author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35634 | 6fdfe37b84d6 |
child 35815 | 10e723e54076 |
child 35828 | 46cfc4b8112e |
permissions | -rw-r--r-- |
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(* Title: Int.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Tobias Nipkow, Florian Haftmann, TU Muenchen |
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Copyright 1994 University of Cambridge |
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*) |
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header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} |
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theory Int |
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Merged theories about wellfoundedness into one: Wellfounded.thy
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imports Equiv_Relations Nat Wellfounded |
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uses |
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("Tools/numeral.ML") |
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("Tools/numeral_syntax.ML") |
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modules numeral_simprocs, nat_numeral_simprocs; proper structures for numeral simprocs
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("Tools/int_arith.ML") |
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begin |
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|
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subsection {* The equivalence relation underlying the integers *} |
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definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where |
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[code del]: "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }" |
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|
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typedef (Integ) |
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int = "UNIV//intrel" |
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by (auto simp add: quotient_def) |
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|
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instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}" |
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begin |
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|
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definition |
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Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})" |
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|
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definition |
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One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})" |
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|
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definition |
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add_int_def [code del]: "z + w = Abs_Integ |
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(\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w. |
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intrel `` {(x + u, y + v)})" |
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|
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definition |
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minus_int_def [code del]: |
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"- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})" |
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|
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definition |
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diff_int_def [code del]: "z - w = z + (-w \<Colon> int)" |
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|
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definition |
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mult_int_def [code del]: "z * w = Abs_Integ |
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(\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w. |
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intrel `` {(x*u + y*v, x*v + y*u)})" |
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|
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definition |
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le_int_def [code del]: |
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"z \<le> w \<longleftrightarrow> (\<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w)" |
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|
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definition |
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less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w" |
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|
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definition |
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zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)" |
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|
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definition |
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zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)" |
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|
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instance .. |
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end |
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69 |
|
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70 |
|
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71 |
subsection{*Construction of the Integers*} |
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72 |
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73 |
lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)" |
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74 |
by (simp add: intrel_def) |
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75 |
|
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lemma equiv_intrel: "equiv UNIV intrel" |
30198 | 77 |
by (simp add: intrel_def equiv_def refl_on_def sym_def trans_def) |
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78 |
|
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79 |
text{*Reduces equality of equivalence classes to the @{term intrel} relation: |
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@{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *} |
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81 |
lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I] |
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82 |
|
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83 |
text{*All equivalence classes belong to set of representatives*} |
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84 |
lemma [simp]: "intrel``{(x,y)} \<in> Integ" |
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85 |
by (auto simp add: Integ_def intrel_def quotient_def) |
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86 |
|
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87 |
text{*Reduces equality on abstractions to equality on representatives: |
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88 |
@{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *} |
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89 |
declare Abs_Integ_inject [simp,noatp] Abs_Integ_inverse [simp,noatp] |
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90 |
|
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91 |
text{*Case analysis on the representation of an integer as an equivalence |
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92 |
class of pairs of naturals.*} |
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93 |
lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]: |
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94 |
"(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P" |
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95 |
apply (rule Abs_Integ_cases [of z]) |
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96 |
apply (auto simp add: Integ_def quotient_def) |
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97 |
done |
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98 |
|
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99 |
|
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100 |
subsection {* Arithmetic Operations *} |
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101 |
|
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102 |
lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})" |
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103 |
proof - |
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104 |
have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel" |
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105 |
by (simp add: congruent_def) |
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106 |
thus ?thesis |
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107 |
by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel]) |
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108 |
qed |
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109 |
|
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110 |
lemma add: |
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111 |
"Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) = |
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112 |
Abs_Integ (intrel``{(x+u, y+v)})" |
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113 |
proof - |
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114 |
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z) |
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115 |
respects2 intrel" |
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116 |
by (simp add: congruent2_def) |
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117 |
thus ?thesis |
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118 |
by (simp add: add_int_def UN_UN_split_split_eq |
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119 |
UN_equiv_class2 [OF equiv_intrel equiv_intrel]) |
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120 |
qed |
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121 |
|
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122 |
text{*Congruence property for multiplication*} |
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123 |
lemma mult_congruent2: |
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124 |
"(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1) |
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125 |
respects2 intrel" |
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126 |
apply (rule equiv_intrel [THEN congruent2_commuteI]) |
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127 |
apply (force simp add: mult_ac, clarify) |
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parents:
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128 |
apply (simp add: congruent_def mult_ac) |
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129 |
apply (rename_tac u v w x y z) |
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|
130 |
apply (subgoal_tac "u*y + x*y = w*y + v*y & u*z + x*z = w*z + v*z") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
131 |
apply (simp add: mult_ac) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
132 |
apply (simp add: add_mult_distrib [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
133 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
134 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
135 |
lemma mult: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
136 |
"Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
137 |
Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
138 |
by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
139 |
UN_equiv_class2 [OF equiv_intrel equiv_intrel]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
140 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
141 |
text{*The integers form a @{text comm_ring_1}*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
142 |
instance int :: comm_ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
143 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
144 |
fix i j k :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
145 |
show "(i + j) + k = i + (j + k)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
146 |
by (cases i, cases j, cases k) (simp add: add add_assoc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
147 |
show "i + j = j + i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
148 |
by (cases i, cases j) (simp add: add_ac add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
149 |
show "0 + i = i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
150 |
by (cases i) (simp add: Zero_int_def add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
151 |
show "- i + i = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
152 |
by (cases i) (simp add: Zero_int_def minus add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
153 |
show "i - j = i + - j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
154 |
by (simp add: diff_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
155 |
show "(i * j) * k = i * (j * k)" |
29667 | 156 |
by (cases i, cases j, cases k) (simp add: mult algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
157 |
show "i * j = j * i" |
29667 | 158 |
by (cases i, cases j) (simp add: mult algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
159 |
show "1 * i = i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
160 |
by (cases i) (simp add: One_int_def mult) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
161 |
show "(i + j) * k = i * k + j * k" |
29667 | 162 |
by (cases i, cases j, cases k) (simp add: add mult algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
163 |
show "0 \<noteq> (1::int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
164 |
by (simp add: Zero_int_def One_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
165 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
166 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
167 |
lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
168 |
by (induct m, simp_all add: Zero_int_def One_int_def add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
169 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
170 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
171 |
subsection {* The @{text "\<le>"} Ordering *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
172 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
173 |
lemma le: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
174 |
"(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
175 |
by (force simp add: le_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
176 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
177 |
lemma less: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
178 |
"(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
179 |
by (simp add: less_int_def le order_less_le) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
180 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
181 |
instance int :: linorder |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
182 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
183 |
fix i j k :: int |
27682 | 184 |
show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j" |
185 |
by (cases i, cases j) (simp add: le) |
|
186 |
show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)" |
|
187 |
by (auto simp add: less_int_def dest: antisym) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
188 |
show "i \<le> i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
189 |
by (cases i) (simp add: le) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
190 |
show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
191 |
by (cases i, cases j, cases k) (simp add: le) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
192 |
show "i \<le> j \<or> j \<le> i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
193 |
by (cases i, cases j) (simp add: le linorder_linear) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
194 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
195 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
196 |
instantiation int :: distrib_lattice |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
197 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
198 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
199 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
200 |
"(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
201 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
202 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
203 |
"(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
204 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
205 |
instance |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
206 |
by intro_classes |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
207 |
(auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
208 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
209 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
210 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
211 |
instance int :: ordered_cancel_ab_semigroup_add |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
212 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
213 |
fix i j k :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
214 |
show "i \<le> j \<Longrightarrow> k + i \<le> k + j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
215 |
by (cases i, cases j, cases k) (simp add: le add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
216 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
217 |
|
25961 | 218 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
219 |
text{*Strict Monotonicity of Multiplication*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
220 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
221 |
text{*strict, in 1st argument; proof is by induction on k>0*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
222 |
lemma zmult_zless_mono2_lemma: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
223 |
"(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
224 |
apply (induct "k", simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
225 |
apply (simp add: left_distrib) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
226 |
apply (case_tac "k=0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
227 |
apply (simp_all add: add_strict_mono) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
228 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
229 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
230 |
lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
231 |
apply (cases k) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
232 |
apply (auto simp add: le add int_def Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
233 |
apply (rule_tac x="x-y" in exI, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
234 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
235 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
236 |
lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
237 |
apply (cases k) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
238 |
apply (simp add: less int_def Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
239 |
apply (rule_tac x="x-y" in exI, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
240 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
241 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
242 |
lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
243 |
apply (drule zero_less_imp_eq_int) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
244 |
apply (auto simp add: zmult_zless_mono2_lemma) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
245 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
246 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
247 |
text{*The integers form an ordered integral domain*} |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
