author | haftmann |
Tue, 27 Oct 2020 16:59:44 +0000 | |
changeset 72512 | 83b5911c0164 |
parent 71837 | dca11678c495 |
child 73109 | 783406dd051e |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Int.thy |
25919
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joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
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|
2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
41959 | 3 |
Author: Tobias Nipkow, Florian Haftmann, TU Muenchen |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff
changeset
|
4 |
*) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
5 |
|
60758 | 6 |
section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close> |
25919
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joined theories IntDef, Numeral, IntArith to theory Int
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diff
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|
7 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
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|
8 |
theory Int |
63652 | 9 |
imports Equiv_Relations Power Quotient Fun_Def |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff
changeset
|
10 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
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|
11 |
|
60758 | 12 |
subsection \<open>Definition of integers as a quotient type\<close> |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
13 |
|
63652 | 14 |
definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" |
15 |
where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)" |
|
48045 | 16 |
|
17 |
lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y" |
|
18 |
by (simp add: intrel_def) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff
changeset
|
19 |
|
48045 | 20 |
quotient_type int = "nat \<times> nat" / "intrel" |
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45607
diff
changeset
|
21 |
morphisms Rep_Integ Abs_Integ |
48045 | 22 |
proof (rule equivpI) |
63652 | 23 |
show "reflp intrel" by (auto simp: reflp_def) |
24 |
show "symp intrel" by (auto simp: symp_def) |
|
25 |
show "transp intrel" by (auto simp: transp_def) |
|
48045 | 26 |
qed |
25919
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haftmann
parents:
diff
changeset
|
27 |
|
48045 | 28 |
lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]: |
63652 | 29 |
"(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P" |
30 |
by (induct z) auto |
|
31 |
||
48045 | 32 |
|
60758 | 33 |
subsection \<open>Integers form a commutative ring\<close> |
48045 | 34 |
|
35 |
instantiation int :: comm_ring_1 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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diff
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36 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff
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37 |
|
51994 | 38 |
lift_definition zero_int :: "int" is "(0, 0)" . |
25919
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parents:
diff
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39 |
|
51994 | 40 |
lift_definition one_int :: "int" is "(1, 0)" . |
25919
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parents:
diff
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41 |
|
48045 | 42 |
lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
43 |
is "\<lambda>(x, y) (u, v). (x + u, y + v)" |
|
44 |
by clarsimp |
|
25919
8b1c0d434824
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parents:
diff
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|
45 |
|
48045 | 46 |
lift_definition uminus_int :: "int \<Rightarrow> int" |
47 |
is "\<lambda>(x, y). (y, x)" |
|
48 |
by clarsimp |
|
25919
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parents:
diff
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|
49 |
|
48045 | 50 |
lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
51 |
is "\<lambda>(x, y) (u, v). (x + v, y + u)" |
|
52 |
by clarsimp |
|
25919
8b1c0d434824
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parents:
diff
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53 |
|
48045 | 54 |
lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int" |
55 |
is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)" |
|
56 |
proof (clarsimp) |
|
57 |
fix s t u v w x y z :: nat |
|
58 |
assume "s + v = u + t" and "w + z = y + x" |
|
63652 | 59 |
then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) = |
60 |
(u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)" |
|
48045 | 61 |
by simp |
63652 | 62 |
then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)" |
48045 | 63 |
by (simp add: algebra_simps) |
64 |
qed |
|
25919
8b1c0d434824
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parents:
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65 |
|
48045 | 66 |
instance |
63652 | 67 |
by standard (transfer; clarsimp simp: algebra_simps)+ |
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68 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
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69 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
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diff
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70 |
|
63652 | 71 |
abbreviation int :: "nat \<Rightarrow> int" |
72 |
where "int \<equiv> of_nat" |
|
44709 | 73 |
|
48045 | 74 |
lemma int_def: "int n = Abs_Integ (n, 0)" |
63652 | 75 |
by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq) |
25919
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haftmann
parents:
diff
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76 |
|
70927 | 77 |
lemma int_transfer [transfer_rule]: |
78 |
includes lifting_syntax |
|
79 |
shows "rel_fun (=) pcr_int (\<lambda>n. (n, 0)) int" |
|
63652 | 80 |
by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def) |
25919
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haftmann
parents:
diff
changeset
|
81 |
|
63652 | 82 |
lemma int_diff_cases: obtains (diff) m n where "z = int m - int n" |
48045 | 83 |
by transfer clarsimp |
84 |
||
63652 | 85 |
|
60758 | 86 |
subsection \<open>Integers are totally ordered\<close> |
25919
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parents:
diff
changeset
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87 |
|
48045 | 88 |
instantiation int :: linorder |
89 |
begin |
|
90 |
||
91 |
lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool" |
|
92 |
is "\<lambda>(x, y) (u, v). x + v \<le> u + y" |
|
93 |
by auto |
|
94 |
||
95 |
lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool" |
|
96 |
is "\<lambda>(x, y) (u, v). x + v < u + y" |
|
97 |
by auto |
|
98 |
||
99 |
instance |
|
61169 | 100 |
by standard (transfer, force)+ |
48045 | 101 |
|
102 |
end |
|
25919
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diff
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103 |
|
8b1c0d434824
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parents:
diff
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104 |
instantiation int :: distrib_lattice |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
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105 |
begin |
8b1c0d434824
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haftmann
parents:
diff
changeset
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106 |
|
63652 | 107 |
definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min" |
25919
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haftmann
parents:
diff
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108 |
|
63652 | 109 |
definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max" |
25919
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110 |
|
8b1c0d434824
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haftmann
parents:
diff
changeset
|
111 |
instance |
63652 | 112 |
by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2) |
25919
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113 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
114 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
115 |
|
60758 | 116 |
subsection \<open>Ordering properties of arithmetic operations\<close> |
48045 | 117 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
118 |
instance int :: ordered_cancel_ab_semigroup_add |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
119 |
proof |
8b1c0d434824
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haftmann
parents:
diff
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|
120 |
fix i j k :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
121 |
show "i \<le> j \<Longrightarrow> k + i \<le> k + j" |
48045 | 122 |
by transfer clarsimp |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
123 |
qed |
8b1c0d434824
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haftmann
parents:
diff
changeset
|
124 |
|
63652 | 125 |
text \<open>Strict Monotonicity of Multiplication.\<close> |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
126 |
|
63652 | 127 |
text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close> |
128 |
lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j" |
|
129 |
for i j :: int |
|
130 |
proof (induct k) |
|
131 |
case 0 |
|
132 |
then show ?case by simp |
|
133 |
next |
|
134 |
case (Suc k) |
|
135 |
then show ?case |
|
136 |
by (cases "k = 0") (simp_all add: distrib_right add_strict_mono) |
|
137 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
138 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
139 |
lemma zero_le_imp_eq_int: |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
140 |
assumes "k \<ge> (0::int)" shows "\<exists>n. k = int n" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
141 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
142 |
have "b \<le> a \<Longrightarrow> \<exists>n::nat. a = n + b" for a b |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
143 |
by (rule_tac x="a - b" in exI) simp |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
144 |
with assms show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
145 |
by transfer auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
146 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
147 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
148 |
lemma zero_less_imp_eq_int: |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
149 |
assumes "k > (0::int)" shows "\<exists>n>0. k = int n" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
150 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
151 |
have "b < a \<Longrightarrow> \<exists>n::nat. n>0 \<and> a = n + b" for a b |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
152 |
by (rule_tac x="a - b" in exI) simp |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
153 |
with assms show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
154 |
by transfer auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
155 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
156 |
|
63652 | 157 |
lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j" |
158 |
for i j k :: int |
|
159 |
by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
160 |
|
63652 | 161 |
|
162 |
text \<open>The integers form an ordered integral domain.\<close> |
|
163 |
||
48045 | 164 |
instantiation int :: linordered_idom |
165 |
begin |
|
166 |
||
63652 | 167 |
definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)" |
48045 | 168 |
|
63652 | 169 |
definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)" |
48045 | 170 |
|
63652 | 171 |
instance |
172 |
proof |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
173 |
fix i j k :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
174 |
show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
175 |
by (rule zmult_zless_mono2) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
176 |
show "\<bar>i\<bar> = (if i < 0 then -i else i)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
177 |
by (simp only: zabs_def) |
61076 | 178 |
show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
179 |
by (simp only: zsgn_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
180 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
181 |
|
48045 | 182 |
end |
183 |
||
63652 | 184 |
lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z" |
185 |
for w z :: int |
|
48045 | 186 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
187 |
|
63652 | 188 |
lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))" |
189 |
for w z :: int |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
190 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
191 |
have "\<And>a b c d. a + d < c + b \<Longrightarrow> \<exists>n. c + b = Suc (a + n + d)" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
192 |
by (rule_tac x="c+b - Suc(a+d)" in exI) arith |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
193 |
then show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
194 |
by transfer auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
195 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
196 |
|
63652 | 197 |
lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs") |
198 |
for z :: int |
|
62347 | 199 |
proof |
63652 | 200 |
assume ?rhs |
201 |
then show ?lhs by simp |
|
62347 | 202 |
next |
63652 | 203 |
assume ?lhs |
204 |
with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp |
|
205 |
then have "\<bar>z\<bar> \<le> 0" by simp |
|
206 |
then show ?rhs by simp |
|
62347 | 207 |
qed |
208 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
209 |
|
61799 | 210 |
subsection \<open>Embedding of the Integers into any \<open>ring_1\<close>: \<open>of_int\<close>\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
211 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
212 |
context ring_1 |
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213 |
begin |
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214 |
|
63652 | 215 |
lift_definition of_int :: "int \<Rightarrow> 'a" |
216 |
is "\<lambda>(i, j). of_nat i - of_nat j" |
|
48045 | 217 |
by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq |
63652 | 218 |
of_nat_add [symmetric] simp del: of_nat_add) |
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219 |
|
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lemma of_int_0 [simp]: "of_int 0 = 0" |
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by transfer simp |
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|
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lemma of_int_1 [simp]: "of_int 1 = 1" |
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by transfer simp |
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|
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lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z" |
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by transfer (clarsimp simp add: algebra_simps) |
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228 |
|
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lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)" |
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by (transfer fixing: uminus) clarsimp |
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231 |
|
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232 |
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z" |
54230
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more simplification rules on unary and binary minus
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233 |
using of_int_add [of w "- z"] by simp |
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234 |
|
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235 |
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" |
63652 | 236 |
by (transfer fixing: times) (clarsimp simp add: algebra_simps) |
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237 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
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|
238 |
lemma mult_of_int_commute: "of_int x * y = y * of_int x" |
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Rounding function, uniform limits, cotangent, binomial identities
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239 |
by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute) |
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240 |
|
63652 | 241 |
text \<open>Collapse nested embeddings.\<close> |
44709 | 242 |
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n" |
63652 | 243 |
by (induct n) auto |
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244 |
|
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245 |
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k" |
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by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric]) |
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247 |
|
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eliminiated neg_numeral in favour of - (numeral _)
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248 |
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k" |
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249 |
by simp |
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250 |
|
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lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n" |
31015 | 252 |
by (induct n) simp_all |
253 |
||
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lemma of_int_of_bool [simp]: |
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"of_int (of_bool P) = of_bool P" |
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more fundamental definition of div and mod on int
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256 |
by auto |
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|
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end |
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|
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context ring_char_0 |
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261 |
begin |
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262 |
|
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lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z" |
264 |
by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add) |
|
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265 |
|
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text \<open>Special cases where either operand is zero.\<close> |
267 |
lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0" |
|
36424 | 268 |
using of_int_eq_iff [of z 0] by simp |
269 |
||
