nat_eq_add_iff1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
881 |
val nat_eq_add_iff2 = thm"nat_eq_add_iff2"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
882 |
val nat_less_add_iff1 = thm"nat_less_add_iff1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
883 |
val nat_less_add_iff2 = thm"nat_less_add_iff2"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
884 |
val nat_le_add_iff1 = thm"nat_le_add_iff1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
885 |
val nat_le_add_iff2 = thm"nat_le_add_iff2"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
886 |
val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
887 |
val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
888 |
val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
889 |
val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
890 |
val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
891 |
val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
892 |
val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj"; |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
893 |
val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj"; |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
894 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14288
diff
changeset
|
895 |
val power_minus_even = thm"power_minus_even"; |
14273
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
896 |
*} |
e33ffff0123c
further simplifications of the integer development; converting more .ML files
paulson
parents:
14272
diff
changeset
|
897 |
|
7032 | 898 |
end |