248 |
instance int :: linordered_idom |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
249 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
250 |
fix i j k :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
251 |
show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
252 |
by (rule zmult_zless_mono2) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
253 |
show "\<bar>i\<bar> = (if i < 0 then -i else i)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
254 |
by (simp only: zabs_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
255 |
show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
256 |
by (simp only: zsgn_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
257 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
258 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
259 |
lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
260 |
apply (cases w, cases z) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
261 |
apply (simp add: less le add One_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
262 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
263 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
264 |
lemma zless_iff_Suc_zadd: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
265 |
"(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
266 |
apply (cases z, cases w) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
267 |
apply (auto simp add: less add int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
268 |
apply (rename_tac a b c d) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
269 |
apply (rule_tac x="a+d - Suc(c+b)" in exI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
270 |
apply arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
271 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
272 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
273 |
lemmas int_distrib = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
274 |
left_distrib [of "z1::int" "z2" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
275 |
right_distrib [of "w::int" "z1" "z2", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
276 |
left_diff_distrib [of "z1::int" "z2" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
277 |
right_diff_distrib [of "w::int" "z1" "z2", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
278 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
279 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
280 |
subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
281 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
282 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
283 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
284 |
|
31015 | 285 |
definition of_int :: "int \<Rightarrow> 'a" where |
28562 | 286 |
[code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
287 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
288 |
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
289 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
290 |
have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel" |
29667 | 291 |
by (simp add: congruent_def algebra_simps of_nat_add [symmetric] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
292 |
del: of_nat_add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
293 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
294 |
by (simp add: of_int_def UN_equiv_class [OF equiv_intrel]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
295 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
296 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
297 |
lemma of_int_0 [simp]: "of_int 0 = 0" |
29667 | 298 |
by (simp add: of_int Zero_int_def) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
299 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
300 |
lemma of_int_1 [simp]: "of_int 1 = 1" |
29667 | 301 |
by (simp add: of_int One_int_def) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
302 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
303 |
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z" |
29667 | 304 |
by (cases w, cases z, simp add: algebra_simps of_int add) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
305 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
306 |
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)" |
29667 | 307 |
by (cases z, simp add: algebra_simps of_int minus) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
308 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
309 |
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z" |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
310 |
by (simp add: diff_minus Groups.diff_minus) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
311 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
312 |
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
313 |
apply (cases w, cases z) |
29667 | 314 |
apply (simp add: algebra_simps of_int mult of_nat_mult) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
315 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
316 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
317 |
text{*Collapse nested embeddings*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
318 |
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n" |
29667 | 319 |
by (induct n) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
320 |
|
31015 | 321 |
lemma of_int_power: |
322 |
"of_int (z ^ n) = of_int z ^ n" |
|
323 |
by (induct n) simp_all |
|
324 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
325 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
326 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
327 |
context linordered_idom |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
328 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
329 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
330 |
lemma of_int_le_iff [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
331 |
"of_int w \<le> of_int z \<longleftrightarrow> w \<le> z" |
29667 | 332 |
by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
333 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
334 |
text{*Special cases where either operand is zero*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
335 |
lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
336 |
lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
337 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
338 |
lemma of_int_less_iff [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
339 |
"of_int w < of_int z \<longleftrightarrow> w < z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
340 |
by (simp add: not_le [symmetric] linorder_not_le [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
341 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
342 |
text{*Special cases where either operand is zero*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
343 |
lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
344 |
lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
345 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
346 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
347 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
348 |
text{*Class for unital rings with characteristic zero. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
349 |
Includes non-ordered rings like the complex numbers.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
350 |
class ring_char_0 = ring_1 + semiring_char_0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
351 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
352 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
353 |
lemma of_int_eq_iff [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
354 |
"of_int w = of_int z \<longleftrightarrow> w = z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
355 |
apply (cases w, cases z, simp add: of_int) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
356 |
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
357 |
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
358 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
359 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
360 |
text{*Special cases where either operand is zero*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
361 |
lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
362 |
lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
363 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
364 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
365 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
366 |
text{*Every @{text linordered_idom} has characteristic zero.*} |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
367 |
subclass (in linordered_idom) ring_char_0 by intro_locales |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
368 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
369 |
lemma of_int_eq_id [simp]: "of_int = id" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
370 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
371 |
fix z show "of_int z = id z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
372 |
by (cases z) (simp add: of_int add minus int_def diff_minus) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
373 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
374 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
375 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
376 |
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
377 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
378 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
379 |
nat :: "int \<Rightarrow> nat" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
380 |
where |
28562 | 381 |
[code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
382 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
383 |
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
384 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
385 |
have "(\<lambda>(x,y). {x-y}) respects intrel" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
386 |
by (simp add: congruent_def) arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
387 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
388 |
by (simp add: nat_def UN_equiv_class [OF equiv_intrel]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
389 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
390 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
391 |
lemma nat_int [simp]: "nat (of_nat n) = n" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
392 |
by (simp add: nat int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
393 |
|
35216 | 394 |
(* FIXME: duplicates nat_0 *) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
395 |
lemma nat_zero [simp]: "nat 0 = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
396 |
by (simp add: Zero_int_def nat) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
397 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
398 |
lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
399 |
by (cases z, simp add: nat le int_def Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
400 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
401 |
corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
402 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
403 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
404 |
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
405 |
by (cases z, simp add: nat le Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
406 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
407 |
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
408 |
apply (cases w, cases z) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
409 |
apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
410 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
411 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
412 |
text{*An alternative condition is @{term "0 \<le> w"} *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
413 |
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
414 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
415 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
416 |
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
417 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
418 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
419 |
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
420 |
apply (cases w, cases z) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
421 |
apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
422 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
423 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
424 |
lemma nonneg_eq_int: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
425 |
fixes z :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
426 |
assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
427 |
shows P |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
428 |
using assms by (blast dest: nat_0_le sym) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
429 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
430 |
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
431 |
by (cases w, simp add: nat le int_def Zero_int_def, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
432 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
433 |
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
434 |
by (simp only: eq_commute [of m] nat_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
435 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
436 |
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
437 |
apply (cases w) |
29700 | 438 |
apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
439 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
440 |
|
29700 | 441 |
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0" |
442 |
by(simp add: nat_eq_iff) arith |
|
443 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
444 |
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
445 |
by (auto simp add: nat_eq_iff2) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
446 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
447 |
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
448 |
by (insert zless_nat_conj [of 0], auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
449 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
450 |
lemma nat_add_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
451 |
"[| (0::int) \<le> z; 0 \<le> z' |] ==> nat (z+z') = nat z + nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
452 |
by (cases z, cases z', simp add: nat add le Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
453 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
454 |
lemma nat_diff_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
455 |
"[| (0::int) \<le> z'; z' \<le> z |] ==> nat (z-z') = nat z - nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
456 |
by (cases z, cases z', |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
457 |
simp add: nat add minus diff_minus le Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
458 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
459 |
lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
460 |
by (simp add: int_def minus nat Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
461 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
462 |
lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
463 |
by (cases z, simp add: nat less int_def, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
464 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
465 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
466 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
467 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
468 |
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
469 |
by (cases z rule: eq_Abs_Integ) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
470 |
(simp add: nat le of_int Zero_int_def of_nat_diff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
471 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
472 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
473 |
|
29779 | 474 |
text {* For termination proofs: *} |
475 |
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" .. |
|
476 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
477 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
478 |
subsection{*Lemmas about the Function @{term of_nat} and Orderings*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
479 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
480 |
lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
481 |
by (simp add: order_less_le del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
482 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
483 |
lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
484 |
by (rule negative_zless_0 [THEN order_less_le_trans], simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
485 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
486 |
lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
487 |
by (simp add: minus_le_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
488 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
489 |
lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
490 |
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
491 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
492 |
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
493 |
by (subst le_minus_iff, simp del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
494 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
495 |
lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
496 |
by (simp add: int_def le minus Zero_int_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
497 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
498 |
lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
499 |
by (simp add: linorder_not_less) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
500 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
501 |
lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
502 |
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
503 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
504 |
lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
505 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
506 |
have "(w \<le> z) = (0 \<le> z - w)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
507 |
by (simp only: le_diff_eq add_0_left) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
508 |
also have "\<dots> = (\<exists>n. z - w = of_nat n)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
509 |
by (auto elim: zero_le_imp_eq_int) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
510 |
also have "\<dots> = (\<exists>n. z = w + of_nat n)" |
29667 | 511 |
by (simp only: algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
512 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
513 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
514 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
515 |
lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
516 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
517 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
518 |
lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
519 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
520 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
521 |
text{*This version is proved for all ordered rings, not just integers! |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
522 |
It is proved here because attribute @{text arith_split} is not available |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
523 |
in theory @{text Rings}. |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
524 |
But is it really better than just rewriting with @{text abs_if}?*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
525 |
lemma abs_split [arith_split,noatp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
526 |
"P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
527 |
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