63652 | 270 |
lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0" |
36424 | 271 |
using of_int_eq_iff [of 0 z] by simp |
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272 |
|
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lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1" |
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using of_int_eq_iff [of z 1] by simp |
275 |
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lemma numeral_power_eq_of_int_cancel_iff [simp]: |
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"numeral x ^ n = of_int y \<longleftrightarrow> numeral x ^ n = y" |
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278 |
using of_int_eq_iff[of "numeral x ^ n" y, unfolded of_int_numeral of_int_power] . |
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279 |
|
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280 |
lemma of_int_eq_numeral_power_cancel_iff [simp]: |
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281 |
"of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n" |
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282 |
using numeral_power_eq_of_int_cancel_iff [of x n y] by (metis (mono_tags)) |
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283 |
|
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generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
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284 |
lemma neg_numeral_power_eq_of_int_cancel_iff [simp]: |
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285 |
"(- numeral x) ^ n = of_int y \<longleftrightarrow> (- numeral x) ^ n = y" |
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286 |
using of_int_eq_iff[of "(- numeral x) ^ n" y] |
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287 |
by simp |
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288 |
|
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289 |
lemma of_int_eq_neg_numeral_power_cancel_iff [simp]: |
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290 |
"of_int y = (- numeral x) ^ n \<longleftrightarrow> y = (- numeral x) ^ n" |
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generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
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parents:
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diff
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291 |
using neg_numeral_power_eq_of_int_cancel_iff[of x n y] by (metis (mono_tags)) |
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292 |
|
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lemma of_int_eq_of_int_power_cancel_iff[simp]: "(of_int b) ^ w = of_int x \<longleftrightarrow> b ^ w = x" |
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294 |
by (metis of_int_power of_int_eq_iff) |
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295 |
|
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296 |
lemma of_int_power_eq_of_int_cancel_iff[simp]: "of_int x = (of_int b) ^ w \<longleftrightarrow> x = b ^ w" |
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297 |
by (metis of_int_eq_of_int_power_cancel_iff) |
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298 |
|
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299 |
end |
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diff
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|
300 |
|
36424 | 301 |
context linordered_idom |
302 |
begin |
|
303 |
||
63652 | 304 |
text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close> |
36424 | 305 |
subclass ring_char_0 .. |
306 |
||
63652 | 307 |
lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z" |
308 |
by (transfer fixing: less_eq) |
|
309 |
(clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add) |
|
36424 | 310 |
|
63652 | 311 |
lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z" |
36424 | 312 |
by (simp add: less_le order_less_le) |
313 |
||
63652 | 314 |
lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z" |
36424 | 315 |
using of_int_le_iff [of 0 z] by simp |
316 |
||
63652 | 317 |
lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0" |
36424 | 318 |
using of_int_le_iff [of z 0] by simp |
319 |
||
63652 | 320 |
lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z" |
36424 | 321 |
using of_int_less_iff [of 0 z] by simp |
322 |
||
63652 | 323 |
lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0" |
36424 | 324 |
using of_int_less_iff [of z 0] by simp |
325 |
||
63652 | 326 |
lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z" |
61234 | 327 |
using of_int_le_iff [of 1 z] by simp |
328 |
||
63652 | 329 |
lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1" |
61234 | 330 |
using of_int_le_iff [of z 1] by simp |
331 |
||
63652 | 332 |
lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z" |
61234 | 333 |
using of_int_less_iff [of 1 z] by simp |
334 |
||
63652 | 335 |
lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1" |
61234 | 336 |
using of_int_less_iff [of z 1] by simp |
337 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
338 |
lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
339 |
by simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
340 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
341 |
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
342 |
by simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
343 |
|
63652 | 344 |
lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>" |
62347 | 345 |
by (auto simp add: abs_if) |
346 |
||
347 |
lemma of_int_lessD: |
|
348 |
assumes "\<bar>of_int n\<bar> < x" |
|
349 |
shows "n = 0 \<or> x > 1" |
|
350 |
proof (cases "n = 0") |
|
63652 | 351 |
case True |
352 |
then show ?thesis by simp |
|
62347 | 353 |
next |
354 |
case False |
|
355 |
then have "\<bar>n\<bar> \<noteq> 0" by simp |
|
356 |
then have "\<bar>n\<bar> > 0" by simp |
|
357 |
then have "\<bar>n\<bar> \<ge> 1" |
|
358 |
using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp |
|
359 |
then have "\<bar>of_int n\<bar> \<ge> 1" |
|
360 |
unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp |
|
361 |
then have "1 < x" using assms by (rule le_less_trans) |
|
362 |
then show ?thesis .. |
|
363 |
qed |
|
364 |
||
365 |
lemma of_int_leD: |
|
366 |
assumes "\<bar>of_int n\<bar> \<le> x" |
|
367 |
shows "n = 0 \<or> 1 \<le> x" |
|
368 |
proof (cases "n = 0") |
|
63652 | 369 |
case True |
370 |
then show ?thesis by simp |
|
62347 | 371 |
next |
372 |
case False |
|
373 |
then have "\<bar>n\<bar> \<noteq> 0" by simp |
|
374 |
then have "\<bar>n\<bar> > 0" by simp |
|
375 |
then have "\<bar>n\<bar> \<ge> 1" |
|
376 |
using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp |
|
377 |
then have "\<bar>of_int n\<bar> \<ge> 1" |
|
378 |
unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp |
|
379 |
then have "1 \<le> x" using assms by (rule order_trans) |
|
380 |
then show ?thesis .. |
|
381 |
qed |
|
382 |
||
66912
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
383 |
lemma numeral_power_le_of_int_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
384 |
"numeral x ^ n \<le> of_int a \<longleftrightarrow> numeral x ^ n \<le> a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
385 |
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_le_iff) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
386 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
387 |
lemma of_int_le_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
388 |
"of_int a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
389 |
by (metis (mono_tags) local.numeral_power_eq_of_int_cancel_iff of_int_le_iff) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
390 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
391 |
lemma numeral_power_less_of_int_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
392 |
"numeral x ^ n < of_int a \<longleftrightarrow> numeral x ^ n < a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
393 |
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
394 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
395 |
lemma of_int_less_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
396 |
"of_int a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
397 |
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
398 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
399 |
lemma neg_numeral_power_le_of_int_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
400 |
"(- numeral x) ^ n \<le> of_int a \<longleftrightarrow> (- numeral x) ^ n \<le> a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
401 |
by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
402 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
403 |
lemma of_int_le_neg_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
404 |
"of_int a \<le> (- numeral x) ^ n \<longleftrightarrow> a \<le> (- numeral x) ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
405 |
by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
406 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
407 |
lemma neg_numeral_power_less_of_int_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
408 |
"(- numeral x) ^ n < of_int a \<longleftrightarrow> (- numeral x) ^ n < a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
409 |
using of_int_less_iff[of "(- numeral x) ^ n" a] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
410 |
by simp |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
411 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
412 |
lemma of_int_less_neg_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
413 |
"of_int a < (- numeral x) ^ n \<longleftrightarrow> a < (- numeral x::int) ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
414 |
using of_int_less_iff[of a "(- numeral x) ^ n"] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
415 |
by simp |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
416 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
417 |
lemma of_int_le_of_int_power_cancel_iff[simp]: "(of_int b) ^ w \<le> of_int x \<longleftrightarrow> b ^ w \<le> x" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
418 |
by (metis (mono_tags) of_int_le_iff of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
419 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
420 |
lemma of_int_power_le_of_int_cancel_iff[simp]: "of_int x \<le> (of_int b) ^ w\<longleftrightarrow> x \<le> b ^ w" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
421 |
by (metis (mono_tags) of_int_le_iff of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
422 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
423 |
lemma of_int_less_of_int_power_cancel_iff[simp]: "(of_int b) ^ w < of_int x \<longleftrightarrow> b ^ w < x" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
424 |
by (metis (mono_tags) of_int_less_iff of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
425 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
426 |
lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w\<longleftrightarrow> x < b ^ w" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
427 |
by (metis (mono_tags) of_int_less_iff of_int_power) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
428 |
|
67969
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
429 |
lemma of_int_max: "of_int (max x y) = max (of_int x) (of_int y)" |
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
430 |
by (auto simp: max_def) |
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
431 |
|
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
432 |
lemma of_int_min: "of_int (min x y) = min (of_int x) (of_int y)" |
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
433 |
by (auto simp: min_def) |
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
434 |
|
36424 | 435 |
end |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
436 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
437 |
context division_ring |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
438 |
begin |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
439 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
440 |
lemmas mult_inverse_of_int_commute = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
441 |
mult_commute_imp_mult_inverse_commute[OF mult_of_int_commute] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
442 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
443 |
end |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69700
diff
changeset
|
444 |
|
69593 | 445 |
text \<open>Comparisons involving \<^term>\<open>of_int\<close>.\<close> |
61234 | 446 |
|
63652 | 447 |
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n" |
61234 | 448 |
using of_int_eq_iff by fastforce |
449 |
||
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
450 |
lemma of_int_le_numeral_iff [simp]: |
63652 | 451 |
"of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n" |
61234 | 452 |
using of_int_le_iff [of z "numeral n"] by simp |
453 |
||
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
454 |
lemma of_int_numeral_le_iff [simp]: |
63652 | 455 |
"(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z" |
61234 | 456 |
using of_int_le_iff [of "numeral n"] by simp |
457 |
||
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
458 |
lemma of_int_less_numeral_iff [simp]: |
63652 | 459 |
"of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n" |
61234 | 460 |
using of_int_less_iff [of z "numeral n"] by simp |
461 |
||
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
462 |
lemma of_int_numeral_less_iff [simp]: |
63652 | 463 |
"(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z" |
61234 | 464 |
using of_int_less_iff [of "numeral n" z] by simp |
465 |
||
63652 | 466 |
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56525
diff
changeset
|
467 |
by (metis of_int_of_nat_eq of_int_less_iff) |
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56525
diff
changeset
|
468 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
469 |
lemma of_int_eq_id [simp]: "of_int = id" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
470 |
proof |
63652 | 471 |
show "of_int z = id z" for z |
472 |
by (cases z rule: int_diff_cases) simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
473 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
474 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
475 |
instance int :: no_top |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
476 |
proof |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
477 |
show "\<And>x::int. \<exists>y. x < y" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
478 |
by (rule_tac x="x + 1" in exI) simp |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
479 |
qed |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
480 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
481 |
instance int :: no_bot |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
482 |
proof |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
483 |
show "\<And>x::int. \<exists>y. y < x" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
484 |
by (rule_tac x="x - 1" in exI) simp |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
485 |
qed |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
486 |
|
63652 | 487 |
|
61799 | 488 |
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
489 |
|
48045 | 490 |
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y" |
491 |
by auto |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
492 |
|
44709 | 493 |
lemma nat_int [simp]: "nat (int n) = n" |
48045 | 494 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
495 |
|
44709 | 496 |
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)" |
48045 | 497 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
498 |
|
63652 | 499 |
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z" |
500 |
by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
501 |
|
63652 | 502 |
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0" |
48045 | 503 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
504 |
|
63652 | 505 |
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z" |
48045 | 506 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
507 |
|
69593 | 508 |
text \<open>An alternative condition is \<^term>\<open>0 \<le> w\<close>.\<close> |
63652 | 509 |
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z" |
510 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
511 |
|
63652 | 512 |
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z" |
513 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
514 |
|
63652 | 515 |
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z" |
48045 | 516 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
517 |
|
64714
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
518 |
lemma nonneg_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
519 |
assumes "0 \<le> k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
520 |
obtains n where "k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
521 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
522 |
from assms have "k = int (nat k)" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
523 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
524 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
525 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
526 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
527 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
528 |
lemma pos_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
529 |
assumes "0 < k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
530 |
obtains n where "k = int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
531 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
532 |
from assms have "0 \<le> k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
533 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
534 |
then obtain n where "k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
535 |
by (rule nonneg_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
536 |
moreover have "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
537 |
using \<open>k = int n\<close> assms by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
538 |
ultimately show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
539 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
540 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
541 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
542 |
lemma nonpos_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
543 |
assumes "k \<le> 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
544 |
obtains n where "k = - int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
545 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
546 |
from assms have "- k \<ge> 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
547 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
548 |
then obtain n where "- k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
549 |
by (rule nonneg_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
550 |
then have "k = - int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
551 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
552 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
553 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
554 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
555 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
556 |
lemma neg_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
557 |
assumes "k < 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
558 |
obtains n where "k = - int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
559 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
560 |
from assms have "- k > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
561 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
562 |
then obtain n where "- k = int n" and "- k > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
563 |
by (blast elim: pos_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
564 |
then have "k = - int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
565 |
by simp_all |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
566 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
567 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
568 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
569 |
|
63652 | 570 |
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
48045 | 571 |
by transfer (clarsimp simp add: le_imp_diff_is_add) |
60162 | 572 |
|
63652 | 573 |
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
54223 | 574 |
using nat_eq_iff [of w m] by auto |
575 |
||
63652 | 576 |
lemma nat_0 [simp]: "nat 0 = 0" |
54223 | 577 |
by (simp add: nat_eq_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
578 |
|
63652 | 579 |
lemma nat_1 [simp]: "nat 1 = Suc 0" |
54223 | 580 |
by (simp add: nat_eq_iff) |
581 |
||
63652 | 582 |
lemma nat_numeral [simp]: "nat (numeral k) = numeral k" |
54223 | 583 |
by (simp add: nat_eq_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
584 |
|
63652 | 585 |
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0" |
54223 | 586 |
by simp |
587 |
||
588 |
lemma nat_2: "nat 2 = Suc (Suc 0)" |
|
589 |
by simp |
|
60162 | 590 |
|
63652 | 591 |
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m" |
48045 | 592 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
593 |
|
44709 | 594 |
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n" |
48045 | 595 |
by transfer (clarsimp simp add: le_diff_conv) |
44707 | 596 |
|
597 |
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y" |
|
48045 | 598 |
by transfer auto |
44707 | 599 |
|
63652 | 600 |
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0" |
601 |
for i :: int |
|
48045 | 602 |
by transfer clarsimp |
29700 | 603 |
|
63652 | 604 |
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z" |
605 |
by (auto simp add: nat_eq_iff2) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
606 |
|
63652 | 607 |
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z" |
608 |
using zless_nat_conj [of 0] by auto |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
609 |
|
63652 | 610 |
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'" |
48045 | 611 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
612 |
|
63652 | 613 |
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y" |
54223 | 614 |
by transfer clarsimp |
60162 | 615 |
|
63652 | 616 |
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'" |
54223 | 617 |
by (rule nat_diff_distrib') auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
618 |
|
44709 | 619 |
lemma nat_zminus_int [simp]: "nat (- int n) = 0" |
48045 | 620 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
621 |
|
63652 | 622 |
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k" |
53065 | 623 |
by transfer auto |
60162 | 624 |
|
63652 | 625 |
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z" |
48045 | 626 |
by transfer (clarsimp simp add: less_diff_conv) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
627 |
|
63652 | 628 |
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z" |
48066
c6783c9b87bf
transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents:
48045
diff
changeset
|
629 |
by transfer (clarsimp simp add: of_nat_diff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
630 |
|
63652 | 631 |
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')" |
54249 | 632 |
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral) |
633 |
||
66886 | 634 |
lemma nat_abs_triangle_ineq: |
635 |
"nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>" |
|
636 |
by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq) |
|
637 |
||
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
638 |
lemma nat_of_bool [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
639 |
"nat (of_bool P) = of_bool P" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
640 |
by auto |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
641 |
|
66836 | 642 |
lemma split_nat [arith_split]: "P (nat i) \<longleftrightarrow> ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))" |
643 |
(is "?P = (?L \<and> ?R)") |
|
644 |
for i :: int |
|
645 |
proof (cases "i < 0") |
|
646 |
case True |
|
647 |
then show ?thesis |
|
648 |
by auto |
|
649 |
next |
|
650 |
case False |
|
651 |
have "?P = ?L" |
|
652 |
proof |
|
653 |
assume ?P |
|
654 |
then show ?L using False by auto |
|
655 |
next |
|
656 |
assume ?L |
|
657 |
moreover from False have "int (nat i) = i" |
|
658 |
by (simp add: not_less) |
|
659 |
ultimately show ?P |
|
660 |
by simp |
|
661 |
qed |
|
662 |
with False show ?thesis by simp |
|
663 |
qed |
|
664 |
||
665 |
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))" |
|
666 |
by (auto split: split_nat) |
|
667 |
||
668 |
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))" |
|
669 |
proof |
|
670 |
assume "\<exists>x. P x" |
|
671 |
then obtain x where "P x" .. |
|
672 |
then have "int x \<ge> 0 \<and> P (nat (int x))" by simp |
|
673 |
then show "\<exists>x\<ge>0. P (nat x)" .. |
|
674 |
next |
|
675 |
assume "\<exists>x\<ge>0. P (nat x)" |
|
676 |
then show "\<exists>x. P x" by auto |
|
677 |
qed |
|
678 |
||
54249 | 679 |
|
60758 | 680 |
text \<open>For termination proofs:\<close> |
63652 | 681 |
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" .. |
29779 | 682 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
683 |
|
69593 | 684 |
subsection \<open>Lemmas about the Function \<^term>\<open>of_nat\<close> and Orderings\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
685 |
|
61076 | 686 |
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)" |
63652 | 687 |
by (simp add: order_less_le del: of_nat_Suc) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
688 |
|
44709 | 689 |
lemma negative_zless [iff]: "- (int (Suc n)) < int m" |
63652 | 690 |
by (rule negative_zless_0 [THEN order_less_le_trans], simp) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
691 |
|
44709 | 692 |
lemma negative_zle_0: "- int n \<le> 0" |
63652 | 693 |
by (simp add: minus_le_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
694 |
|
44709 | 695 |
lemma negative_zle [iff]: "- int n \<le> int m" |
63652 | 696 |
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
697 |
|
63652 | 698 |
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)" |
699 |
by (subst le_minus_iff) (simp del: of_nat_Suc) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
700 |
|
63652 | 701 |
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0" |
48045 | 702 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
703 |
|
63652 | 704 |
lemma not_int_zless_negative [simp]: "\<not> int n < - int m" |
705 |
by (simp add: linorder_not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
706 |
|
63652 | 707 |
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0" |
708 |
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
709 |
|
63652 | 710 |
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)" |
711 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
62348 | 712 |
proof |
63652 | 713 |
assume ?rhs |
714 |
then show ?lhs by auto |
|
62348 | 715 |
next |
63652 | 716 |
assume ?lhs |
62348 | 717 |
then have "0 \<le> z - w" by simp |
718 |
then obtain n where "z - w = int n" |
|
719 |
using zero_le_imp_eq_int [of "z - w"] by blast |
|
63652 | 720 |
then have "z = w + int n" by simp |
721 |
then show ?rhs .. |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
722 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
723 |
|
44709 | 724 |
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z" |
63652 | 725 |
by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
726 |
|
63652 | 727 |
text \<open> |
728 |
This version is proved for all ordered rings, not just integers! |
|
729 |
It is proved here because attribute \<open>arith_split\<close> is not available |
|
730 |
in theory \<open>Rings\<close>. |
|
731 |
But is it really better than just rewriting with \<open>abs_if\<close>? |
|
732 |
\<close> |
|
733 |
lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))" |
|
734 |
for a :: "'a::linordered_idom" |
|
735 |
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
736 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
737 |
lemma negD: |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
738 |
assumes "x < 0" shows "\<exists>n. x = - (int (Suc n))" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
739 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
740 |
have "\<And>a b. a < b \<Longrightarrow> \<exists>n. Suc (a + n) = b" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
741 |
by (rule_tac x="b - Suc a" in exI) arith |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
742 |
with assms show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
743 |
by transfer auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
744 |
qed |
63652 | 745 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
746 |
|
60758 | 747 |
subsection \<open>Cases and induction\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
748 |
|
63652 | 749 |
text \<open> |
750 |
Now we replace the case analysis rule by a more conventional one: |
|
751 |
whether an integer is negative or not. |
|
752 |
\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
753 |
|
63652 | 754 |
text \<open>This version is symmetric in the two subgoals.\<close> |
755 |
lemma int_cases2 [case_names nonneg nonpos, cases type: int]: |
|
756 |
"(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P" |
|
757 |
by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym]) |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59582
diff
changeset
|
758 |
|
63652 | 759 |
text \<open>This is the default, with a negative case.\<close> |
760 |
lemma int_cases [case_names nonneg neg, cases type: int]: |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
761 |
assumes pos: "\<And>n. z = int n \<Longrightarrow> P" and neg: "\<And>n. z = - (int (Suc n)) \<Longrightarrow> P" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
762 |
shows P |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
763 |
proof (cases "z < 0") |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
764 |
case True |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
765 |
with neg show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
766 |
by (blast dest!: negD) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
767 |
next |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
768 |
case False |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
769 |
with pos show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
770 |
by (force simp add: linorder_not_less dest: nat_0_le [THEN sym]) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
771 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
772 |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
773 |
lemma int_cases3 [case_names zero pos neg]: |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
774 |
fixes k :: int |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
775 |
assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P" |
61204 | 776 |
and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P" |
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
777 |
shows "P" |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
778 |
proof (cases k "0::int" rule: linorder_cases) |
63652 | 779 |
case equal |
780 |
with assms(1) show P by simp |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
781 |
next |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
782 |
case greater |
63539 | 783 |
then have *: "nat k > 0" by simp |
784 |
moreover from * have "k = int (nat k)" by auto |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
785 |
ultimately show P using assms(2) by blast |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
786 |
next |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
787 |
case less |
63539 | 788 |
then have *: "nat (- k) > 0" by simp |
789 |
moreover from * have "k = - int (nat (- k))" by auto |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
790 |
ultimately show P using assms(3) by blast |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
791 |
qed |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
792 |
|
63652 | 793 |
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]: |
794 |
"(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z" |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
795 |
by (cases z) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
796 |
|
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
797 |
lemma sgn_mult_dvd_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
798 |
"sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
799 |
by (cases r rule: int_cases3) auto |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
800 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
801 |
lemma mult_sgn_dvd_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
802 |
"l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
803 |
using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
804 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
805 |
lemma dvd_sgn_mult_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
806 |
"l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
807 |
by (cases r rule: int_cases3) simp_all |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
808 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
809 |
lemma dvd_mult_sgn_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
810 |
"l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
811 |
using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
812 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
813 |
lemma int_sgnE: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
814 |
fixes k :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
815 |
obtains n and l where "k = sgn l * int n" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
816 |
proof - |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
817 |
have "k = sgn k * int (nat \<bar>k\<bar>)" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
818 |
by (simp add: sgn_mult_abs) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
819 |
then show ?thesis .. |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
820 |
qed |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
821 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
822 |
|
60758 | 823 |
subsubsection \<open>Binary comparisons\<close> |
28958 | 824 |
|
60758 | 825 |
text \<open>Preliminaries\<close> |
28958 | 826 |
|
60162 | 827 |
lemma le_imp_0_less: |
63652 | 828 |
fixes z :: int |
28958 | 829 |
assumes le: "0 \<le> z" |
63652 | 830 |
shows "0 < 1 + z" |
28958 | 831 |
proof - |
832 |
have "0 \<le> z" by fact |
|
63652 | 833 |
also have "\<dots> < z + 1" by (rule less_add_one) |
834 |
also have "\<dots> = 1 + z" by (simp add: ac_simps) |
|
28958 | 835 |
finally show "0 < 1 + z" . |
836 |
qed |
|
837 |
||
63652 | 838 |
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0" |
839 |
for z :: int |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
840 |
proof (cases z) |
28958 | 841 |
case (nonneg n) |
63652 | 842 |
then show ?thesis |
843 |
by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le]) |
|
28958 | 844 |
next |
845 |
case (neg n) |
|
63652 | 846 |
then show ?thesis |
847 |
by (simp del: of_nat_Suc of_nat_add of_nat_1 |
|
848 |
add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric]) |
|
28958 | 849 |
qed |
850 |
||
63652 | 851 |
|
60758 | 852 |
subsubsection \<open>Comparisons, for Ordered Rings\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
853 |
|
63652 | 854 |
lemma odd_nonzero: "1 + z + z \<noteq> 0" |
855 |
for z :: int |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
856 |
proof (cases z) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
857 |
case (nonneg n) |
63652 | 858 |
have le: "0 \<le> z + z" |
859 |
by (simp add: nonneg add_increasing) |
|
860 |
then show ?thesis |
|
67116 | 861 |
using le_imp_0_less [OF le] by (auto simp: ac_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
862 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
863 |
case (neg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
864 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
865 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
866 |
assume eq: "1 + z + z = 0" |
63652 | 867 |
have "0 < 1 + (int n + int n)" |
60162 | 868 |
by (simp add: le_imp_0_less add_increasing) |
63652 | 869 |
also have "\<dots> = - (1 + z + z)" |
60162 | 870 |
by (simp add: neg add.assoc [symmetric]) |
63652 | 871 |
also have "\<dots> = 0" by (simp add: eq) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
872 |
finally have "0<0" .. |
63652 | 873 |
then show False by blast |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
874 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
875 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
876 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
877 |
|
60758 | 878 |
subsection \<open>The Set of Integers\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
879 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