528 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
529 |
lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
530 |
apply (cases x) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
531 |
apply (auto simp add: le minus Zero_int_def int_def order_less_le) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
532 |
apply (rule_tac x="y - Suc x" in exI, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
533 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
534 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
535 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
536 |
subsection {* Cases and induction *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
537 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
538 |
text{*Now we replace the case analysis rule by a more conventional one: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
539 |
whether an integer is negative or not.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
540 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
541 |
theorem int_cases [cases type: int, case_names nonneg neg]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
542 |
"[|!! n. (z \<Colon> int) = of_nat n ==> P; !! n. z = - (of_nat (Suc n)) ==> P |] ==> P" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
543 |
apply (cases "z < 0", blast dest!: negD) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
544 |
apply (simp add: linorder_not_less del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
545 |
apply auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
546 |
apply (blast dest: nat_0_le [THEN sym]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
547 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
548 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
549 |
theorem int_induct [induct type: int, case_names nonneg neg]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
550 |
"[|!! n. P (of_nat n \<Colon> int); !!n. P (- (of_nat (Suc n))) |] ==> P z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
551 |
by (cases z rule: int_cases) auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
552 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
553 |
text{*Contributed by Brian Huffman*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
554 |
theorem int_diff_cases: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
555 |
obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
556 |
apply (cases z rule: eq_Abs_Integ) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
557 |
apply (rule_tac m=x and n=y in diff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
558 |
apply (simp add: int_def diff_def minus add) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
559 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
560 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
561 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
562 |
subsection {* Binary representation *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
563 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
564 |
text {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
565 |
This formalization defines binary arithmetic in terms of the integers |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
566 |
rather than using a datatype. This avoids multiple representations (leading |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
567 |
zeroes, etc.) See @{text "ZF/Tools/twos-compl.ML"}, function @{text |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
568 |
int_of_binary}, for the numerical interpretation. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
569 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
570 |
The representation expects that @{text "(m mod 2)"} is 0 or 1, |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
571 |
even if m is negative; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
572 |
For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
573 |
@{text "-5 = (-3)*2 + 1"}. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
574 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
575 |
This two's complement binary representation derives from the paper |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
576 |
"An Efficient Representation of Arithmetic for Term Rewriting" by |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
577 |
Dave Cohen and Phil Watson, Rewriting Techniques and Applications, |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
578 |
Springer LNCS 488 (240-251), 1991. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
579 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
580 |
|
28958 | 581 |
subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *} |
582 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
583 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
584 |
Pls :: int where |
28562 | 585 |
[code del]: "Pls = 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
586 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
587 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
588 |
Min :: int where |
28562 | 589 |
[code del]: "Min = - 1" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
590 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
591 |
definition |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
592 |
Bit0 :: "int \<Rightarrow> int" where |
28562 | 593 |
[code del]: "Bit0 k = k + k" |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
594 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
595 |
definition |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
596 |
Bit1 :: "int \<Rightarrow> int" where |
28562 | 597 |
[code del]: "Bit1 k = 1 + k + k" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
598 |
|
29608 | 599 |
class number = -- {* for numeric types: nat, int, real, \dots *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
600 |
fixes number_of :: "int \<Rightarrow> 'a" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
601 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
602 |
use "Tools/numeral.ML" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
603 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
604 |
syntax |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
605 |
"_Numeral" :: "num_const \<Rightarrow> 'a" ("_") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
606 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
607 |
use "Tools/numeral_syntax.ML" |
35123 | 608 |
setup Numeral_Syntax.setup |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
609 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
610 |
abbreviation |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
611 |
"Numeral0 \<equiv> number_of Pls" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
612 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
613 |
abbreviation |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
614 |
"Numeral1 \<equiv> number_of (Bit1 Pls)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
615 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
616 |
lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
617 |
-- {* Unfold all @{text let}s involving constants *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
618 |
unfolding Let_def .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
619 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
620 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
621 |
succ :: "int \<Rightarrow> int" where |
28562 | 622 |
[code del]: "succ k = k + 1" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
623 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
624 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
625 |
pred :: "int \<Rightarrow> int" where |
28562 | 626 |
[code del]: "pred k = k - 1" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
627 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
628 |
lemmas |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
629 |
max_number_of [simp] = max_def |
35216 | 630 |
[of "number_of u" "number_of v", standard] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
631 |
and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
632 |
min_number_of [simp] = min_def |
35216 | 633 |
[of "number_of u" "number_of v", standard] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
634 |
-- {* unfolding @{text minx} and @{text max} on numerals *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
635 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
636 |
lemmas numeral_simps = |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
637 |
succ_def pred_def Pls_def Min_def Bit0_def Bit1_def |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
638 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
639 |
text {* Removal of leading zeroes *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
640 |
|
31998
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31100
diff
changeset
|
641 |
lemma Bit0_Pls [simp, code_post]: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
642 |
"Bit0 Pls = Pls" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
643 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
644 |
|
31998
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31100
diff
changeset
|
645 |
lemma Bit1_Min [simp, code_post]: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
646 |
"Bit1 Min = Min" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
647 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
648 |
|
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
649 |
lemmas normalize_bin_simps = |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
650 |
Bit0_Pls Bit1_Min |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
651 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
652 |
|
28958 | 653 |
subsubsection {* Successor and predecessor functions *} |
654 |
||
655 |
text {* Successor *} |
|
656 |
||
657 |
lemma succ_Pls: |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
658 |
"succ Pls = Bit1 Pls" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
659 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
660 |
|
28958 | 661 |
lemma succ_Min: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
662 |
"succ Min = Pls" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
663 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
664 |
|
28958 | 665 |
lemma succ_Bit0: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
666 |
"succ (Bit0 k) = Bit1 k" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
667 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
668 |
|
28958 | 669 |
lemma succ_Bit1: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
670 |
"succ (Bit1 k) = Bit0 (succ k)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
671 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
672 |
|
28958 | 673 |
lemmas succ_bin_simps [simp] = |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
674 |
succ_Pls succ_Min succ_Bit0 succ_Bit1 |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
675 |
|
28958 | 676 |
text {* Predecessor *} |
677 |
||
678 |
lemma pred_Pls: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
679 |
"pred Pls = Min" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
680 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
681 |
|
28958 | 682 |
lemma pred_Min: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
683 |
"pred Min = Bit0 Min" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
684 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
685 |
|
28958 | 686 |
lemma pred_Bit0: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
687 |
"pred (Bit0 k) = Bit1 (pred k)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
688 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
689 |
|
28958 | 690 |
lemma pred_Bit1: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
691 |
"pred (Bit1 k) = Bit0 k" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
692 |
unfolding numeral_simps by simp |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
693 |
|
28958 | 694 |
lemmas pred_bin_simps [simp] = |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
695 |
pred_Pls pred_Min pred_Bit0 pred_Bit1 |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
696 |
|
28958 | 697 |
|
698 |
subsubsection {* Binary arithmetic *} |
|
699 |
||
700 |
text {* Addition *} |
|
701 |
||
702 |
lemma add_Pls: |
|
703 |
"Pls + k = k" |
|
704 |
unfolding numeral_simps by simp |
|
705 |
||
706 |
lemma add_Min: |
|
707 |
"Min + k = pred k" |
|
708 |
unfolding numeral_simps by simp |
|
709 |
||
710 |
lemma add_Bit0_Bit0: |
|
711 |
"(Bit0 k) + (Bit0 l) = Bit0 (k + l)" |
|
712 |
unfolding numeral_simps by simp |
|
713 |
||
714 |
lemma add_Bit0_Bit1: |
|
715 |
"(Bit0 k) + (Bit1 l) = Bit1 (k + l)" |
|
716 |
unfolding numeral_simps by simp |
|
717 |
||
718 |
lemma add_Bit1_Bit0: |
|
719 |
"(Bit1 k) + (Bit0 l) = Bit1 (k + l)" |
|
720 |
unfolding numeral_simps by simp |
|
721 |
||
722 |
lemma add_Bit1_Bit1: |
|
723 |
"(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)" |
|
724 |
unfolding numeral_simps by simp |
|
725 |
||
726 |
lemma add_Pls_right: |
|
727 |
"k + Pls = k" |
|
728 |
unfolding numeral_simps by simp |
|
729 |
||
730 |
lemma add_Min_right: |
|
731 |
"k + Min = pred k" |
|
732 |
unfolding numeral_simps by simp |
|
733 |
||
734 |
lemmas add_bin_simps [simp] = |
|
735 |
add_Pls add_Min add_Pls_right add_Min_right |
|
736 |
add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1 |
|
737 |
||
738 |
text {* Negation *} |
|
739 |
||
740 |
lemma minus_Pls: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
741 |
"- Pls = Pls" |
28958 | 742 |
unfolding numeral_simps by simp |
743 |
||
744 |
lemma minus_Min: |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
745 |
"- Min = Bit1 Pls" |
28958 | 746 |
unfolding numeral_simps by simp |
747 |
||
748 |
lemma minus_Bit0: |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
749 |
"- (Bit0 k) = Bit0 (- k)" |
28958 | 750 |
unfolding numeral_simps by simp |
751 |
||
752 |
lemma minus_Bit1: |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
753 |
"- (Bit1 k) = Bit1 (pred (- k))" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
754 |
unfolding numeral_simps by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
755 |
|
28958 | 756 |
lemmas minus_bin_simps [simp] = |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
757 |
minus_Pls minus_Min minus_Bit0 minus_Bit1 |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
758 |
|
28958 | 759 |
text {* Subtraction *} |
760 |
||
29046 | 761 |
lemma diff_bin_simps [simp]: |
762 |
"k - Pls = k" |
|
763 |
"k - Min = succ k" |
|
764 |
"Pls - (Bit0 l) = Bit0 (Pls - l)" |
|
765 |
"Pls - (Bit1 l) = Bit1 (Min - l)" |
|
766 |
"Min - (Bit0 l) = Bit1 (Min - l)" |
|
767 |
"Min - (Bit1 l) = Bit0 (Min - l)" |
|
28958 | 768 |
"(Bit0 k) - (Bit0 l) = Bit0 (k - l)" |
769 |
"(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)" |
|
770 |
"(Bit1 k) - (Bit0 l) = Bit1 (k - l)" |
|
771 |
"(Bit1 k) - (Bit1 l) = Bit0 (k - l)" |
|
29046 | 772 |
unfolding numeral_simps by simp_all |
28958 | 773 |
|
774 |
text {* Multiplication *} |
|
775 |
||
776 |
lemma mult_Pls: |
|
777 |
"Pls * w = Pls" |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
778 |
unfolding numeral_simps by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
779 |
|
28958 | 780 |
lemma mult_Min: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
781 |
"Min * k = - k" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
782 |
unfolding numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
783 |
|
28958 | 784 |
lemma mult_Bit0: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
785 |
"(Bit0 k) * l = Bit0 (k * l)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
786 |
unfolding numeral_simps int_distrib by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
787 |
|
28958 | 788 |
lemma mult_Bit1: |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
789 |
"(Bit1 k) * l = (Bit0 (k * l)) + l" |
28958 | 790 |
unfolding numeral_simps int_distrib by simp |
791 |
||
792 |
lemmas mult_bin_simps [simp] = |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
793 |
mult_Pls mult_Min mult_Bit0 mult_Bit1 |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
794 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
795 |
|
28958 | 796 |
subsubsection {* Binary comparisons *} |
797 |
||
798 |
text {* Preliminaries *} |
|
799 |
||
800 |
lemma even_less_0_iff: |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
801 |
"a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)" |
28958 | 802 |
proof - |
803 |
have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib) |
|
804 |
also have "(1+1)*a < 0 \<longleftrightarrow> a < 0" |
|
805 |
by (simp add: mult_less_0_iff zero_less_two |
|
806 |
order_less_not_sym [OF zero_less_two]) |
|
807 |
finally show ?thesis . |
|
808 |
qed |
|
809 |
||
810 |
lemma le_imp_0_less: |
|
811 |
assumes le: "0 \<le> z" |
|
812 |
shows "(0::int) < 1 + z" |
|
813 |
proof - |
|
814 |
have "0 \<le> z" by fact |
|
815 |
also have "... < z + 1" by (rule less_add_one) |
|
816 |
also have "... = 1 + z" by (simp add: add_ac) |
|
817 |
finally show "0 < 1 + z" . |
|
818 |
qed |
|
819 |
||
820 |
lemma odd_less_0_iff: |
|
821 |
"(1 + z + z < 0) = (z < (0::int))" |
|
822 |
proof (cases z rule: int_cases) |
|
823 |
case (nonneg n) |
|
824 |
thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing |
|
825 |
le_imp_0_less [THEN order_less_imp_le]) |
|
826 |
next |
|
827 |
case (neg n) |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
828 |
thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1 |
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
829 |
add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric]) |
28958 | 830 |
qed |
831 |
||
28985
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
832 |
lemma bin_less_0_simps: |
28958 | 833 |
"Pls < 0 \<longleftrightarrow> False" |
834 |
"Min < 0 \<longleftrightarrow> True" |
|
835 |
"Bit0 w < 0 \<longleftrightarrow> w < 0" |
|
836 |
"Bit1 w < 0 \<longleftrightarrow> w < 0" |
|
837 |
unfolding numeral_simps |
|
838 |
by (simp_all add: even_less_0_iff odd_less_0_iff) |
|
839 |
||
840 |
lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)" |
|
841 |
by simp |
|
842 |
||
843 |
lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l" |
|
844 |
unfolding numeral_simps |
|
845 |
proof |
|
846 |
have "k - 1 < k" by simp |
|
847 |
also assume "k \<le> l" |
|
848 |
finally show "k - 1 < l" . |
|
849 |
next |
|
850 |
assume "k - 1 < l" |
|
851 |
hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle) |
|
852 |
thus "k \<le> l" by simp |
|
853 |
qed |
|
854 |
||
855 |
lemma succ_pred: "succ (pred x) = x" |
|
856 |
unfolding numeral_simps by simp |
|
857 |
||
858 |
text {* Less-than *} |
|
859 |
||
860 |
lemma less_bin_simps [simp]: |
|
861 |
"Pls < Pls \<longleftrightarrow> False" |
|
862 |
"Pls < Min \<longleftrightarrow> False" |
|
863 |
"Pls < Bit0 k \<longleftrightarrow> Pls < k" |
|
864 |
"Pls < Bit1 k \<longleftrightarrow> Pls \<le> k" |
|
865 |
"Min < Pls \<longleftrightarrow> True" |
|
866 |
"Min < Min \<longleftrightarrow> False" |
|
867 |
"Min < Bit0 k \<longleftrightarrow> Min < k" |
|
868 |
"Min < Bit1 k \<longleftrightarrow> Min < k" |
|
869 |
"Bit0 k < Pls \<longleftrightarrow> k < Pls" |
|
870 |
"Bit0 k < Min \<longleftrightarrow> k \<le> Min" |
|
871 |
"Bit1 k < Pls \<longleftrightarrow> k < Pls" |
|
872 |
"Bit1 k < Min \<longleftrightarrow> k < Min" |
|
873 |
"Bit0 k < Bit0 l \<longleftrightarrow> k < l" |
|
874 |
"Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l" |
|
875 |
"Bit1 k < Bit0 l \<longleftrightarrow> k < l" |
|
876 |
"Bit1 k < Bit1 l \<longleftrightarrow> k < l" |
|
877 |
unfolding le_iff_pred_less |
|
878 |
less_bin_lemma [of Pls] |
|
879 |
less_bin_lemma [of Min] |
|
880 |
less_bin_lemma [of "k"] |
|
881 |
less_bin_lemma [of "Bit0 k"] |
|
882 |
less_bin_lemma [of "Bit1 k"] |
|
883 |
less_bin_lemma [of "pred Pls"] |
|
884 |
less_bin_lemma [of "pred k"] |
|
28985
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
885 |