880 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
881 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
882 |
|
61070 | 883 |
definition Ints :: "'a set" ("\<int>") |
884 |
where "\<int> = range of_int" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
885 |
|
35634 | 886 |
lemma Ints_of_int [simp]: "of_int z \<in> \<int>" |
887 |
by (simp add: Ints_def) |
|
888 |
||
889 |
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>" |
|
45533 | 890 |
using Ints_of_int [of "of_nat n"] by simp |
35634 | 891 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
892 |
lemma Ints_0 [simp]: "0 \<in> \<int>" |
45533 | 893 |
using Ints_of_int [of "0"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
894 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
895 |
lemma Ints_1 [simp]: "1 \<in> \<int>" |
45533 | 896 |
using Ints_of_int [of "1"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
897 |
|
61552
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
898 |
lemma Ints_numeral [simp]: "numeral n \<in> \<int>" |
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
899 |
by (subst of_nat_numeral [symmetric], rule Ints_of_nat) |
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
900 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
901 |
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
902 |
by (force simp add: Ints_def simp flip: of_int_add intro: range_eqI) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
903 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
904 |
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
905 |
by (force simp add: Ints_def simp flip: of_int_minus intro: range_eqI) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
906 |
|
68721 | 907 |
lemma minus_in_Ints_iff: "-x \<in> \<int> \<longleftrightarrow> x \<in> \<int>" |
908 |
using Ints_minus[of x] Ints_minus[of "-x"] by auto |
|
909 |
||
35634 | 910 |
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
911 |
by (force simp add: Ints_def simp flip: of_int_diff intro: range_eqI) |
35634 | 912 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
913 |
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
914 |
by (force simp add: Ints_def simp flip: of_int_mult intro: range_eqI) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
915 |
|
35634 | 916 |
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>" |
63652 | 917 |
by (induct n) simp_all |
35634 | 918 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
919 |
lemma Ints_cases [cases set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
920 |
assumes "q \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
921 |
obtains (of_int) z where "q = of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
922 |
unfolding Ints_def |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
923 |
proof - |
60758 | 924 |
from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def . |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
925 |
then obtain z where "q = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
926 |
then show thesis .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
927 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
928 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
929 |
lemma Ints_induct [case_names of_int, induct set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
930 |
"q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
931 |
by (rule Ints_cases) auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
932 |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
933 |
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
934 |
unfolding Nats_def Ints_def |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
935 |
by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
936 |
|
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
937 |
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
938 |
proof (intro subsetI equalityI) |
63652 | 939 |
fix x :: 'a |
940 |
assume "x \<in> {of_int n |n. n \<ge> 0}" |
|
941 |
then obtain n where "x = of_int n" "n \<ge> 0" |
|
942 |
by (auto elim!: Ints_cases) |
|
943 |
then have "x = of_nat (nat n)" |
|
944 |
by (subst of_nat_nat) simp_all |
|
945 |
then show "x \<in> \<nat>" |
|
946 |
by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
947 |
next |
63652 | 948 |
fix x :: 'a |
949 |
assume "x \<in> \<nat>" |
|
950 |
then obtain n where "x = of_nat n" |
|
951 |
by (auto elim!: Nats_cases) |
|
952 |
then have "x = of_int (int n)" by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
953 |
also have "int n \<ge> 0" by simp |
63652 | 954 |
then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast |
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
955 |
finally show "x \<in> {of_int n |n. n \<ge> 0}" . |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
956 |
qed |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
957 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
958 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
959 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
960 |
lemma (in linordered_idom) Ints_abs [simp]: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
961 |
shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
962 |
by (auto simp: abs_if) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
963 |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
964 |
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
965 |
proof (intro subsetI equalityI) |
63652 | 966 |
fix x :: 'a |
967 |
assume "x \<in> {n \<in> \<int>. n \<ge> 0}" |
|
968 |
then obtain n where "x = of_int n" "n \<ge> 0" |
|
969 |
by (auto elim!: Ints_cases) |
|
970 |
then have "x = of_nat (nat n)" |
|
971 |
by (subst of_nat_nat) simp_all |
|
972 |
then show "x \<in> \<nat>" |
|
973 |
by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
974 |
qed (auto elim!: Nats_cases) |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
975 |
|
64849 | 976 |
lemma (in idom_divide) of_int_divide_in_Ints: |
977 |
"of_int a div of_int b \<in> \<int>" if "b dvd a" |
|
978 |
proof - |
|
979 |
from that obtain c where "a = b * c" .. |
|
980 |
then show ?thesis |
|
981 |
by (cases "of_int b = 0") simp_all |
|
982 |
qed |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
983 |
|
69593 | 984 |
text \<open>The premise involving \<^term>\<open>Ints\<close> prevents \<^term>\<open>a = 1/2\<close>.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
985 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
986 |
lemma Ints_double_eq_0_iff: |
63652 | 987 |
fixes a :: "'a::ring_char_0" |
61070 | 988 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 989 |
shows "a + a = 0 \<longleftrightarrow> a = 0" |
990 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
991 |
proof - |
63652 | 992 |
from in_Ints have "a \<in> range of_int" |
993 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
994 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
995 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
996 |
proof |
63652 | 997 |
assume ?rhs |
998 |
then show ?lhs by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
999 |
next |
63652 | 1000 |
assume ?lhs |
1001 |
with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp |
|
1002 |
then have "z + z = 0" by (simp only: of_int_eq_iff) |
|
67116 | 1003 |
then have "z = 0" by (simp only: double_zero) |
63652 | 1004 |
with a show ?rhs by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1005 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1006 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1007 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1008 |
lemma Ints_odd_nonzero: |
63652 | 1009 |
fixes a :: "'a::ring_char_0" |
61070 | 1010 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 1011 |
shows "1 + a + a \<noteq> 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1012 |
proof - |
63652 | 1013 |
from in_Ints have "a \<in> range of_int" |
1014 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1015 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1016 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1017 |
proof |
63652 | 1018 |
assume "1 + a + a = 0" |
1019 |
with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp |
|
1020 |
then have "1 + z + z = 0" by (simp only: of_int_eq_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1021 |
with odd_nonzero show False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1022 |
qed |
60162 | 1023 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1024 |
|
61070 | 1025 |
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1026 |
using of_nat_in_Nats [of "numeral w"] by simp |
35634 | 1027 |
|
60162 | 1028 |
lemma Ints_odd_less_0: |
63652 | 1029 |
fixes a :: "'a::linordered_idom" |
61070 | 1030 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 1031 |
shows "1 + a + a < 0 \<longleftrightarrow> a < 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1032 |
proof - |
63652 | 1033 |
from in_Ints have "a \<in> range of_int" |
1034 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1035 |
then obtain z where a: "a = of_int z" .. |
63652 | 1036 |
with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)" |
1037 |
by simp |
|
1038 |
also have "\<dots> \<longleftrightarrow> z < 0" |
|
1039 |
by (simp only: of_int_less_iff odd_less_0_iff) |
|
1040 |
also have "\<dots> \<longleftrightarrow> a < 0" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1041 |
by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1042 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1043 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1044 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1045 |
|
69593 | 1046 |
subsection \<open>\<^term>\<open>sum\<close> and \<^term>\<open>prod\<close>\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1047 |
|
69182 | 1048 |
context semiring_1 |
1049 |
begin |
|
1050 |
||
1051 |
lemma of_nat_sum [simp]: |
|
1052 |
"of_nat (sum f A) = (\<Sum>x\<in>A. of_nat (f x))" |
|
1053 |
by (induction A rule: infinite_finite_induct) auto |
|
1054 |
||
1055 |
end |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1056 |
|
69182 | 1057 |
context ring_1 |
1058 |
begin |
|
1059 |
||
1060 |
lemma of_int_sum [simp]: |
|
1061 |
"of_int (sum f A) = (\<Sum>x\<in>A. of_int (f x))" |
|
1062 |
by (induction A rule: infinite_finite_induct) auto |
|
1063 |
||
1064 |
end |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1065 |
|
69182 | 1066 |
context comm_semiring_1 |
1067 |
begin |
|
1068 |
||
1069 |
lemma of_nat_prod [simp]: |
|
1070 |
"of_nat (prod f A) = (\<Prod>x\<in>A. of_nat (f x))" |
|
1071 |
by (induction A rule: infinite_finite_induct) auto |
|
1072 |
||
1073 |
end |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1074 |
|
69182 | 1075 |
context comm_ring_1 |
1076 |
begin |
|
1077 |
||
1078 |
lemma of_int_prod [simp]: |
|
1079 |
"of_int (prod f A) = (\<Prod>x\<in>A. of_int (f x))" |
|
1080 |
by (induction A rule: infinite_finite_induct) auto |
|
1081 |
||
1082 |
end |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1083 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1084 |
|
60758 | 1085 |
subsection \<open>Setting up simplification procedures\<close> |
30802 | 1086 |
|
70356
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1087 |
ML_file \<open>Tools/int_arith.ML\<close> |
54249 | 1088 |
|
70356
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1089 |
declaration \<open>K ( |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1090 |
Lin_Arith.add_discrete_type \<^type_name>\<open>Int.int\<close> |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1091 |
#> Lin_Arith.add_lessD @{thm zless_imp_add1_zle} |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1092 |
#> Lin_Arith.add_inj_thms @{thms of_nat_le_iff [THEN iffD2] of_nat_eq_iff [THEN iffD2]} |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1093 |
#> Lin_Arith.add_inj_const (\<^const_name>\<open>of_nat\<close>, \<^typ>\<open>nat \<Rightarrow> int\<close>) |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1094 |
#> Lin_Arith.add_simps |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1095 |
@{thms of_int_0 of_int_1 of_int_add of_int_mult of_int_numeral of_int_neg_numeral nat_0 nat_1 diff_nat_numeral nat_numeral |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1096 |
neg_less_iff_less |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1097 |
True_implies_equals |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1098 |
distrib_left [where a = "numeral v" for v] |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1099 |
distrib_left [where a = "- numeral v" for v] |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1100 |
div_by_1 div_0 |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1101 |
times_divide_eq_right times_divide_eq_left |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1102 |
minus_divide_left [THEN sym] minus_divide_right [THEN sym] |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1103 |
add_divide_distrib diff_divide_distrib |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1104 |
of_int_minus of_int_diff |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1105 |
of_int_of_nat_eq} |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1106 |
#> Lin_Arith.add_simprocs [Int_Arith.zero_one_idom_simproc] |
4a327c061870
streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents:
70354
diff
changeset
|
1107 |
)\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1108 |
|
63652 | 1109 |
simproc_setup fast_arith |
1110 |
("(m::'a::linordered_idom) < n" | |
|
1111 |
"(m::'a::linordered_idom) \<le> n" | |
|
1112 |
"(m::'a::linordered_idom) = n") = |
|
61144 | 1113 |
\<open>K Lin_Arith.simproc\<close> |
43595 | 1114 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1115 |
|
60758 | 1116 |
subsection\<open>More Inequality Reasoning\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1117 |
|
63652 | 1118 |
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z" |
1119 |
for w z :: int |
|
1120 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1121 |
|
63652 | 1122 |
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z" |
1123 |
for w z :: int |
|
1124 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1125 |
|
63652 | 1126 |
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z" |
1127 |
for w z :: int |
|
1128 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1129 |
|
63652 | 1130 |
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z" |
1131 |
for w z :: int |
|
1132 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1133 |
|
63652 | 1134 |
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z" |
1135 |
for z :: int |
|
1136 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1137 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1138 |
lemma Ints_nonzero_abs_ge1: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1139 |
fixes x:: "'a :: linordered_idom" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1140 |
assumes "x \<in> Ints" "x \<noteq> 0" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1141 |
shows "1 \<le> abs x" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1142 |
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>]) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1143 |
fix z::int |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1144 |
assume "x = of_int z" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1145 |
with \<open>x \<noteq> 0\<close> |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1146 |
show "1 \<le> \<bar>x\<bar>" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1147 |
apply (auto simp add: abs_if) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1148 |
by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1149 |
qed |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1150 |
|
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1151 |
lemma Ints_nonzero_abs_less1: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1152 |
fixes x:: "'a :: linordered_idom" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1153 |
shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1154 |
using Ints_nonzero_abs_ge1 [of x] by auto |
70365
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1155 |
|
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1156 |
lemma Ints_eq_abs_less1: |
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1157 |
fixes x:: "'a :: linordered_idom" |
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1158 |
shows "\<lbrakk>x \<in> Ints; y \<in> Ints\<rbrakk> \<Longrightarrow> x = y \<longleftrightarrow> abs (x-y) < 1" |
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1159 |
using eq_iff_diff_eq_0 by (fastforce intro: Ints_nonzero_abs_less1) |
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70356
diff
changeset
|
1160 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1161 |
|
69593 | 1162 |
subsection \<open>The functions \<^term>\<open>nat\<close> and \<^term>\<open>int\<close>\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1163 |
|
69593 | 1164 |
text \<open>Simplify the term \<^term>\<open>w + - z\<close>.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1165 |
|
63652 | 1166 |
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z" |
60162 | 1167 |
using zless_nat_conj [of 1 z] by auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1168 |
|
67116 | 1169 |
lemma int_eq_iff_numeral [simp]: |
1170 |
"int m = numeral v \<longleftrightarrow> m = numeral v" |
|
1171 |
by (simp add: int_eq_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1172 |
|
67116 | 1173 |
lemma nat_abs_int_diff: |
1174 |
"nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)" |
|
59000 | 1175 |
by auto |
1176 |
||
1177 |
lemma nat_int_add: "nat (int a + int b) = a + b" |
|
1178 |
by auto |
|
1179 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1180 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1181 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1182 |
|
33056
791a4655cae3
renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents:
32437
diff
changeset
|
1183 |
lemma of_int_of_nat [nitpick_simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1184 |
"of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1185 |
proof (cases "k < 0") |
63652 | 1186 |
case True |
1187 |
then have "0 \<le> - k" by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1188 |
then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1189 |
with True show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1190 |
next |
63652 | 1191 |
case False |
1192 |