by (simp_all add: bin_less_0_simps succ_pred) |
28958 | 886 |
|
887 |
text {* Less-than-or-equal *} |
|
888 |
||
889 |
lemma le_bin_simps [simp]: |
|
890 |
"Pls \<le> Pls \<longleftrightarrow> True" |
|
891 |
"Pls \<le> Min \<longleftrightarrow> False" |
|
892 |
"Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k" |
|
893 |
"Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k" |
|
894 |
"Min \<le> Pls \<longleftrightarrow> True" |
|
895 |
"Min \<le> Min \<longleftrightarrow> True" |
|
896 |
"Min \<le> Bit0 k \<longleftrightarrow> Min < k" |
|
897 |
"Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k" |
|
898 |
"Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls" |
|
899 |
"Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min" |
|
900 |
"Bit1 k \<le> Pls \<longleftrightarrow> k < Pls" |
|
901 |
"Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min" |
|
902 |
"Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l" |
|
903 |
"Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l" |
|
904 |
"Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l" |
|
905 |
"Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l" |
|
906 |
unfolding not_less [symmetric] |
|
907 |
by (simp_all add: not_le) |
|
908 |
||
909 |
text {* Equality *} |
|
910 |
||
911 |
lemma eq_bin_simps [simp]: |
|
912 |
"Pls = Pls \<longleftrightarrow> True" |
|
913 |
"Pls = Min \<longleftrightarrow> False" |
|
914 |
"Pls = Bit0 l \<longleftrightarrow> Pls = l" |
|
915 |
"Pls = Bit1 l \<longleftrightarrow> False" |
|
916 |
"Min = Pls \<longleftrightarrow> False" |
|
917 |
"Min = Min \<longleftrightarrow> True" |
|
918 |
"Min = Bit0 l \<longleftrightarrow> False" |
|
919 |
"Min = Bit1 l \<longleftrightarrow> Min = l" |
|
920 |
"Bit0 k = Pls \<longleftrightarrow> k = Pls" |
|
921 |
"Bit0 k = Min \<longleftrightarrow> False" |
|
922 |
"Bit1 k = Pls \<longleftrightarrow> False" |
|
923 |
"Bit1 k = Min \<longleftrightarrow> k = Min" |
|
924 |
"Bit0 k = Bit0 l \<longleftrightarrow> k = l" |
|
925 |
"Bit0 k = Bit1 l \<longleftrightarrow> False" |
|
926 |
"Bit1 k = Bit0 l \<longleftrightarrow> False" |
|
927 |
"Bit1 k = Bit1 l \<longleftrightarrow> k = l" |
|
928 |
unfolding order_eq_iff [where 'a=int] |
|
929 |
by (simp_all add: not_less) |
|
930 |
||
931 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
932 |
subsection {* Converting Numerals to Rings: @{term number_of} *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
933 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
934 |
class number_ring = number + comm_ring_1 + |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
935 |
assumes number_of_eq: "number_of k = of_int k" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
936 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
937 |
text {* self-embedding of the integers *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
938 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
939 |
instantiation int :: number_ring |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
940 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
941 |
|
28724 | 942 |
definition int_number_of_def [code del]: |
943 |
"number_of w = (of_int w \<Colon> int)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
944 |
|
28724 | 945 |
instance proof |
946 |
qed (simp only: int_number_of_def) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
947 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
948 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
949 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
950 |
lemma number_of_is_id: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
951 |
"number_of (k::int) = k" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
952 |
unfolding int_number_of_def by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
953 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
954 |
lemma number_of_succ: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
955 |
"number_of (succ k) = (1 + number_of k ::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
956 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
957 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
958 |
lemma number_of_pred: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
959 |
"number_of (pred w) = (- 1 + number_of w ::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
960 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
961 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
962 |
lemma number_of_minus: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
963 |
"number_of (uminus w) = (- (number_of w)::'a::number_ring)" |
28958 | 964 |
unfolding number_of_eq by (rule of_int_minus) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
965 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
966 |
lemma number_of_add: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
967 |
"number_of (v + w) = (number_of v + number_of w::'a::number_ring)" |
28958 | 968 |
unfolding number_of_eq by (rule of_int_add) |
969 |
||
970 |
lemma number_of_diff: |
|
971 |
"number_of (v - w) = (number_of v - number_of w::'a::number_ring)" |
|
972 |
unfolding number_of_eq by (rule of_int_diff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
973 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
974 |
lemma number_of_mult: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
975 |
"number_of (v * w) = (number_of v * number_of w::'a::number_ring)" |
28958 | 976 |
unfolding number_of_eq by (rule of_int_mult) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
977 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
978 |
text {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
979 |
The correctness of shifting. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
980 |
But it doesn't seem to give a measurable speed-up. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
981 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
982 |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
983 |
lemma double_number_of_Bit0: |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
984 |
"(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
985 |
unfolding number_of_eq numeral_simps left_distrib by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
986 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
987 |
text {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
988 |
Converting numerals 0 and 1 to their abstract versions. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
989 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
990 |
|
32272
cc1bf9077167
added numeral code postprocessor rules on type int
haftmann
parents:
32069
diff
changeset
|
991 |
lemma numeral_0_eq_0 [simp, code_post]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
992 |
"Numeral0 = (0::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
993 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
994 |
|
32272
cc1bf9077167
added numeral code postprocessor rules on type int
haftmann
parents:
32069
diff
changeset
|
995 |
lemma numeral_1_eq_1 [simp, code_post]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
996 |
"Numeral1 = (1::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
997 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
998 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
999 |
text {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1000 |
Special-case simplification for small constants. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1001 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1002 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1003 |
text{* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1004 |
Unary minus for the abstract constant 1. Cannot be inserted |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1005 |
as a simprule until later: it is @{text number_of_Min} re-oriented! |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1006 |
*} |
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 numeral_m1_eq_minus_1: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1009 |
"(-1::'a::number_ring) = - 1" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1010 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1011 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1012 |
lemma mult_minus1 [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1013 |
"-1 * z = -(z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1014 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1015 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1016 |
lemma mult_minus1_right [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1017 |
"z * -1 = -(z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1018 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1019 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1020 |
(*Negation of a coefficient*) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1021 |
lemma minus_number_of_mult [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1022 |
"- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1023 |
unfolding number_of_eq by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1024 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1025 |
text {* Subtraction *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1026 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1027 |
lemma diff_number_of_eq: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1028 |
"number_of v - number_of w = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1029 |
(number_of (v + uminus w)::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1030 |
unfolding number_of_eq by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1031 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1032 |
lemma number_of_Pls: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1033 |
"number_of Pls = (0::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1034 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1035 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1036 |
lemma number_of_Min: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1037 |
"number_of Min = (- 1::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1038 |
unfolding number_of_eq numeral_simps by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1039 |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1040 |
lemma number_of_Bit0: |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1041 |
"number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1042 |
unfolding number_of_eq numeral_simps by simp |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1043 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1044 |
lemma number_of_Bit1: |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1045 |
"number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1046 |
unfolding number_of_eq numeral_simps by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1047 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1048 |
|
28958 | 1049 |
subsubsection {* Equality of Binary Numbers *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1050 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1051 |
text {* First version by Norbert Voelker *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1052 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1053 |
definition (*for simplifying equalities*) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1054 |
iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1055 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1056 |
"iszero z \<longleftrightarrow> z = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1057 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1058 |
lemma iszero_0: "iszero 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1059 |
by (simp add: iszero_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1060 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1061 |
lemma not_iszero_1: "~ iszero 1" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1062 |
by (simp add: iszero_def eq_commute) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1063 |
|
35216 | 1064 |
lemma eq_number_of_eq [simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1065 |
"((number_of x::'a::number_ring) = number_of y) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1066 |
iszero (number_of (x + uminus y) :: 'a)" |
29667 | 1067 |
unfolding iszero_def number_of_add number_of_minus |
1068 |
by (simp add: algebra_simps) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1069 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1070 |
lemma iszero_number_of_Pls: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1071 |
"iszero ((number_of Pls)::'a::number_ring)" |
29667 | 1072 |
unfolding iszero_def numeral_0_eq_0 .. |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1073 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1074 |
lemma nonzero_number_of_Min: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1075 |
"~ iszero ((number_of Min)::'a::number_ring)" |
29667 | 1076 |
unfolding iszero_def numeral_m1_eq_minus_1 by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1077 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1078 |
|
28958 | 1079 |
subsubsection {* Comparisons, for Ordered Rings *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1080 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1081 |
lemmas double_eq_0_iff = double_zero |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1082 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1083 |
lemma odd_nonzero: |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1084 |
"1 + z + z \<noteq> (0::int)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1085 |
proof (cases z rule: int_cases) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1086 |
case (nonneg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1087 |
have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1088 |
thus ?thesis using le_imp_0_less [OF le] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1089 |
by (auto simp add: add_assoc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1090 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1091 |
case (neg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1092 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1093 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1094 |
assume eq: "1 + z + z = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1095 |
have "(0::int) < 1 + (of_nat n + of_nat n)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1096 |
by (simp add: le_imp_0_less add_increasing) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1097 |
also have "... = - (1 + z + z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1098 |
by (simp add: neg add_assoc [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1099 |
also have "... = 0" by (simp add: eq) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1100 |
finally have "0<0" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1101 |
thus False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1102 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1103 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1104 |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1105 |
lemma iszero_number_of_Bit0: |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1106 |
"iszero (number_of (Bit0 w)::'a) = |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1107 |
iszero (number_of w::'a::{ring_char_0,number_ring})" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1108 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1109 |
have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1110 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1111 |
assume eq: "of_int w + of_int w = (0::'a)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1112 |
then have "of_int (w + w) = (of_int 0 :: 'a)" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1113 |
then have "w + w = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1114 |
then show "w = 0" by (simp only: double_eq_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1115 |
qed |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1116 |
thus ?thesis |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1117 |
by (auto simp add: iszero_def number_of_eq numeral_simps) |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1118 |
qed |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1119 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1120 |
lemma iszero_number_of_Bit1: |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1121 |
"~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1122 |
proof - |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1123 |
have "1 + of_int w + of_int w \<noteq> (0::'a)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1124 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1125 |
assume eq: "1 + of_int w + of_int w = (0::'a)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1126 |
hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1127 |
hence "1 + w + w = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1128 |
with odd_nonzero show False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1129 |
qed |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1130 |
thus ?thesis |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
1131 |
by (auto simp add: iszero_def number_of_eq numeral_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1132 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1133 |
|
35216 | 1134 |
lemmas iszero_simps [simp] = |
28985
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
1135 |
iszero_0 not_iszero_1 |
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
1136 |
iszero_number_of_Pls nonzero_number_of_Min |
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
1137 |
iszero_number_of_Bit0 iszero_number_of_Bit1 |
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
1138 |
(* iszero_number_of_Pls would never normally be used |
af325cd29b15
add named lemma lists: neg_simps and iszero_simps
huffman
parents:
28984
diff
changeset
|
1139 |
because its lhs simplifies to "iszero 0" *) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1140 |
|
28958 | 1141 |
subsubsection {* The Less-Than Relation *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1142 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1143 |
lemma double_less_0_iff: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1144 |
"(a + a < 0) = (a < (0::'a::linordered_idom))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1145 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1146 |
have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1147 |
also have "... = (a < 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1148 |
by (simp add: mult_less_0_iff zero_less_two |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1149 |
order_less_not_sym [OF zero_less_two]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1150 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1151 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1152 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1153 |
lemma odd_less_0: |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1154 |
"(1 + z + z < 0) = (z < (0::int))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1155 |
proof (cases z rule: int_cases) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1156 |
case (nonneg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1157 |
thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1158 |
le_imp_0_less [THEN order_less_imp_le]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1159 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1160 |
case (neg n) |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1161 |
thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1 |
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1162 |
add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1163 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1164 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1165 |
text {* Less-Than or Equals *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1166 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1167 |
text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1168 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1169 |
lemmas le_number_of_eq_not_less = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1170 |
linorder_not_less [of "number_of w" "number_of v", symmetric, |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1171 |
standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1172 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1173 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1174 |
text {* Absolute value (@{term abs}) *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1175 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1176 |
lemma abs_number_of: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1177 |
"abs(number_of x::'a::{linordered_idom,number_ring}) = |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1178 |
(if number_of x < (0::'a) then -number_of x else number_of x)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1179 |
by (simp add: abs_if) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1180 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1181 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1182 |
text {* Re-orientation of the equation nnn=x *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1183 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1184 |
lemma number_of_reorient: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1185 |
"(number_of w = x) = (x = number_of w)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1186 |
by auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1187 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1188 |
|
28958 | 1189 |
subsubsection {* Simplification of arithmetic operations on integer constants. *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1190 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1191 |
lemmas arith_extra_simps [standard, simp] = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1192 |
number_of_add [symmetric] |
28958 | 1193 |
number_of_minus [symmetric] |
1194 |
numeral_m1_eq_minus_1 [symmetric] |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1195 |
number_of_mult [symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1196 |
diff_number_of_eq abs_number_of |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1197 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1198 |
text {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1199 |
For making a minimal simpset, one must include these default simprules. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1200 |
Also include @{text simp_thms}. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1201 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1202 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1203 |
lemmas arith_simps = |
26075
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
1204 |
normalize_bin_simps pred_bin_simps succ_bin_simps |
815f3ccc0b45
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
26072
diff
changeset
|
1205 |
add_bin_simps minus_bin_simps mult_bin_simps |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1206 |
abs_zero abs_one arith_extra_simps |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1207 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1208 |
text {* Simplification of relational operations *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1209 |
|
28962
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1210 |
lemma less_number_of [simp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1211 |
"(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y" |
28962
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1212 |
unfolding number_of_eq by (rule of_int_less_iff) |
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1213 |
|
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1214 |
lemma le_number_of [simp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1215 |
"(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y" |
28962
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1216 |
unfolding number_of_eq by (rule of_int_le_iff) |
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1217 |
|
28967
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28962
diff
changeset
|
1218 |
lemma eq_number_of [simp]: |
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28962
diff
changeset
|
1219 |
"(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y" |
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28962
diff
changeset
|
1220 |
unfolding number_of_eq by (rule of_int_eq_iff) |
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28962
diff
changeset
|
1221 |
|
35216 | 1222 |
lemmas rel_simps = |
28962
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1223 |
less_number_of less_bin_simps |
f603183f7a5c
enable le_bin_simps and less_bin_simps for simplifying inequalities on numerals
huffman
parents:
28958
diff
changeset
|
1224 |
le_number_of le_bin_simps |
28988
13d6f120992b
revert to using eq_number_of_eq for simplification (Groebner_Examples.thy was broken)
huffman
parents:
28985
diff
changeset
|
1225 |
eq_number_of_eq eq_bin_simps |
29039 | 1226 |
iszero_simps |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1227 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1228 |
|
28958 | 1229 |
subsubsection {* Simplification of arithmetic when nested to the right. *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1230 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1231 |
lemma add_number_of_left [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1232 |
"number_of v + (number_of w + z) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1233 |
(number_of(v + w) + z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1234 |
by (simp add: add_assoc [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1235 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1236 |
lemma mult_number_of_left [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1237 |
"number_of v * (number_of w * z) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1238 |
(number_of(v * w) * z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1239 |
by (simp add: mult_assoc [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1240 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1241 |
lemma add_number_of_diff1: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1242 |
"number_of v + (number_of w - c) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1243 |
number_of(v + w) - (c::'a::number_ring)" |
35216 | 1244 |
by (simp add: diff_minus) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1245 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1246 |
lemma add_number_of_diff2 [simp]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1247 |
"number_of v + (c - number_of w) = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1248 |
number_of (v + uminus w) + (c::'a::number_ring)" |
29667 | 1249 |
by (simp add: algebra_simps diff_number_of_eq [symmetric]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1250 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1251 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1252 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1253 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1254 |
subsection {* The Set of Integers *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1255 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1256 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1257 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1258 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1259 |
definition Ints :: "'a set" where |
28562 | 1260 |
[code del]: "Ints = range of_int" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1261 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1262 |
notation (xsymbols) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1263 |
Ints ("\<int>") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1264 |
|
35634 | 1265 |
lemma Ints_of_int [simp]: "of_int z \<in> \<int>" |
1266 |
by (simp add: Ints_def) |
|
1267 |
||
1268 |
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>" |
|
1269 |
apply (simp add: Ints_def) |
|
1270 |
apply (rule range_eqI) |
|
1271 |
apply (rule of_int_of_nat_eq [symmetric]) |
|
1272 |
done |
|
1273 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1274 |
lemma Ints_0 [simp]: "0 \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1275 |
apply (simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1276 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1277 |
apply (rule of_int_0 [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1278 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1279 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1280 |
lemma Ints_1 [simp]: "1 \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1281 |
apply (simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1282 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1283 |
apply (rule of_int_1 [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1284 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1285 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1286 |
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1287 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1288 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1289 |
apply (rule of_int_add [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1290 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1291 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1292 |
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1293 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1294 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1295 |
apply (rule of_int_minus [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1296 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1297 |
|
35634 | 1298 |
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>" |
1299 |
apply (auto simp add: Ints_def) |
|
1300 |
apply (rule range_eqI) |
|
1301 |
apply (rule of_int_diff [symmetric]) |
|
1302 |
done |
|
1303 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1304 |
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1305 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1306 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1307 |
apply (rule of_int_mult [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1308 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1309 |
|
35634 | 1310 |
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>" |
1311 |
by (induct n) simp_all |
|
1312 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1313 |
lemma Ints_cases [cases set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1314 |
assumes "q \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1315 |
obtains (of_int) z where "q = of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1316 |
unfolding Ints_def |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1317 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1318 |
from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1319 |
then obtain z where "q = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1320 |
then show thesis .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1321 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1322 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1323 |
lemma Ints_induct [case_names of_int, induct set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1324 |
"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
|
1325 |
by (rule Ints_cases) auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1326 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1327 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1328 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1329 |
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1330 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1331 |
lemma Ints_double_eq_0_iff: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1332 |
assumes in_Ints: "a \<in> Ints" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1333 |
shows "(a + a = 0) = (a = (0::'a::ring_char_0))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1334 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1335 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1336 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1337 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1338 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1339 |
assume "a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1340 |
thus "a + a = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1341 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1342 |
assume eq: "a + a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1343 |
hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1344 |
hence "z + z = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1345 |
hence "z = 0" by (simp only: double_eq_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1346 |
thus "a = 0" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1347 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1348 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1349 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1350 |
lemma Ints_odd_nonzero: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1351 |
assumes in_Ints: "a \<in> Ints" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1352 |
shows "1 + a + a \<noteq> (0::'a::ring_char_0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1353 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1354 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1355 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1356 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1357 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1358 |
assume eq: "1 + a + a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1359 |
hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1360 |
hence "1 + z + z = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1361 |
with odd_nonzero show False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1362 |
qed |
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 |
|
35634 | 1365 |
lemma Ints_number_of [simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1366 |
"(number_of w :: 'a::number_ring) \<in> Ints" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1367 |
unfolding number_of_eq Ints_def by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1368 |
|
35634 | 1369 |
lemma Nats_number_of [simp]: |
1370 |
"Int.Pls \<le> w \<Longrightarrow> (number_of w :: 'a::number_ring) \<in> Nats" |
|
1371 |
unfolding Int.Pls_def number_of_eq |
|
1372 |
by (simp only: of_nat_nat [symmetric] of_nat_in_Nats) |
|
1373 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1374 |
lemma Ints_odd_less_0: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1375 |
assumes in_Ints: "a \<in> Ints" |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1376 |
shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1377 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1378 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1379 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1380 |
hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1381 |
by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1382 |
also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1383 |
also have "... = (a < 0)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1384 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1385 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1386 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1387 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1388 |
subsection {* @{term setsum} and @{term setprod} *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1389 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1390 |
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1391 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1392 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1393 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1394 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1395 |
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1396 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1397 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1398 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1399 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1400 |
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1401 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1402 |
apply (erule finite_induct, auto simp add: of_nat_mult) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1403 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1404 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1405 |
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1406 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1407 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1408 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1409 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1410 |
lemmas int_setsum = of_nat_setsum [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1411 |
lemmas int_setprod = of_nat_setprod [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1412 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1413 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1414 |
subsection{*Inequality Reasoning for the Arithmetic Simproc*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1415 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1416 |
lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1417 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1418 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1419 |
lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1420 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1421 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1422 |
lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1423 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1424 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1425 |
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1426 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1427 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1428 |
lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1429 |
by simp |
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 |
lemma inverse_numeral_1: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1432 |
"inverse Numeral1 = (Numeral1::'a::{number_ring,field})" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1433 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1434 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1435 |
text{*Theorem lists for the cancellation simprocs. The use of binary numerals |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1436 |
for 0 and 1 reduces the number of special cases.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1437 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1438 |
lemmas add_0s = add_numeral_0 add_numeral_0_right |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1439 |
lemmas mult_1s = mult_numeral_1 mult_numeral_1_right |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1440 |
mult_minus1 mult_minus1_right |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1441 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1442 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1443 |
subsection{*Special Arithmetic Rules for Abstract 0 and 1*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1444 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1445 |
text{*Arithmetic computations are defined for binary literals, which leaves 0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1446 |
and 1 as special cases. Addition already has rules for 0, but not 1. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1447 |