then show ?thesis by (simp add: not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1193 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1194 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1195 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1196 |
|
64014 | 1197 |
lemma transfer_rule_of_int: |
70927 | 1198 |
includes lifting_syntax |
64014 | 1199 |
fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool" |
1200 |
assumes [transfer_rule]: "R 0 0" "R 1 1" |
|
70927 | 1201 |
"(R ===> R ===> R) (+) (+)" |
1202 |
"(R ===> R) uminus uminus" |
|
1203 |
shows "((=) ===> R) of_int of_int" |
|
64014 | 1204 |
proof - |
70927 | 1205 |
note assms |
64014 | 1206 |
note transfer_rule_of_nat [transfer_rule] |
70927 | 1207 |
have [transfer_rule]: "((=) ===> R) of_nat of_nat" |
64014 | 1208 |
by transfer_prover |
1209 |
show ?thesis |
|
1210 |
by (unfold of_int_of_nat [abs_def]) transfer_prover |
|
1211 |
qed |
|
1212 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1213 |
lemma nat_mult_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1214 |
fixes z z' :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1215 |
assumes "0 \<le> z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1216 |
shows "nat (z * z') = nat z * nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1217 |
proof (cases "0 \<le> z'") |
63652 | 1218 |
case False |
1219 |
with assms have "z * z' \<le> 0" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1220 |
by (simp add: not_le mult_le_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1221 |
then have "nat (z * z') = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1222 |
moreover from False have "nat z' = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1223 |
ultimately show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1224 |
next |
63652 | 1225 |
case True |
1226 |
with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1227 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1228 |
by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1229 |
(simp only: of_nat_mult of_nat_nat [OF True] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1230 |
of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1231 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1232 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1233 |
lemma nat_mult_distrib_neg: |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1234 |
assumes "z \<le> (0::int)" shows "nat (z * z') = nat (- z) * nat (- z')" (is "?L = ?R") |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1235 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1236 |
have "?L = nat (- z * - z')" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1237 |
using assms by auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1238 |
also have "... = ?R" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1239 |
by (rule nat_mult_distrib) (use assms in auto) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1240 |
finally show ?thesis . |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1241 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1242 |
|
61944 | 1243 |
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>" |
63652 | 1244 |
by (cases "z = 0 \<or> w = 0") |
1245 |
(auto simp add: abs_if nat_mult_distrib [symmetric] |
|
1246 |
nat_mult_distrib_neg [symmetric] mult_less_0_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1247 |
|
63652 | 1248 |
lemma int_in_range_abs [simp]: "int n \<in> range abs" |
60570 | 1249 |
proof (rule range_eqI) |
63652 | 1250 |
show "int n = \<bar>int n\<bar>" by simp |
60570 | 1251 |
qed |
1252 |
||
63652 | 1253 |
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)" |
60570 | 1254 |
proof - |
1255 |
have "\<bar>k\<bar> \<in> \<nat>" for k :: int |
|
1256 |
by (cases k) simp_all |
|
1257 |
moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int |
|
1258 |
using that by induct simp |
|
1259 |
ultimately show ?thesis by blast |
|
61204 | 1260 |
qed |
60570 | 1261 |
|
63652 | 1262 |
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)" |
1263 |
for z :: int |
|
1264 |
by (rule sym) (simp add: nat_eq_iff) |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1265 |
|
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1266 |
lemma diff_nat_eq_if: |
63652 | 1267 |
"nat z - nat z' = |
1268 |
(if z' < 0 then nat z |
|
1269 |
else |
|
1270 |
let d = z - z' |
|
1271 |
in if d < 0 then 0 else nat d)" |
|
1272 |
by (simp add: Let_def nat_diff_distrib [symmetric]) |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1273 |
|
63652 | 1274 |
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)" |
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1275 |
using diff_nat_numeral [of v Num.One] by simp |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1276 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1277 |
|
63652 | 1278 |
subsection \<open>Induction principles for int\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1279 |
|
63652 | 1280 |
text \<open>Well-founded segments of the integers.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1281 |
|
63652 | 1282 |
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set" |
1283 |
where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1284 |
|
63652 | 1285 |
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1286 |
proof - |
63652 | 1287 |
have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1288 |
by (auto simp add: int_ge_less_than_def) |
63652 | 1289 |
then show ?thesis |
60162 | 1290 |
by (rule wf_subset [OF wf_measure]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1291 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1292 |
|
63652 | 1293 |
text \<open> |
1294 |
This variant looks odd, but is typical of the relations suggested |
|
1295 |
by RankFinder.\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1296 |
|
63652 | 1297 |
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set" |
1298 |
where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1299 |
|
63652 | 1300 |
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1301 |
proof - |
63652 | 1302 |
have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1303 |
by (auto simp add: int_ge_less_than2_def) |
63652 | 1304 |
then show ?thesis |
60162 | 1305 |
by (rule wf_subset [OF wf_measure]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1306 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1307 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1308 |
(* `set:int': dummy construction *) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1309 |
theorem int_ge_induct [case_names base step, induct set: int]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1310 |
fixes i :: int |
63652 | 1311 |
assumes ge: "k \<le> i" |
1312 |
and base: "P k" |
|
1313 |
and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1314 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1315 |
proof - |
63652 | 1316 |
have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n |
1317 |
proof (induct n) |
|
1318 |
case 0 |
|
1319 |
then have "i = k" by arith |
|
1320 |
with base show "P i" by simp |
|
1321 |
next |
|
1322 |
case (Suc n) |
|
1323 |
then have "n = nat ((i - 1) - k)" by arith |
|
1324 |
moreover have k: "k \<le> i - 1" using Suc.prems by arith |
|
1325 |
ultimately have "P (i - 1)" by (rule Suc.hyps) |
|
1326 |
from step [OF k this] show ?case by simp |
|
1327 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1328 |
with ge show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1329 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1330 |
|
25928 | 1331 |
(* `set:int': dummy construction *) |
1332 |
theorem int_gr_induct [case_names base step, induct set: int]: |
|
63652 | 1333 |
fixes i k :: int |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1334 |
assumes "k < i" "P (k + 1)" "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1335 |
shows "P i" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1336 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1337 |
have "k+1 \<le> i" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1338 |
using assms by auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1339 |
then show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1340 |
by (induction i rule: int_ge_induct) (auto simp: assms) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1341 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1342 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1343 |
theorem int_le_induct [consumes 1, case_names base step]: |
63652 | 1344 |
fixes i k :: int |
1345 |
assumes le: "i \<le> k" |
|
1346 |
and base: "P k" |
|
1347 |
and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1348 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1349 |
proof - |
63652 | 1350 |
have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n |
1351 |
proof (induct n) |
|
1352 |
case 0 |
|
1353 |
then have "i = k" by arith |
|
1354 |
with base show "P i" by simp |
|
1355 |
next |
|
1356 |
case (Suc n) |
|
1357 |
then have "n = nat (k - (i + 1))" by arith |
|
1358 |
moreover have k: "i + 1 \<le> k" using Suc.prems by arith |
|
1359 |
ultimately have "P (i + 1)" by (rule Suc.hyps) |
|
1360 |
from step[OF k this] show ?case by simp |
|
1361 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1362 |
with le show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1363 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1364 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1365 |
theorem int_less_induct [consumes 1, case_names base step]: |
63652 | 1366 |
fixes i k :: int |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1367 |
assumes "i < k" "P (k - 1)" "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1368 |
shows "P i" |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1369 |
proof - |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1370 |
have "i \<le> k-1" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1371 |
using assms by auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1372 |
then show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1373 |
by (induction i rule: int_le_induct) (auto simp: assms) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1374 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1375 |
|
36811
4ab4aa5bee1c
renamed former Int.int_induct to Int.int_of_nat_induct, former Presburger.int_induct to Int.int_induct: is more conservative and more natural than the intermediate solution
haftmann
parents:
36801
diff
changeset
|
1376 |
theorem int_induct [case_names base step1 step2]: |
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1377 |
fixes k :: int |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1378 |
assumes base: "P k" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1379 |
and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1380 |
and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1381 |
shows "P i" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1382 |
proof - |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1383 |
have "i \<le> k \<or> i \<ge> k" by arith |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1384 |
then show ?thesis |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1385 |
proof |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1386 |
assume "i \<ge> k" |
63652 | 1387 |
then show ?thesis |
1388 |
using base by (rule int_ge_induct) (fact step1) |
|
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1389 |
next |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1390 |
assume "i \<le> k" |
63652 | 1391 |
then show ?thesis |
1392 |
using base by (rule int_le_induct) (fact step2) |
|
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1393 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1394 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1395 |
|
63652 | 1396 |
|
1397 |
subsection \<open>Intermediate value theorems\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1398 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1399 |
lemma nat_ivt_aux: |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1400 |
"\<lbrakk>\<forall>i<n. \<bar>f (Suc i) - f i\<bar> \<le> 1; f 0 \<le> k; k \<le> f n\<rbrakk> \<Longrightarrow> \<exists>i \<le> n. f i = k" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1401 |
for m n :: nat and k :: int |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1402 |
proof (induct n) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1403 |
case (Suc n) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1404 |
show ?case |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1405 |
proof (cases "k = f (Suc n)") |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1406 |
case False |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1407 |
with Suc have "k \<le> f n" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1408 |
by auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1409 |
with Suc show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1410 |
by (auto simp add: abs_if split: if_split_asm intro: le_SucI) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1411 |
qed (use Suc in auto) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1412 |
qed auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1413 |
|
67116 | 1414 |
lemma nat_intermed_int_val: |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1415 |
fixes m n :: nat and k :: int |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1416 |
assumes "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (Suc i) - f i\<bar> \<le> 1" "m \<le> n" "f m \<le> k" "k \<le> f n" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1417 |
shows "\<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k" |
67116 | 1418 |
proof - |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1419 |
obtain i where "i \<le> n - m" "k = f (m + i)" |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1420 |
using nat_ivt_aux [of "n - m" "f \<circ> plus m" k] assms by auto |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1421 |
with assms show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1422 |
by (rule_tac x = "m + i" in exI) auto |
67116 | 1423 |
qed |
1424 |
||
1425 |
lemma nat0_intermed_int_val: |
|
1426 |
"\<exists>i\<le>n. f i = k" |
|
1427 |
if "\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1" "f 0 \<le> k" "k \<le> f n" |
|
63652 | 1428 |
for n :: nat and k :: int |
67116 | 1429 |
using nat_intermed_int_val [of 0 n f k] that by auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1430 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1431 |
|
63652 | 1432 |
subsection \<open>Products and 1, by T. M. Rasmussen\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1433 |
|
34055 | 1434 |
lemma abs_zmult_eq_1: |
63652 | 1435 |
fixes m n :: int |
34055 | 1436 |
assumes mn: "\<bar>m * n\<bar> = 1" |
63652 | 1437 |
shows "\<bar>m\<bar> = 1" |
34055 | 1438 |
proof - |
63652 | 1439 |
from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto |
1440 |
have "\<not> 2 \<le> \<bar>m\<bar>" |
|
34055 | 1441 |
proof |
1442 |
assume "2 \<le> \<bar>m\<bar>" |
|
63652 | 1443 |
then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0) |
1444 |
also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult) |
|
1445 |
also from mn have "\<dots> = 1" by simp |
|
1446 |
finally have "2 * \<bar>n\<bar> \<le> 1" . |
|
1447 |
with 0 show "False" by arith |
|
34055 | 1448 |
qed |
63652 | 1449 |
with 0 show ?thesis by auto |
34055 | 1450 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1451 |
|
63652 | 1452 |
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1" |
1453 |
for m n :: int |
|
1454 |
using abs_zmult_eq_1 [of m n] by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1455 |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1456 |
lemma pos_zmult_eq_1_iff: |
63652 | 1457 |
fixes m n :: int |
1458 |
assumes "0 < m" |
|
1459 |
shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1460 |
proof - |
63652 | 1461 |
from assms have "m * n = 1 \<Longrightarrow> m = 1" |
1462 |
by (auto dest: pos_zmult_eq_1_iff_lemma) |
|
1463 |
then show ?thesis |
|
1464 |
by (auto dest: pos_zmult_eq_1_iff_lemma) |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1465 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1466 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1467 |
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)" (is "?L = ?R") |
63652 | 1468 |
for m n :: int |
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1469 |
proof |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1470 |
assume L: ?L show ?R |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1471 |
using pos_zmult_eq_1_iff_lemma [OF L] L by force |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1472 |
qed auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1473 |
|
69700
7a92cbec7030
new material about summations and powers, along with some tweaks
paulson <lp15@cam.ac.uk>
parents:
69605
diff
changeset
|
1474 |
lemma infinite_UNIV_int [simp]: "\<not> finite (UNIV::int set)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1475 |
proof |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1476 |
assume "finite (UNIV::int set)" |
61076 | 1477 |
moreover have "inj (\<lambda>i::int. 2 * i)" |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1478 |
by (rule injI) simp |
61076 | 1479 |
ultimately have "surj (\<lambda>i::int. 2 * i)" |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1480 |
by (rule finite_UNIV_inj_surj) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1481 |
then obtain i :: int where "1 = 2 * i" by (rule surjE) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1482 |
then show False by (simp add: pos_zmult_eq_1_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1483 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1484 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1485 |
|
60758 | 1486 |
subsection \<open>The divides relation\<close> |
33320 | 1487 |
|
63652 | 1488 |
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n" |
1489 |
for m n :: int |
|
1490 |
by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff) |
|
33320 | 1491 |
|
63652 | 1492 |
lemma zdvd_antisym_abs: |
1493 |
fixes a b :: int |
|
1494 |
assumes "a dvd b" and "b dvd a" |
|
33320 | 1495 |
shows "\<bar>a\<bar> = \<bar>b\<bar>" |
63652 | 1496 |
proof (cases "a = 0") |
1497 |
case True |
|
1498 |
with assms show ?thesis by simp |
|
33657 | 1499 |
next |
63652 | 1500 |
case False |
1501 |
from \<open>a dvd b\<close> obtain k where k: "b = a * k" |
|
1502 |
unfolding dvd_def by blast |
|
1503 |
from \<open>b dvd a\<close> obtain k' where k': "a = b * k'" |
|
1504 |
unfolding dvd_def by blast |
|
1505 |
from k k' have "a = a * k * k'" by simp |
|
1506 |
with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1" |
|
1507 |
using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc) |
|
1508 |
then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" |
|
1509 |
by (simp add: zmult_eq_1_iff) |
|
1510 |
with k k' show ?thesis by auto |
|
33320 | 1511 |
qed |
1512 |
||
63652 | 1513 |
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m" |
1514 |
for k m n :: int |
|
60162 | 1515 |
using dvd_add_right_iff [of k "- n" m] by simp |
33320 | 1516 |
|
63652 | 1517 |
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n" |
1518 |
for k m n :: int |
|
58649
a62065b5e1e2
generalized and consolidated some theorems concerning divisibility
haftmann
parents:
58512
diff
changeset
|
1519 |
using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps) |
33320 | 1520 |
|
1521 |
lemma dvd_imp_le_int: |
|
1522 |
fixes d i :: int |
|
1523 |
assumes "i \<noteq> 0" and "d dvd i" |
|
1524 |
shows "\<bar>d\<bar> \<le> \<bar>i\<bar>" |
|
1525 |
proof - |
|
60758 | 1526 |
from \<open>d dvd i\<close> obtain k where "i = d * k" .. |
1527 |
with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto |
|
33320 | 1528 |
then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto |
1529 |
then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono) |
|
60758 | 1530 |
with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult) |
33320 | 1531 |
qed |
1532 |
||
1533 |
lemma zdvd_not_zless: |
|
1534 |
fixes m n :: int |
|
1535 |
assumes "0 < m" and "m < n" |
|
1536 |
shows "\<not> n dvd m" |
|
1537 |
proof |
|
1538 |
from assms have "0 < n" by auto |
|
1539 |
assume "n dvd m" then obtain k where k: "m = n * k" .. |
|
60758 | 1540 |
with \<open>0 < m\<close> have "0 < n * k" by auto |
1541 |
with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff) |
|
1542 |
with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp |
|
1543 |
with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto |
|
33320 | 1544 |
qed |
1545 |
||
63652 | 1546 |
lemma zdvd_mult_cancel: |
1547 |
fixes k m n :: int |
|
1548 |
assumes d: "k * m dvd k * n" |
|
1549 |
and "k \<noteq> 0" |
|
33320 | 1550 |
shows "m dvd n" |
63652 | 1551 |
proof - |
1552 |
from d obtain h where h: "k * n = k * m * h" |
|
1553 |
unfolding dvd_def by blast |
|
1554 |
have "n = m * h" |
|
1555 |
proof (rule ccontr) |
|
1556 |
assume "\<not> ?thesis" |
|
1557 |
with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp |
|
1558 |
with h show False |
|
1559 |
by (simp add: mult.assoc) |
|
1560 |
qed |
|
1561 |
then show ?thesis by simp |
|
33320 | 1562 |
qed |
1563 |
||
67118 | 1564 |
lemma int_dvd_int_iff [simp]: |
1565 |
"int m dvd int n \<longleftrightarrow> m dvd n" |
|
33320 | 1566 |
proof - |
67118 | 1567 |
have "m dvd n" if "int n = int m * k" for k |
63652 | 1568 |
proof (cases k) |
67118 | 1569 |
case (nonneg q) |
1570 |
with that have "n = m * q" |
|
63652 | 1571 |
by (simp del: of_nat_mult add: of_nat_mult [symmetric]) |
1572 |
then show ?thesis .. |
|
1573 |
next |
|
67118 | 1574 |
case (neg q) |
1575 |
with that have "int n = int m * (- int (Suc q))" |
|
63652 | 1576 |
by simp |
67118 | 1577 |
also have "\<dots> = - (int m * int (Suc q))" |
63652 | 1578 |
by (simp only: mult_minus_right) |
67118 | 1579 |
also have "\<dots> = - int (m * Suc q)" |
63652 | 1580 |
by (simp only: of_nat_mult [symmetric]) |
67118 | 1581 |
finally have "- int (m * Suc q) = int n" .. |
63652 | 1582 |
then show ?thesis |
1583 |
by (simp only: negative_eq_positive) auto |
|
33320 | 1584 |
qed |
67118 | 1585 |
then show ?thesis by (auto simp add: dvd_def) |
33320 | 1586 |
qed |
1587 |
||
67118 | 1588 |
lemma dvd_nat_abs_iff [simp]: |
1589 |
"n dvd nat \<bar>k\<bar> \<longleftrightarrow> int n dvd k" |
|
1590 |
proof - |
|
1591 |
have "n dvd nat \<bar>k\<bar> \<longleftrightarrow> int n dvd int (nat \<bar>k\<bar>)" |
|
1592 |
by (simp only: int_dvd_int_iff) |
|
1593 |
then show ?thesis |
|
1594 |
by simp |
|
1595 |
qed |
|
1596 |
||
1597 |
lemma nat_abs_dvd_iff [simp]: |
|
1598 |
"nat \<bar>k\<bar> dvd n \<longleftrightarrow> k dvd int n" |
|
1599 |
proof - |
|
1600 |
have "nat \<bar>k\<bar> dvd n \<longleftrightarrow> int (nat \<bar>k\<bar>) dvd int n" |
|
1601 |
by (simp only: int_dvd_int_iff) |
|
1602 |
then show ?thesis |
|
1603 |
by simp |
|
1604 |
qed |
|
1605 |
||
1606 |
lemma zdvd1_eq [simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1" (is "?lhs \<longleftrightarrow> ?rhs") |
|
63652 | 1607 |
for x :: int |
33320 | 1608 |
proof |
63652 | 1609 |
assume ?lhs |
67118 | 1610 |
then have "nat \<bar>x\<bar> dvd nat \<bar>1\<bar>" |
1611 |
by (simp only: nat_abs_dvd_iff) simp |
|
1612 |
then have "nat \<bar>x\<bar> = 1" |
|
1613 |
by simp |
|
1614 |
then show ?rhs |
|
1615 |
by (cases "x < 0") simp_all |
|
33320 | 1616 |
next |
63652 | 1617 |
assume ?rhs |
67118 | 1618 |
then have "x = 1 \<or> x = - 1" |
1619 |
by auto |
|
1620 |
then show ?lhs |
|
1621 |
by (auto intro: dvdI) |
|
33320 | 1622 |
qed |
1623 |
||
60162 | 1624 |
lemma zdvd_mult_cancel1: |
63652 | 1625 |
fixes m :: int |
1626 |
assumes mp: "m \<noteq> 0" |
|
1627 |
shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1" |
|
1628 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
33320 | 1629 |
proof |
63652 | 1630 |
assume ?rhs |
1631 |
then show ?lhs |
|
1632 |
by (cases "n > 0") (auto simp add: minus_equation_iff) |
|
33320 | 1633 |
next |
63652 | 1634 |
assume ?lhs |
1635 |
then have "m * n dvd m * 1" by simp |
|
1636 |
from zdvd_mult_cancel[OF this mp] show ?rhs |
|
1637 |
by (simp only: zdvd1_eq) |
|
33320 | 1638 |
qed |
1639 |
||
63652 | 1640 |
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)" |
67118 | 1641 |
using nat_abs_dvd_iff [of z m] by (cases "z \<ge> 0") auto |
33320 | 1642 |
|
63652 | 1643 |
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'" |
67116 | 1644 |
by (auto elim: nonneg_int_cases) |
33341 | 1645 |
|
63652 | 1646 |
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n" |
33341 | 1647 |
by (induct n) (simp_all add: nat_mult_distrib) |
1648 |
||
66912
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1649 |
lemma numeral_power_eq_nat_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1650 |
"numeral x ^ n = nat y \<longleftrightarrow> numeral x ^ n = y" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1651 |
using nat_eq_iff2 by auto |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1652 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1653 |
lemma nat_eq_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1654 |
"nat y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1655 |
using numeral_power_eq_nat_cancel_iff[of x n y] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1656 |
by (metis (mono_tags)) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1657 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1658 |
lemma numeral_power_le_nat_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1659 |
"numeral x ^ n \<le> nat a \<longleftrightarrow> numeral x ^ n \<le> a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1660 |
using nat_le_eq_zle[of "numeral x ^ n" a] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1661 |
by (auto simp: nat_power_eq) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1662 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1663 |
lemma nat_le_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1664 |
"nat a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1665 |
by (simp add: nat_le_iff) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1666 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1667 |
lemma numeral_power_less_nat_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1668 |
"numeral x ^ n < nat a \<longleftrightarrow> numeral x ^ n < a" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1669 |
using nat_less_eq_zless[of "numeral x ^ n" a] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1670 |
by (auto simp: nat_power_eq) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1671 |
|
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1672 |
lemma nat_less_numeral_power_cancel_iff [simp]: |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1673 |
"nat a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n" |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1674 |
using nat_less_eq_zless[of a "numeral x ^ n"] |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1675 |
by (cases "a < 0") (auto simp: nat_power_eq less_le_trans[where y=0]) |
a99a7cbf0fb5
generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents:
66886
diff
changeset
|
1676 |
|
71616
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1677 |
lemma zdvd_imp_le: "z \<le> n" if "z dvd n" "0 < n" for n z :: int |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1678 |
proof (cases n) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1679 |
case (nonneg n) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1680 |
show ?thesis |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1681 |
by (cases z) (use nonneg dvd_imp_le that in auto) |
a9de39608b1a
more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents:
70927
diff
changeset
|
1682 |
qed (use that in auto) |
33320 | 1683 |
|
36749 | 1684 |
lemma zdvd_period: |
1685 |
fixes a d :: int |
|
1686 |
assumes "a dvd d" |
|
1687 |
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" |
|
63652 | 1688 |
(is "?lhs \<longleftrightarrow> ?rhs") |
36749 | 1689 |
proof - |
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1690 |
from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1691 |
by (simp add: dvd_add_left_iff) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1692 |
then show ?thesis |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1693 |
by (simp add: ac_simps) |
36749 | 1694 |
qed |
1695 |
||
33320 | 1696 |
|
71837
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1697 |
subsection \<open>Powers with integer exponents\<close> |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1698 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1699 |
text \<open> |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1700 |
The following allows writing powers with an integer exponent. While the type signature |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1701 |
is very generic, most theorems will assume that the underlying type is a division ring or |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1702 |
a field. |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1703 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1704 |
The notation `powi' is inspired by the `powr' notation for real/complex exponentiation. |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1705 |
\<close> |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1706 |
definition power_int :: "'a :: {inverse, power} \<Rightarrow> int \<Rightarrow> 'a" (infixr "powi" 80) where |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1707 |
"power_int x n = (if n \<ge> 0 then x ^ nat n else inverse x ^ (nat (-n)))" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1708 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1709 |
lemma power_int_0_right [simp]: "power_int x 0 = 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1710 |
and power_int_1_right [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1711 |
"power_int (y :: 'a :: {power, inverse, monoid_mult}) 1 = y" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1712 |
and power_int_minus1_right [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1713 |
"power_int (y :: 'a :: {power, inverse, monoid_mult}) (-1) = inverse y" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1714 |
by (simp_all add: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1715 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1716 |
lemma power_int_of_nat [simp]: "power_int x (int n) = x ^ n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1717 |
by (simp add: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1718 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1719 |
lemma power_int_numeral [simp]: "power_int x (numeral n) = x ^ numeral n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1720 |
by (simp add: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1721 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1722 |
lemma int_cases4 [case_names nonneg neg]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1723 |
fixes m :: int |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1724 |
obtains n where "m = int n" | n where "n > 0" "m = -int n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1725 |
proof (cases "m \<ge> 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1726 |
case True |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1727 |
thus ?thesis using that(1)[of "nat m"] by auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1728 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1729 |
case False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1730 |
thus ?thesis using that(2)[of "nat (-m)"] by auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1731 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1732 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1733 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1734 |
context |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1735 |
assumes "SORT_CONSTRAINT('a::division_ring)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1736 |
begin |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1737 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1738 |
lemma power_int_minus: "power_int (x::'a) (-n) = inverse (power_int x n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1739 |
by (auto simp: power_int_def power_inverse) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1740 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1741 |
lemma power_int_eq_0_iff [simp]: "power_int (x::'a) n = 0 \<longleftrightarrow> x = 0 \<and> n \<noteq> 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1742 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1743 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1744 |
lemma power_int_0_left_If: "power_int (0 :: 'a) m = (if m = 0 then 1 else 0)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1745 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1746 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1747 |
lemma power_int_0_left [simp]: "m \<noteq> 0 \<Longrightarrow> power_int (0 :: 'a) m = 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1748 |
by (simp add: power_int_0_left_If) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1749 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1750 |
lemma power_int_1_left [simp]: "power_int 1 n = (1 :: 'a :: division_ring)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1751 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1752 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1753 |
lemma power_diff_conv_inverse: "x \<noteq> 0 \<Longrightarrow> m \<le> n \<Longrightarrow> (x :: 'a) ^ (n - m) = x ^ n * inverse x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1754 |
by (simp add: field_simps flip: power_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1755 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1756 |
lemma power_mult_inverse_distrib: "x ^ m * inverse (x :: 'a) = inverse x * x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1757 |
proof (cases "x = 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1758 |
case [simp]: False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1759 |
show ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1760 |
proof (cases m) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1761 |
case (Suc m') |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1762 |
have "x ^ Suc m' * inverse x = x ^ m'" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1763 |
by (subst power_Suc2) (auto simp: mult.assoc) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1764 |
also have "\<dots> = inverse x * x ^ Suc m'" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1765 |
by (subst power_Suc) (auto simp: mult.assoc [symmetric]) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1766 |
finally show ?thesis using Suc by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1767 |
qed auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1768 |
qed auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1769 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1770 |
lemma power_mult_power_inverse_commute: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1771 |
"x ^ m * inverse (x :: 'a) ^ n = inverse x ^ n * x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1772 |
proof (induction n) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1773 |
case (Suc n) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1774 |
have "x ^ m * inverse x ^ Suc n = (x ^ m * inverse x ^ n) * inverse x" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1775 |
by (simp only: power_Suc2 mult.assoc) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1776 |
also have "x ^ m * inverse x ^ n = inverse x ^ n * x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1777 |
by (rule Suc) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1778 |
also have "\<dots> * inverse x = (inverse x ^ n * inverse x) * x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1779 |
by (simp add: mult.assoc power_mult_inverse_distrib) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1780 |
also have "\<dots> = inverse x ^ (Suc n) * x ^ m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1781 |
by (simp only: power_Suc2) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1782 |
finally show ?case . |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1783 |
qed auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1784 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1785 |
lemma power_int_add: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1786 |
assumes "x \<noteq> 0 \<or> m + n \<noteq> 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1787 |
shows "power_int (x::'a) (m + n) = power_int x m * power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1788 |
proof (cases "x = 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1789 |
case True |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1790 |