Multiplication and unary minus already have rules for both 0 and 1.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1448 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1449 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1450 |
lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1451 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1452 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1453 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1454 |
lemmas add_number_of_eq = number_of_add [symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1455 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1456 |
text{*Allow 1 on either or both sides*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1457 |
lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)" |
35216 | 1458 |
by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1459 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1460 |
lemmas add_special = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1461 |
one_add_one_is_two |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1462 |
binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1463 |
binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1464 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1465 |
text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1466 |
lemmas diff_special = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1467 |
binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1468 |
binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1469 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1470 |
text{*Allow 0 or 1 on either side with a binary numeral on the other*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1471 |
lemmas eq_special = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1472 |
binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1473 |
binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1474 |
binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1475 |
binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1476 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1477 |
text{*Allow 0 or 1 on either side with a binary numeral on the other*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1478 |
lemmas less_special = |
28984 | 1479 |
binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard] |
1480 |
binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard] |
|
1481 |
binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard] |
|
1482 |
binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard] |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1483 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1484 |
text{*Allow 0 or 1 on either side with a binary numeral on the other*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1485 |
lemmas le_special = |
28984 | 1486 |
binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard] |
1487 |
binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard] |
|
1488 |
binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard] |
|
1489 |
binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard] |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1490 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1491 |
lemmas arith_special[simp] = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1492 |
add_special diff_special eq_special less_special le_special |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1493 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1494 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1495 |
text {* Legacy theorems *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1496 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1497 |
lemmas zle_int = of_nat_le_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1498 |
lemmas int_int_eq = of_nat_eq_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1499 |
|
30802 | 1500 |
subsection {* Setting up simplification procedures *} |
1501 |
||
1502 |
lemmas int_arith_rules = |
|
1503 |
neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1 |
|
1504 |
minus_zero diff_minus left_minus right_minus |
|
1505 |
mult_zero_left mult_zero_right mult_Bit1 mult_1_right |
|
1506 |
mult_minus_left mult_minus_right |
|
1507 |
minus_add_distrib minus_minus mult_assoc |
|
1508 |
of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult |
|
1509 |
of_int_0 of_int_1 of_int_add of_int_mult |
|
1510 |
||
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28724
diff
changeset
|
1511 |
use "Tools/int_arith.ML" |
31100 | 1512 |
setup {* Int_Arith.global_setup *} |
30496
7cdcc9dd95cb
vague cleanup in arith proof tools setup: deleted dead code, more proper structures, clearer arrangement
haftmann
parents:
30273
diff
changeset
|
1513 |
declaration {* K Int_Arith.setup *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1514 |
|
31024
0fdf666e08bf
reimplement reorientation simproc using theory data
huffman
parents:
31021
diff
changeset
|
1515 |
setup {* |
33523 | 1516 |
Reorient_Proc.add |
31065 | 1517 |
(fn Const (@{const_name number_of}, _) $ _ => true | _ => false) |
31024
0fdf666e08bf
reimplement reorientation simproc using theory data
huffman
parents:
31021
diff
changeset
|
1518 |
*} |
0fdf666e08bf
reimplement reorientation simproc using theory data
huffman
parents:
31021
diff
changeset
|
1519 |
|
33523 | 1520 |
simproc_setup reorient_numeral ("number_of w = x") = Reorient_Proc.proc |
31024
0fdf666e08bf
reimplement reorientation simproc using theory data
huffman
parents:
31021
diff
changeset
|
1521 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1522 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1523 |
subsection{*Lemmas About Small Numerals*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1524 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1525 |
lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1526 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1527 |
have "(of_int -1 :: 'a) = of_int (- 1)" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1528 |
also have "... = - of_int 1" by (simp only: of_int_minus) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1529 |
also have "... = -1" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1530 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1531 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1532 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1533 |
lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1534 |
by (simp add: abs_if) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1535 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1536 |
lemma abs_power_minus_one [simp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1537 |
"abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1538 |
by (simp add: power_abs) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1539 |
|
30000 | 1540 |
lemma of_int_number_of_eq [simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1541 |
"of_int (number_of v) = (number_of v :: 'a :: number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1542 |
by (simp add: number_of_eq) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1543 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1544 |
text{*Lemmas for specialist use, NOT as default simprules*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1545 |
lemma mult_2: "2 * z = (z+z::'a::number_ring)" |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1546 |
unfolding one_add_one_is_two [symmetric] left_distrib by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1547 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1548 |
lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1549 |
by (subst mult_commute, rule mult_2) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1550 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1551 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1552 |
subsection{*More Inequality Reasoning*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1553 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1554 |
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1555 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1556 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1557 |
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1558 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1559 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1560 |
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1561 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1562 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1563 |
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1564 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1565 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1566 |
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1567 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1568 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1569 |
|
28958 | 1570 |
subsection{*The functions @{term nat} and @{term int}*} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1571 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1572 |
text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1573 |
@{term "w + - z"}*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1574 |
declare Zero_int_def [symmetric, simp] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1575 |
declare One_int_def [symmetric, simp] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1576 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1577 |
lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1578 |
|
35216 | 1579 |
(* FIXME: duplicates nat_zero *) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1580 |
lemma nat_0: "nat 0 = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1581 |
by (simp add: nat_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1582 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1583 |
lemma nat_1: "nat 1 = Suc 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1584 |
by (subst nat_eq_iff, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1585 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1586 |
lemma nat_2: "nat 2 = Suc (Suc 0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1587 |
by (subst nat_eq_iff, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1588 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1589 |
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1590 |
apply (insert zless_nat_conj [of 1 z]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1591 |
apply (auto simp add: nat_1) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1592 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1593 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1594 |
text{*This simplifies expressions of the form @{term "int n = z"} where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1595 |
z is an integer literal.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1596 |
lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1597 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1598 |
lemma split_nat [arith_split]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1599 |
"P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1600 |
(is "?P = (?L & ?R)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1601 |
proof (cases "i < 0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1602 |
case True thus ?thesis by auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1603 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1604 |
case False |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1605 |
have "?P = ?L" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1606 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1607 |
assume ?P thus ?L using False by clarsimp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1608 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1609 |
assume ?L thus ?P using False by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1610 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1611 |
with False show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1612 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1613 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1614 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1615 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1616 |
|
33056
791a4655cae3
renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents:
32437
diff
changeset
|
1617 |
lemma of_int_of_nat [nitpick_simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1618 |
"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
|
1619 |
proof (cases "k < 0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1620 |
case True then have "0 \<le> - k" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1621 |
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
|
1622 |
with True show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1623 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1624 |
case False then show ?thesis by (simp add: not_less of_nat_nat) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1625 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1626 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1627 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1628 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1629 |
lemma nat_mult_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1630 |
fixes z z' :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1631 |
assumes "0 \<le> z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1632 |
shows "nat (z * z') = nat z * nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1633 |
proof (cases "0 \<le> z'") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1634 |
case False with assms have "z * z' \<le> 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1635 |
by (simp add: not_le mult_le_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1636 |
then have "nat (z * z') = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1637 |
moreover from False have "nat z' = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1638 |
ultimately show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1639 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1640 |
case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1641 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1642 |
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
|
1643 |
(simp only: of_nat_mult of_nat_nat [OF True] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1644 |
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
|
1645 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1646 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1647 |
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1648 |
apply (rule trans) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1649 |
apply (rule_tac [2] nat_mult_distrib, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1650 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1651 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1652 |
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1653 |
apply (cases "z=0 | w=0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1654 |
apply (auto simp add: abs_if nat_mult_distrib [symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1655 |
nat_mult_distrib_neg [symmetric] mult_less_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1656 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1657 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1658 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1659 |
subsection "Induction principles for int" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1660 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1661 |
text{*Well-founded segments of the integers*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1662 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1663 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1664 |
int_ge_less_than :: "int => (int * int) set" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1665 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1666 |
"int_ge_less_than d = {(z',z). d \<le> z' & z' < z}" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1667 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1668 |
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1669 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1670 |
have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1671 |
by (auto simp add: int_ge_less_than_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1672 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1673 |
by (rule wf_subset [OF wf_measure]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1674 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1675 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1676 |
text{*This variant looks odd, but is typical of the relations suggested |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1677 |
by RankFinder.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1678 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1679 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1680 |
int_ge_less_than2 :: "int => (int * int) set" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1681 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1682 |
"int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1683 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1684 |
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1685 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1686 |
have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1687 |
by (auto simp add: int_ge_less_than2_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1688 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1689 |
by (rule wf_subset [OF wf_measure]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1690 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1691 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1692 |
abbreviation |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1693 |
int :: "nat \<Rightarrow> int" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1694 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1695 |
"int \<equiv> of_nat" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1696 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1697 |
(* `set:int': dummy construction *) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1698 |
theorem int_ge_induct [case_names base step, induct set: int]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1699 |
fixes i :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1700 |
assumes ge: "k \<le> i" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1701 |
base: "P k" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1702 |
step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1703 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1704 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1705 |
{ fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1706 |
proof (induct n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1707 |
case 0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1708 |
hence "i = k" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1709 |
thus "P i" using base by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1710 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1711 |
case (Suc n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1712 |
then have "n = nat((i - 1) - k)" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1713 |
moreover |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1714 |
have ki1: "k \<le> i - 1" using Suc.prems by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1715 |
ultimately |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1716 |
have "P(i - 1)" by(rule Suc.hyps) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1717 |
from step[OF ki1 this] show ?case by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1718 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1719 |
} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1720 |