thus ?thesis using assms by (auto simp: power_int_0_left_If) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1791 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1792 |
case [simp]: False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1793 |
show ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1794 |
proof (cases m n rule: int_cases4[case_product int_cases4]) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1795 |
case (nonneg_nonneg a b) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1796 |
thus ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1797 |
by (auto simp: power_int_def nat_add_distrib power_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1798 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1799 |
case (nonneg_neg a b) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1800 |
thus ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1801 |
by (auto simp: power_int_def nat_diff_distrib not_le power_diff_conv_inverse |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1802 |
power_mult_power_inverse_commute) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1803 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1804 |
case (neg_nonneg a b) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1805 |
thus ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1806 |
by (auto simp: power_int_def nat_diff_distrib not_le power_diff_conv_inverse |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1807 |
power_mult_power_inverse_commute) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1808 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1809 |
case (neg_neg a b) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1810 |
thus ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1811 |
by (auto simp: power_int_def nat_add_distrib add.commute simp flip: power_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1812 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1813 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1814 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1815 |
lemma power_int_add_1: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1816 |
assumes "x \<noteq> 0 \<or> m \<noteq> -1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1817 |
shows "power_int (x::'a) (m + 1) = power_int x m * x" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1818 |
using assms by (subst power_int_add) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1819 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1820 |
lemma power_int_add_1': |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1821 |
assumes "x \<noteq> 0 \<or> m \<noteq> -1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1822 |
shows "power_int (x::'a) (m + 1) = x * power_int x m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1823 |
using assms by (subst add.commute, subst power_int_add) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1824 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1825 |
lemma power_int_commutes: "power_int (x :: 'a) n * x = x * power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1826 |
by (cases "x = 0") (auto simp flip: power_int_add_1 power_int_add_1') |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1827 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1828 |
lemma power_int_inverse [field_simps, field_split_simps, divide_simps]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1829 |
"power_int (inverse (x :: 'a)) n = inverse (power_int x n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1830 |
by (auto simp: power_int_def power_inverse) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1831 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1832 |
lemma power_int_mult: "power_int (x :: 'a) (m * n) = power_int (power_int x m) n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1833 |
by (auto simp: power_int_def zero_le_mult_iff simp flip: power_mult power_inverse nat_mult_distrib) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1834 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1835 |
end |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1836 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1837 |
context |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1838 |
assumes "SORT_CONSTRAINT('a::field)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1839 |
begin |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1840 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1841 |
lemma power_int_diff: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1842 |
assumes "x \<noteq> 0 \<or> m \<noteq> n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1843 |
shows "power_int (x::'a) (m - n) = power_int x m / power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1844 |
using power_int_add[of x m "-n"] assms by (auto simp: field_simps power_int_minus) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1845 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1846 |
lemma power_int_minus_mult: "x \<noteq> 0 \<or> n \<noteq> 0 \<Longrightarrow> power_int (x :: 'a) (n - 1) * x = power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1847 |
by (auto simp flip: power_int_add_1) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1848 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1849 |
lemma power_int_mult_distrib: "power_int (x * y :: 'a) m = power_int x m * power_int y m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1850 |
by (auto simp: power_int_def power_mult_distrib) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1851 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1852 |
lemmas power_int_mult_distrib_numeral1 = power_int_mult_distrib [where x = "numeral w" for w, simp] |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1853 |
lemmas power_int_mult_distrib_numeral2 = power_int_mult_distrib [where y = "numeral w" for w, simp] |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1854 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1855 |
lemma power_int_divide_distrib: "power_int (x / y :: 'a) m = power_int x m / power_int y m" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1856 |
using power_int_mult_distrib[of x "inverse y" m] unfolding power_int_inverse |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1857 |
by (simp add: field_simps) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1858 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1859 |
end |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1860 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1861 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1862 |
lemma power_int_add_numeral [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1863 |
"power_int x (numeral m) * power_int x (numeral n) = power_int x (numeral (m + n))" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1864 |
for x :: "'a :: division_ring" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1865 |
by (simp add: power_int_add [symmetric]) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1866 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1867 |
lemma power_int_add_numeral2 [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1868 |
"power_int x (numeral m) * (power_int x (numeral n) * b) = power_int x (numeral (m + n)) * b" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1869 |
for x :: "'a :: division_ring" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1870 |
by (simp add: mult.assoc [symmetric]) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1871 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1872 |
lemma power_int_mult_numeral [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1873 |
"power_int (power_int x (numeral m)) (numeral n) = power_int x (numeral (m * n))" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1874 |
for x :: "'a :: division_ring" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1875 |
by (simp only: numeral_mult power_int_mult) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1876 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1877 |
lemma power_int_not_zero: "(x :: 'a :: division_ring) \<noteq> 0 \<or> n = 0 \<Longrightarrow> power_int x n \<noteq> 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1878 |
by (subst power_int_eq_0_iff) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1879 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1880 |
lemma power_int_one_over [field_simps, field_split_simps, divide_simps]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1881 |
"power_int (1 / x :: 'a :: division_ring) n = 1 / power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1882 |
using power_int_inverse[of x] by (simp add: divide_inverse) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1883 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1884 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1885 |
context |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1886 |
assumes "SORT_CONSTRAINT('a :: linordered_field)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1887 |
begin |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1888 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1889 |
lemma power_int_numeral_neg_numeral [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1890 |
"power_int (numeral m) (-numeral n) = (inverse (numeral (Num.pow m n)) :: 'a)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1891 |
by (simp add: power_int_minus) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1892 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1893 |
lemma zero_less_power_int [simp]: "0 < (x :: 'a) \<Longrightarrow> 0 < power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1894 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1895 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1896 |
lemma zero_le_power_int [simp]: "0 \<le> (x :: 'a) \<Longrightarrow> 0 \<le> power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1897 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1898 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1899 |
lemma power_int_mono: "(x :: 'a) \<le> y \<Longrightarrow> n \<ge> 0 \<Longrightarrow> 0 \<le> x \<Longrightarrow> power_int x n \<le> power_int y n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1900 |
by (cases n rule: int_cases4) (auto intro: power_mono) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1901 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1902 |
lemma one_le_power_int [simp]: "1 \<le> (x :: 'a) \<Longrightarrow> n \<ge> 0 \<Longrightarrow> 1 \<le> power_int x n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1903 |
using power_int_mono [of 1 x n] by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1904 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1905 |
lemma power_int_le_one: "0 \<le> (x :: 'a) \<Longrightarrow> n \<ge> 0 \<Longrightarrow> x \<le> 1 \<Longrightarrow> power_int x n \<le> 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1906 |
using power_int_mono [of x 1 n] by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1907 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1908 |
lemma power_int_le_imp_le_exp: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1909 |
assumes gt1: "1 < (x :: 'a :: linordered_field)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1910 |
assumes "power_int x m \<le> power_int x n" "n \<ge> 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1911 |
shows "m \<le> n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1912 |
proof (cases "m < 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1913 |
case True |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1914 |
with \<open>n \<ge> 0\<close> show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1915 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1916 |
case False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1917 |
with assms have "x ^ nat m \<le> x ^ nat n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1918 |
by (simp add: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1919 |
from gt1 and this show ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1920 |
using False \<open>n \<ge> 0\<close> by auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1921 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1922 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1923 |
lemma power_int_le_imp_less_exp: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1924 |
assumes gt1: "1 < (x :: 'a :: linordered_field)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1925 |
assumes "power_int x m < power_int x n" "n \<ge> 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1926 |
shows "m < n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1927 |
proof (cases "m < 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1928 |
case True |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1929 |
with \<open>n \<ge> 0\<close> show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1930 |
next |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1931 |
case False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1932 |
with assms have "x ^ nat m < x ^ nat n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1933 |
by (simp add: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1934 |
from gt1 and this show ?thesis |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1935 |
using False \<open>n \<ge> 0\<close> by auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1936 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1937 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1938 |
lemma power_int_strict_mono: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1939 |
"(a :: 'a :: linordered_field) < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> power_int a n < power_int b n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1940 |
by (auto simp: power_int_def intro!: power_strict_mono) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1941 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1942 |
lemma power_int_mono_iff [simp]: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1943 |
fixes a b :: "'a :: linordered_field" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1944 |
shows "\<lbrakk>a \<ge> 0; b \<ge> 0; n > 0\<rbrakk> \<Longrightarrow> power_int a n \<le> power_int b n \<longleftrightarrow> a \<le> b" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1945 |
by (auto simp: power_int_def intro!: power_strict_mono) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1946 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1947 |
lemma power_int_strict_increasing: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1948 |
fixes a :: "'a :: linordered_field" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1949 |
assumes "n < N" "1 < a" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1950 |
shows "power_int a N > power_int a n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1951 |
proof - |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1952 |
have *: "a ^ nat (N - n) > a ^ 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1953 |
using assms by (intro power_strict_increasing) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1954 |
have "power_int a N = power_int a n * power_int a (N - n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1955 |
using assms by (simp flip: power_int_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1956 |
also have "\<dots> > power_int a n * 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1957 |
using assms * |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1958 |
by (intro mult_strict_left_mono zero_less_power_int) (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1959 |
finally show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1960 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1961 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1962 |
lemma power_int_increasing: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1963 |
fixes a :: "'a :: linordered_field" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1964 |
assumes "n \<le> N" "a \<ge> 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1965 |
shows "power_int a N \<ge> power_int a n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1966 |
proof - |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1967 |
have *: "a ^ nat (N - n) \<ge> a ^ 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1968 |
using assms by (intro power_increasing) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1969 |
have "power_int a N = power_int a n * power_int a (N - n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1970 |
using assms by (simp flip: power_int_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1971 |
also have "\<dots> \<ge> power_int a n * 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1972 |
using assms * by (intro mult_left_mono) (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1973 |
finally show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1974 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1975 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1976 |
lemma power_int_strict_decreasing: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1977 |
fixes a :: "'a :: linordered_field" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1978 |
assumes "n < N" "0 < a" "a < 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1979 |
shows "power_int a N < power_int a n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1980 |
proof - |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1981 |
have *: "a ^ nat (N - n) < a ^ 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1982 |
using assms by (intro power_strict_decreasing) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1983 |
have "power_int a N = power_int a n * power_int a (N - n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1984 |
using assms by (simp flip: power_int_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1985 |
also have "\<dots> < power_int a n * 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1986 |
using assms * |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1987 |
by (intro mult_strict_left_mono zero_less_power_int) (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1988 |
finally show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1989 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1990 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1991 |
lemma power_int_decreasing: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1992 |