with ge show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1721 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1722 |
|
25928 | 1723 |
(* `set:int': dummy construction *) |
1724 |
theorem int_gr_induct [case_names base step, induct set: int]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1725 |
assumes gr: "k < (i::int)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1726 |
base: "P(k+1)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1727 |
step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1728 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1729 |
apply(rule int_ge_induct[of "k + 1"]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1730 |
using gr apply arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1731 |
apply(rule base) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1732 |
apply (rule step, simp+) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1733 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1734 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1735 |
theorem int_le_induct[consumes 1,case_names base step]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1736 |
assumes le: "i \<le> (k::int)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1737 |
base: "P(k)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1738 |
step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1739 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1740 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1741 |
{ fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1742 |
proof (induct n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1743 |
case 0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1744 |
hence "i = k" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1745 |
thus "P i" using base by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1746 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1747 |
case (Suc n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1748 |
hence "n = nat(k - (i+1))" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1749 |
moreover |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1750 |
have ki1: "i + 1 \<le> k" using Suc.prems by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1751 |
ultimately |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1752 |
have "P(i+1)" by(rule Suc.hyps) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1753 |
from step[OF ki1 this] show ?case by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1754 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1755 |
} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1756 |
with le show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1757 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1758 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1759 |
theorem int_less_induct [consumes 1,case_names base step]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1760 |
assumes less: "(i::int) < k" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1761 |
base: "P(k - 1)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1762 |
step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1763 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1764 |
apply(rule int_le_induct[of _ "k - 1"]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1765 |
using less apply arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1766 |
apply(rule base) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1767 |
apply (rule step, simp+) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1768 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1769 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1770 |
subsection{*Intermediate value theorems*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1771 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1772 |
lemma int_val_lemma: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1773 |
"(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) --> |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1774 |
f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))" |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1775 |
unfolding One_nat_def |
27106 | 1776 |
apply (induct n, simp) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1777 |
apply (intro strip) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1778 |
apply (erule impE, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1779 |
apply (erule_tac x = n in allE, simp) |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1780 |
apply (case_tac "k = f (Suc n)") |
27106 | 1781 |
apply force |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1782 |
apply (erule impE) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1783 |
apply (simp add: abs_if split add: split_if_asm) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1784 |
apply (blast intro: le_SucI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1785 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1786 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1787 |
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1788 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1789 |
lemma nat_intermed_int_val: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1790 |
"[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1791 |
f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1792 |
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1793 |
in int_val_lemma) |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1794 |
unfolding One_nat_def |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1795 |
apply simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1796 |
apply (erule exE) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1797 |
apply (rule_tac x = "i+m" in exI, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1798 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1799 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1800 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1801 |
subsection{*Products and 1, by T. M. Rasmussen*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1802 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1803 |
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1804 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1805 |
|
34055 | 1806 |
lemma abs_zmult_eq_1: |
1807 |
assumes mn: "\<bar>m * n\<bar> = 1" |
|
1808 |
shows "\<bar>m\<bar> = (1::int)" |
|
1809 |
proof - |
|
1810 |
have 0: "m \<noteq> 0 & n \<noteq> 0" using mn |
|
1811 |
by auto |
|
1812 |
have "~ (2 \<le> \<bar>m\<bar>)" |
|
1813 |
proof |
|
1814 |
assume "2 \<le> \<bar>m\<bar>" |
|
1815 |
hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>" |
|
1816 |
by (simp add: mult_mono 0) |
|
1817 |
also have "... = \<bar>m*n\<bar>" |
|
1818 |
by (simp add: abs_mult) |
|
1819 |
also have "... = 1" |
|
1820 |
by (simp add: mn) |
|
1821 |
finally have "2*\<bar>n\<bar> \<le> 1" . |
|
1822 |
thus "False" using 0 |
|
1823 |
by auto |
|
1824 |
qed |
|
1825 |
thus ?thesis using 0 |
|
1826 |
by auto |
|
1827 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1828 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1829 |
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1830 |
by (insert abs_zmult_eq_1 [of m n], arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1831 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1832 |
lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1833 |
apply (auto dest: pos_zmult_eq_1_iff_lemma) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1834 |
apply (simp add: mult_commute [of m]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1835 |
apply (frule pos_zmult_eq_1_iff_lemma, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1836 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1837 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1838 |
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1839 |
apply (rule iffI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1840 |
apply (frule pos_zmult_eq_1_iff_lemma) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1841 |
apply (simp add: mult_commute [of m]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1842 |
apply (frule pos_zmult_eq_1_iff_lemma, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1843 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1844 |
|
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1845 |
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1846 |
proof |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1847 |
assume "finite (UNIV::int set)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1848 |
moreover have "inj (\<lambda>i\<Colon>int. 2 * i)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1849 |
by (rule injI) simp |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1850 |
ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1851 |
by (rule finite_UNIV_inj_surj) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1852 |
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
|
1853 |
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
|
1854 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1855 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1856 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1857 |
subsection {* Further theorems on numerals *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1858 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1859 |
subsubsection{*Special Simplification for Constants*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1860 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1861 |
text{*These distributive laws move literals inside sums and differences.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1862 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1863 |
lemmas left_distrib_number_of [simp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1864 |
left_distrib [of _ _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1865 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1866 |
lemmas right_distrib_number_of [simp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1867 |
right_distrib [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1868 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1869 |
lemmas left_diff_distrib_number_of [simp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1870 |
left_diff_distrib [of _ _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1871 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1872 |
lemmas right_diff_distrib_number_of [simp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1873 |
right_diff_distrib [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1874 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1875 |
text{*These are actually for fields, like real: but where else to put them?*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1876 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1877 |
lemmas zero_less_divide_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1878 |
zero_less_divide_iff [of "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1879 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1880 |
lemmas divide_less_0_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1881 |
divide_less_0_iff [of "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1882 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1883 |
lemmas zero_le_divide_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1884 |
zero_le_divide_iff [of "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1885 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1886 |
lemmas divide_le_0_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1887 |
divide_le_0_iff [of "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1888 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1889 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1890 |
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1891 |
strange, but then other simprocs simplify the quotient.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1892 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1893 |
lemmas inverse_eq_divide_number_of [simp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1894 |
inverse_eq_divide [of "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1895 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1896 |
text {*These laws simplify inequalities, moving unary minus from a term |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1897 |
into the literal.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1898 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1899 |
lemmas less_minus_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1900 |
less_minus_iff [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1901 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1902 |
lemmas le_minus_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1903 |
le_minus_iff [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1904 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1905 |
lemmas equation_minus_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1906 |
equation_minus_iff [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1907 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1908 |
lemmas minus_less_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1909 |
minus_less_iff [of _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1910 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1911 |
lemmas minus_le_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1912 |
minus_le_iff [of _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1913 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1914 |
lemmas minus_equation_iff_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1915 |
minus_equation_iff [of _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1916 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1917 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1918 |
text{*To Simplify Inequalities Where One Side is the Constant 1*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1919 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1920 |
lemma less_minus_iff_1 [simp,noatp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1921 |
fixes b::"'b::{linordered_idom,number_ring}" |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1922 |
shows "(1 < - b) = (b < -1)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1923 |
by auto |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1924 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1925 |
lemma le_minus_iff_1 [simp,noatp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1926 |
fixes b::"'b::{linordered_idom,number_ring}" |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1927 |
shows "(1 \<le> - b) = (b \<le> -1)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1928 |
by auto |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1929 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1930 |
lemma equation_minus_iff_1 [simp,noatp]: |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1931 |
fixes b::"'b::number_ring" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1932 |
shows "(1 = - b) = (b = -1)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1933 |
by (subst equation_minus_iff, auto) |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1934 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1935 |
lemma minus_less_iff_1 [simp,noatp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1936 |
fixes a::"'b::{linordered_idom,number_ring}" |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1937 |
shows "(- a < 1) = (-1 < a)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1938 |
by auto |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1939 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1940 |
lemma minus_le_iff_1 [simp,noatp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
1941 |
fixes a::"'b::{linordered_idom,number_ring}" |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1942 |
shows "(- a \<le> 1) = (-1 \<le> a)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1943 |
by auto |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1944 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1945 |
lemma minus_equation_iff_1 [simp,noatp]: |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1946 |
fixes a::"'b::number_ring" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1947 |
shows "(- a = 1) = (a = -1)" |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1948 |
by (subst minus_equation_iff, auto) |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1949 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1950 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1951 |
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1952 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1953 |
lemmas mult_less_cancel_left_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1954 |
mult_less_cancel_left [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1955 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1956 |
lemmas mult_less_cancel_right_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1957 |
mult_less_cancel_right [of _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1958 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1959 |
lemmas mult_le_cancel_left_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1960 |
mult_le_cancel_left [of "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1961 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1962 |
lemmas mult_le_cancel_right_number_of [simp, noatp] = |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1963 |
mult_le_cancel_right [of _ "number_of v", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1964 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1965 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1966 |
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1967 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1968 |
lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1969 |
lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1970 |
lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1971 |
lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1972 |
lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard] |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1973 |
lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard] |
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changeset
|
1974 |
|
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diff
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|
1975 |
|
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|
1976 |
subsubsection{*Optional Simplification Rules Involving Constants*} |
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changeset
|
1977 |
|
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|
1978 |
text{*Simplify quotients that are compared with a literal constant.*} |
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|
1979 |
|
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|
1980 |
lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard] |
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parents:
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diff
changeset
|
1981 |
lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard] |
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haftmann
parents:
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diff
changeset
|
1982 |
lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard] |
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haftmann
parents:
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diff
changeset
|
1983 |
lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard] |
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haftmann
parents:
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diff
changeset
|
1984 |
lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard] |
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haftmann
parents:
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diff