fixes a :: "'a :: linordered_field" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1993 |
assumes "n \<le> N" "0 \<le> a" "a \<le> 1" "a \<noteq> 0 \<or> N \<noteq> 0 \<or> n = 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1994 |
shows "power_int a N \<le> power_int a n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1995 |
proof (cases "a = 0") |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1996 |
case False |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1997 |
have *: "a ^ nat (N - n) \<le> a ^ 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1998 |
using assms by (intro power_decreasing) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
1999 |
have "power_int a N = power_int a n * power_int a (N - n)" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2000 |
using assms False by (simp flip: power_int_add) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2001 |
also have "\<dots> \<le> power_int a n * 1" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2002 |
using assms * by (intro mult_left_mono) (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2003 |
finally show ?thesis by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2004 |
qed (use assms in \<open>auto simp: power_int_0_left_If\<close>) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2005 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2006 |
lemma one_less_power_int: "1 < (a :: 'a) \<Longrightarrow> 0 < n \<Longrightarrow> 1 < power_int a n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2007 |
using power_int_strict_increasing[of 0 n a] by simp |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2008 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2009 |
lemma power_int_abs: "\<bar>power_int a n :: 'a\<bar> = power_int \<bar>a\<bar> n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2010 |
by (auto simp: power_int_def power_abs) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2011 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2012 |
lemma power_int_sgn [simp]: "sgn (power_int a n :: 'a) = power_int (sgn a) n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2013 |
by (auto simp: power_int_def) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2014 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2015 |
lemma abs_power_int_minus [simp]: "\<bar>power_int (- a) n :: 'a\<bar> = \<bar>power_int a n\<bar>" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2016 |
by (simp add: power_int_abs) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2017 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2018 |
lemma power_int_strict_antimono: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2019 |
assumes "(a :: 'a :: linordered_field) < b" "0 < a" "n < 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2020 |
shows "power_int a n > power_int b n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2021 |
proof - |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2022 |
have "inverse (power_int a (-n)) > inverse (power_int b (-n))" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2023 |
using assms by (intro less_imp_inverse_less power_int_strict_mono zero_less_power_int) auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2024 |
thus ?thesis by (simp add: power_int_minus) |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2025 |
qed |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2026 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2027 |
lemma power_int_antimono: |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2028 |
assumes "(a :: 'a :: linordered_field) \<le> b" "0 < a" "n < 0" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2029 |
shows "power_int a n \<ge> power_int b n" |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2030 |
using power_int_strict_antimono[of a b n] assms by (cases "a = b") auto |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2031 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2032 |
end |
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2033 |
|
dca11678c495
new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents:
71616
diff
changeset
|
2034 |
|
60758 | 2035 |
subsection \<open>Finiteness of intervals\<close> |
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2036 |
|
63652 | 2037 |
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}" |
2038 |
proof (cases "a \<le> b") |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2039 |
case True |
63652 | 2040 |
then show ?thesis |
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2041 |
proof (induct b rule: int_ge_induct) |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2042 |
case base |
63652 | 2043 |
have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto |
2044 |
then show ?case by simp |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2045 |
next |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2046 |
case (step b) |
63652 | 2047 |
then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto |
2048 |
with step show ?case by simp |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2049 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2050 |
next |
63652 | 2051 |
case False |
2052 |
then show ?thesis |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2053 |
by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans) |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2054 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2055 |
|
63652 | 2056 |
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}" |
2057 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2058 |
|
63652 | 2059 |
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}" |
2060 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2061 |
|
63652 | 2062 |
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}" |
2063 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2064 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
2065 |
|
60758 | 2066 |
subsection \<open>Configuration of the code generator\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2067 |
|
60758 | 2068 |
text \<open>Constructors\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2069 |
|
63652 | 2070 |
definition Pos :: "num \<Rightarrow> int" |
2071 |
where [simp, code_abbrev]: "Pos = numeral" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2072 |
|
63652 | 2073 |
definition Neg :: "num \<Rightarrow> int" |
2074 |
where [simp, code_abbrev]: "Neg n = - (Pos n)" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2075 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2076 |
code_datatype "0::int" Pos Neg |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2077 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2078 |
|
63652 | 2079 |
text \<open>Auxiliary operations.\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2080 |
|
63652 | 2081 |
definition dup :: "int \<Rightarrow> int" |
2082 |
where [simp]: "dup k = k + k" |
|
26507 | 2083 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2084 |
lemma dup_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2085 |
"dup 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2086 |
"dup (Pos n) = Pos (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2087 |
"dup (Neg n) = Neg (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2088 |
by (simp_all add: numeral_Bit0) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2089 |
|
63652 | 2090 |
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" |
2091 |
where [simp]: "sub m n = numeral m - numeral n" |
|
26507 | 2092 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2093 |
lemma sub_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2094 |
"sub Num.One Num.One = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2095 |
"sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2096 |
"sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2097 |
"sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2098 |
"sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2099 |
"sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2100 |
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2101 |
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2102 |
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1" |
66035
de6cd60b1226
replace non-arithmetic terms by fresh variables before replaying linear-arithmetic proofs: avoid failed proof replays due to an overambitious simpset which may cause proof replay to diverge from the pre-computed proof trace
boehmes
parents:
64996
diff
changeset
|
2103 |
by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2104 |
|
72512 | 2105 |
lemma sub_BitM_One_eq: |
2106 |
\<open>(Num.sub (Num.BitM n) num.One) = 2 * (Num.sub n Num.One :: int)\<close> |
|
2107 |
by (cases n) simp_all |
|
2108 |
||
63652 | 2109 |
text \<open>Implementations.\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2110 |
|
64996 | 2111 |
lemma one_int_code [code]: "1 = Pos Num.One" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2112 |
by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2113 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2114 |
lemma plus_int_code [code]: |
63652 | 2115 |
"k + 0 = k" |
2116 |
"0 + l = l" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2117 |
"Pos m + Pos n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2118 |
"Pos m + Neg n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2119 |
"Neg m + Pos n = sub n m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2120 |
"Neg m + Neg n = Neg (m + n)" |
63652 | 2121 |
for k l :: int |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2122 |
by simp_all |
26507 | 2123 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2124 |
lemma uminus_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2125 |
"uminus 0 = (0::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2126 |
"uminus (Pos m) = Neg m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2127 |
"uminus (Neg m) = Pos m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2128 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2129 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2130 |
lemma minus_int_code [code]: |
63652 | 2131 |
"k - 0 = k" |
2132 |
"0 - l = uminus l" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2133 |
"Pos m - Pos n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2134 |
"Pos m - Neg n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2135 |
"Neg m - Pos n = Neg (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2136 |
"Neg m - Neg n = sub n m" |
63652 | 2137 |
for k l :: int |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2138 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2139 |
|
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|
2140 |
lemma times_int_code [code]: |
63652 | 2141 |
"k * 0 = 0" |
2142 |
"0 * l = 0" |
|
47108
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changeset
|
2143 |
"Pos m * Pos n = Pos (m * n)" |
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changeset
|
2144 |
"Pos m * Neg n = Neg (m * n)" |
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changeset
|
2145 |
"Neg m * Pos n = Neg (m * n)" |
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parents:
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diff
changeset
|
2146 |
"Neg m * Neg n = Pos (m * n)" |
63652 | 2147 |
for k l :: int |
47108
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changeset
|
2148 |
by simp_all |
26507 | 2149 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
37887
diff
changeset
|
2150 |
instantiation int :: equal |
26507 | 2151 |
begin |
2152 |
||
63652 | 2153 |
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)" |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
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diff
changeset
|
2154 |
|
61169 | 2155 |
instance |
2156 |
by standard (rule equal_int_def) |
|
26507 | 2157 |
|
2158 |
end |
|
2159 |
||
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diff
changeset
|
2160 |
lemma equal_int_code [code]: |
2a1953f0d20d
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parents:
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diff
changeset
|
2161 |
"HOL.equal 0 (0::int) \<longleftrightarrow> True" |
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diff
changeset
|
2162 |
"HOL.equal 0 (Pos l) \<longleftrightarrow> False" |
2a1953f0d20d
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parents:
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diff
changeset
|
2163 |
"HOL.equal 0 (Neg l) \<longleftrightarrow> False" |
2a1953f0d20d
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parents:
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diff
changeset
|
2164 |
"HOL.equal (Pos k) 0 \<longleftrightarrow> False" |
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parents:
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diff
changeset
|
2165 |
"HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l" |
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parents:
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diff
changeset
|
2166 |
"HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False" |
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parents:
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diff
changeset
|
2167 |
"HOL.equal (Neg k) 0 \<longleftrightarrow> False" |
2a1953f0d20d
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parents:
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diff
changeset
|
2168 |
"HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False" |
2a1953f0d20d
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parents:
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diff
changeset
|
2169 |
"HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l" |
2a1953f0d20d
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parents:
46756
diff
changeset
|
2170 |
by (auto simp add: equal) |
26507 | 2171 |
|
63652 | 2172 |
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True" |
2173 |
for k :: int |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
2174 |
by (fact equal_refl) |
26507 | 2175 |
|
28562 | 2176 |
lemma less_eq_int_code [code]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
2177 |
"0 \<le> (0::int) \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2178 |
"0 \<le> Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2179 |
"0 \<le> Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2180 |
"Pos k \<le> 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2181 |
"Pos k \<le> Pos l \<longleftrightarrow> k \<le> l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2182 |
"Pos k \<le> Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2183 |
"Neg k \<le> 0 \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2184 |
"Neg k \<le> Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2185 |
"Neg k \<le> Neg l \<longleftrightarrow> l \<le> k" |
28958 | 2186 |
by simp_all |
26507 | 2187 |
|
28562 | 2188 |
lemma less_int_code [code]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2189 |
"0 < (0::int) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2190 |
"0 < Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2191 |
"0 < Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2192 |
"Pos k < 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2193 |
"Pos k < Pos l \<longleftrightarrow> k < l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2194 |
"Pos k < Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2195 |
"Neg k < 0 \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2196 |
"Neg k < Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2197 |
"Neg k < Neg l \<longleftrightarrow> l < k" |
28958 | 2198 |
by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2199 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
2200 |
lemma nat_code [code]: |
2a1953f0d20d
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parents:
46756
diff
changeset
|
2201 |
"nat (Int.Neg k) = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2202 |
"nat 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2203 |
"nat (Int.Pos k) = nat_of_num k" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
2204 |
by (simp_all add: nat_of_num_numeral) |
25928 | 2205 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2206 |
lemma (in ring_1) of_int_code [code]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
2207 |
"of_int (Int.Neg k) = - numeral k" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46756
diff
changeset
|
2208 |
"of_int 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2209 |
"of_int (Int.Pos k) = numeral k" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2210 |
by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2211 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
2212 |
|
63652 | 2213 |
text \<open>Serializer setup.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2214 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51994
diff
changeset
|
2215 |
code_identifier |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51994
diff
changeset
|
2216 |
code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2217 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2218 |
quickcheck_params [default_type = int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2219 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
2220 |
hide_const (open) Pos Neg sub dup |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2221 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2222 |
|
61799 | 2223 |
text \<open>De-register \<open>int\<close> as a quotient type:\<close> |
48045 | 2224 |
|
53652
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53065
diff
changeset
|
2225 |
lifting_update int.lifting |
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53065
diff
changeset
|
2226 |
lifting_forget int.lifting |
48045 | 2227 |
|
67116 | 2228 |
|
2229 |
subsection \<open>Duplicates\<close> |
|
2230 |
||
2231 |
lemmas int_sum = of_nat_sum [where 'a=int] |
|
2232 |
lemmas int_prod = of_nat_prod [where 'a=int] |
|
2233 |
lemmas zle_int = of_nat_le_iff [where 'a=int] |
|
2234 |
lemmas int_int_eq = of_nat_eq_iff [where 'a=int] |
|
2235 |
lemmas nonneg_eq_int = nonneg_int_cases |
|
2236 |
lemmas double_eq_0_iff = double_zero |
|
2237 |
||
2238 |
lemmas int_distrib = |
|
2239 |
distrib_right [of z1 z2 w] |
|
2240 |
distrib_left [of w z1 z2] |
|
2241 |
left_diff_distrib [of z1 z2 w] |
|
2242 |
right_diff_distrib [of w z1 z2] |
|
2243 |
for z1 z2 w :: int |
|
2244 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2245 |
end |
67116 | 2246 |