changeset
|
1985 |
lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard] |
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haftmann
parents:
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diff
changeset
|
1986 |
|
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parents:
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diff
changeset
|
1987 |
|
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|
1988 |
text{*Not good as automatic simprules because they cause case splits.*} |
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parents:
30496
diff
changeset
|
1989 |
lemmas divide_const_simps = |
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haftmann
parents:
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diff
changeset
|
1990 |
le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of |
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changeset
|
1991 |
divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of |
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haftmann
parents:
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changeset
|
1992 |
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 |
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changeset
|
1993 |
|
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parents:
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changeset
|
1994 |
text{*Division By @{text "-1"}*} |
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parents:
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diff
changeset
|
1995 |
|
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|
1996 |
lemma divide_minus1 [simp]: |
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parents:
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|
1997 |
"x/-1 = -(x::'a::{field,division_by_zero,number_ring})" |
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parents:
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|
1998 |
by simp |
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parents:
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|
1999 |
|
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|
2000 |
lemma minus1_divide [simp]: |
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|
2001 |
"-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)" |
35216 | 2002 |
by (simp add: divide_inverse) |
30652
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|
2003 |
|
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|
2004 |
lemma half_gt_zero_iff: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
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|
2005 |
"(0 < r/2) = (0 < (r::'a::{linordered_field,division_by_zero,number_ring}))" |
30652
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|
2006 |
by auto |
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|
2007 |
|
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parents:
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changeset
|
2008 |
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard] |
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parents:
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|
2009 |
|
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|
2010 |
|
33320 | 2011 |
subsection {* The divides relation *} |
2012 |
||
33657 | 2013 |
lemma zdvd_antisym_nonneg: |
2014 |
"0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)" |
|
33320 | 2015 |
apply (simp add: dvd_def, auto) |
33657 | 2016 |
apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff) |
33320 | 2017 |
done |
2018 |
||
33657 | 2019 |
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" |
33320 | 2020 |
shows "\<bar>a\<bar> = \<bar>b\<bar>" |
33657 | 2021 |
proof cases |
2022 |
assume "a = 0" with assms show ?thesis by simp |
|
2023 |
next |
|
2024 |
assume "a \<noteq> 0" |
|
33320 | 2025 |
from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast |
2026 |
from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast |
|
2027 |
from k k' have "a = a*k*k'" by simp |
|
2028 |
with mult_cancel_left1[where c="a" and b="k*k'"] |
|
2029 |
have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc) |
|
2030 |
hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff) |
|
2031 |
thus ?thesis using k k' by auto |
|
2032 |
qed |
|
2033 |
||
2034 |
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)" |
|
2035 |
apply (subgoal_tac "m = n + (m - n)") |
|
2036 |
apply (erule ssubst) |
|
2037 |
apply (blast intro: dvd_add, simp) |
|
2038 |
done |
|
2039 |
||
2040 |
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))" |
|
2041 |
apply (rule iffI) |
|
2042 |
apply (erule_tac [2] dvd_add) |
|
2043 |
apply (subgoal_tac "n = (n + k * m) - k * m") |
|
2044 |
apply (erule ssubst) |
|
2045 |
apply (erule dvd_diff) |
|
2046 |
apply(simp_all) |
|
2047 |
done |
|
2048 |
||
2049 |
lemma dvd_imp_le_int: |
|
2050 |
fixes d i :: int |
|
2051 |
assumes "i \<noteq> 0" and "d dvd i" |
|
2052 |
shows "\<bar>d\<bar> \<le> \<bar>i\<bar>" |
|
2053 |
proof - |
|
2054 |
from `d dvd i` obtain k where "i = d * k" .. |
|
2055 |
with `i \<noteq> 0` have "k \<noteq> 0" by auto |
|
2056 |
then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto |
|
2057 |
then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono) |
|
2058 |
with `i = d * k` show ?thesis by (simp add: abs_mult) |
|
2059 |
qed |
|
2060 |
||
2061 |
lemma zdvd_not_zless: |
|
2062 |
fixes m n :: int |
|
2063 |
assumes "0 < m" and "m < n" |
|
2064 |
shows "\<not> n dvd m" |
|
2065 |
proof |
|
2066 |
from assms have "0 < n" by auto |
|
2067 |
assume "n dvd m" then obtain k where k: "m = n * k" .. |
|
2068 |
with `0 < m` have "0 < n * k" by auto |
|
2069 |
with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff) |
|
2070 |
with k `0 < n` `m < n` have "n * k < n * 1" by simp |
|
2071 |
with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto |
|
2072 |
qed |
|
2073 |
||
2074 |
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)" |
|
2075 |
shows "m dvd n" |
|
2076 |
proof- |
|
2077 |
from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast |
|
2078 |
{assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp |
|
2079 |
with h have False by (simp add: mult_assoc)} |
|
2080 |
hence "n = m * h" by blast |
|
2081 |
thus ?thesis by simp |
|
2082 |
qed |
|
2083 |
||
2084 |
theorem zdvd_int: "(x dvd y) = (int x dvd int y)" |
|
2085 |
proof - |
|
2086 |
have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y" |
|
2087 |
proof - |
|
2088 |
fix k |
|
2089 |
assume A: "int y = int x * k" |
|
2090 |
then show "x dvd y" proof (cases k) |
|
2091 |
case (1 n) with A have "y = x * n" by (simp add: of_nat_mult [symmetric]) |
|
2092 |
then show ?thesis .. |
|
2093 |
next |
|
2094 |
case (2 n) with A have "int y = int x * (- int (Suc n))" by simp |
|
2095 |
also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right) |
|
2096 |
also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric]) |
|
2097 |
finally have "- int (x * Suc n) = int y" .. |
|
2098 |
then show ?thesis by (simp only: negative_eq_positive) auto |
|
2099 |
qed |
|
2100 |
qed |
|
2101 |
then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult) |
|
2102 |
qed |
|
2103 |
||
2104 |
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)" |
|
2105 |
proof |
|
2106 |
assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp |
|
2107 |
hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int) |
|
2108 |
hence "nat \<bar>x\<bar> = 1" by simp |
|
2109 |
thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto) |
|
2110 |
next |
|
2111 |
assume "\<bar>x\<bar>=1" |
|
2112 |
then have "x = 1 \<or> x = -1" by auto |
|
2113 |
then show "x dvd 1" by (auto intro: dvdI) |
|
2114 |
qed |
|
2115 |
||
2116 |
lemma zdvd_mult_cancel1: |
|
2117 |
assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)" |
|
2118 |
proof |
|
2119 |
assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" |
|
35216 | 2120 |
by (cases "n >0", auto simp add: minus_equation_iff) |
33320 | 2121 |
next |
2122 |
assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp |
|
2123 |
from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq) |
|
2124 |
qed |
|
2125 |
||
2126 |
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))" |
|
2127 |
unfolding zdvd_int by (cases "z \<ge> 0") simp_all |
|
2128 |
||
2129 |
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)" |
|
2130 |
unfolding zdvd_int by (cases "z \<ge> 0") simp_all |
|
2131 |
||
2132 |
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)" |
|
2133 |
by (auto simp add: dvd_int_iff) |
|
2134 |
||
33341 | 2135 |
lemma eq_nat_nat_iff: |
2136 |
"0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'" |
|
2137 |
by (auto elim!: nonneg_eq_int) |
|
2138 |
||
2139 |
lemma nat_power_eq: |
|
2140 |
"0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n" |
|
2141 |
by (induct n) (simp_all add: nat_mult_distrib) |
|
2142 |
||
33320 | 2143 |
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)" |
2144 |
apply (rule_tac z=n in int_cases) |
|
2145 |
apply (auto simp add: dvd_int_iff) |
|
2146 |
apply (rule_tac z=z in int_cases) |
|
2147 |
apply (auto simp add: dvd_imp_le) |
|
2148 |
done |
|
2149 |
||
2150 |
||
25919
8b1c0d434824
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haftmann
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diff
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|
2151 |
subsection {* Configuration of the code generator *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2152 |
|
26507 | 2153 |
code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int" |
2154 |
||
28562 | 2155 |
lemmas pred_succ_numeral_code [code] = |
26507 | 2156 |
pred_bin_simps succ_bin_simps |
2157 |
||
28562 | 2158 |
lemmas plus_numeral_code [code] = |
26507 | 2159 |
add_bin_simps |
2160 |
arith_extra_simps(1) [where 'a = int] |
|
2161 |
||
28562 | 2162 |
lemmas minus_numeral_code [code] = |
26507 | 2163 |
minus_bin_simps |
2164 |
arith_extra_simps(2) [where 'a = int] |
|
2165 |
arith_extra_simps(5) [where 'a = int] |
|
2166 |
||
28562 | 2167 |
lemmas times_numeral_code [code] = |
26507 | 2168 |
mult_bin_simps |
2169 |
arith_extra_simps(4) [where 'a = int] |
|
2170 |
||
2171 |
instantiation int :: eq |
|
2172 |
begin |
|
2173 |
||
28562 | 2174 |
definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)" |
26507 | 2175 |
|
2176 |
instance by default (simp add: eq_int_def) |
|
2177 |
||
2178 |
end |
|
2179 |
||
28562 | 2180 |
lemma eq_number_of_int_code [code]: |
26732 | 2181 |
"eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l" |
26507 | 2182 |
unfolding eq_int_def number_of_is_id .. |
2183 |
||
28562 | 2184 |
lemma eq_int_code [code]: |
26732 | 2185 |
"eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True" |
2186 |
"eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False" |
|
2187 |
"eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2" |
|
2188 |
"eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False" |
|
2189 |
"eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False" |
|
2190 |
"eq_class.eq Int.Min Int.Min \<longleftrightarrow> True" |
|
2191 |
"eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False" |
|
2192 |
"eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2" |
|
28958 | 2193 |
"eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls" |
26732 | 2194 |
"eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False" |
2195 |
"eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False" |
|
28958 | 2196 |
"eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min" |
26732 | 2197 |
"eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2" |
2198 |
"eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False" |
|
2199 |
"eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False" |
|
2200 |
"eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2" |
|
28958 | 2201 |
unfolding eq_equals by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2202 |
|
28351 | 2203 |
lemma eq_int_refl [code nbe]: |
2204 |
"eq_class.eq (k::int) k \<longleftrightarrow> True" |
|
2205 |
by (rule HOL.eq_refl) |
|
2206 |
||
28562 | 2207 |
lemma less_eq_number_of_int_code [code]: |
26507 | 2208 |
"(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l" |
2209 |
unfolding number_of_is_id .. |
|
2210 |
||
28562 | 2211 |
lemma less_eq_int_code [code]: |
26507 | 2212 |
"Int.Pls \<le> Int.Pls \<longleftrightarrow> True" |
2213 |
"Int.Pls \<le> Int.Min \<longleftrightarrow> False" |
|
2214 |
"Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k" |
|
2215 |
"Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k" |
|
2216 |
"Int.Min \<le> Int.Pls \<longleftrightarrow> True" |
|
2217 |
"Int.Min \<le> Int.Min \<longleftrightarrow> True" |
|
2218 |
"Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k" |
|
2219 |
"Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k" |
|
2220 |
"Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls" |
|
2221 |
"Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls" |
|
2222 |
"Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min" |
|
2223 |
"Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min" |
|
2224 |
"Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2" |
|
2225 |
"Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2" |
|
2226 |
"Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2" |
|
2227 |
"Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2" |
|
28958 | 2228 |
by simp_all |
26507 | 2229 |
|
28562 | 2230 |
lemma less_number_of_int_code [code]: |
26507 | 2231 |
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l" |
2232 |
unfolding number_of_is_id .. |
|
2233 |
||
28562 | 2234 |
lemma less_int_code [code]: |
26507 | 2235 |
"Int.Pls < Int.Pls \<longleftrightarrow> False" |
2236 |
"Int.Pls < Int.Min \<longleftrightarrow> False" |
|
2237 |
"Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k" |
|
2238 |
"Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k" |
|
2239 |
"Int.Min < Int.Pls \<longleftrightarrow> True" |
|
2240 |
"Int.Min < Int.Min \<longleftrightarrow> False" |
|
2241 |
"Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k" |
|
2242 |
"Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k" |
|
2243 |
"Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls" |
|
2244 |
"Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls" |
|
2245 |
"Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min" |
|
2246 |
"Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min" |
|
2247 |
"Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2" |
|
2248 |
"Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2" |
|
2249 |
"Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2" |
|
2250 |
"Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2" |
|
28958 | 2251 |
by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2252 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2253 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2254 |
nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2255 |
"nat_aux i n = nat i + n" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2256 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2257 |
lemma [code]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2258 |
"nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))" -- {* tail recursive *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2259 |
by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2260 |
dest: zless_imp_add1_zle) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2261 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2262 |
lemma [code]: "nat i = nat_aux i 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2263 |
by (simp add: nat_aux_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2264 |
|
28514 | 2265 |
hide (open) const nat_aux |
25928 | 2266 |
|
32069
6d28bbd33e2c
prefer code_inline over code_unfold; use code_unfold_post where appropriate
haftmann
parents:
31998
diff
changeset
|
2267 |
lemma zero_is_num_zero [code, code_unfold_post]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2268 |
"(0\<Colon>int) = Numeral0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2269 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2270 |
|
32069
6d28bbd33e2c
prefer code_inline over code_unfold; use code_unfold_post where appropriate
haftmann
parents:
31998
diff
changeset
|
2271 |
lemma one_is_num_one [code, code_unfold_post]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2272 |
"(1\<Colon>int) = Numeral1" |
25961 | 2273 |
by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2274 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2275 |
code_modulename SML |
33364 | 2276 |
Int Arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2277 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2278 |
code_modulename OCaml |
33364 | 2279 |
Int Arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2280 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2281 |
code_modulename Haskell |
33364 | 2282 |
Int Arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2283 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2284 |
types_code |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2285 |
"int" ("int") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2286 |
attach (term_of) {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2287 |
val term_of_int = HOLogic.mk_number HOLogic.intT; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2288 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2289 |
attach (test) {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2290 |
fun gen_int i = |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2291 |
let val j = one_of [~1, 1] * random_range 0 i |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2292 |
in (j, fn () => term_of_int j) end; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2293 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2294 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2295 |
setup {* |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2296 |
let |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2297 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2298 |
fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2299 |
| strip_number_of t = t; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2300 |
|
28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28514
diff
changeset
|
2301 |
fun numeral_codegen thy defs dep module b t gr = |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2302 |
let val i = HOLogic.dest_numeral (strip_number_of t) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2303 |
in |
28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28514
diff
changeset
|
2304 |
SOME (Codegen.str (string_of_int i), |
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28514
diff
changeset
|
2305 |
snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr)) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2306 |
end handle TERM _ => NONE; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2307 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2308 |
in |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2309 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2310 |
Codegen.add_codegen "numeral_codegen" numeral_codegen |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2311 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2312 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2313 |
*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2314 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2315 |
consts_code |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2316 |
"number_of :: int \<Rightarrow> int" ("(_)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2317 |
"0 :: int" ("0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2318 |
"1 :: int" ("1") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2319 |
"uminus :: int => int" ("~") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2320 |
"op + :: int => int => int" ("(_ +/ _)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2321 |
"op * :: int => int => int" ("(_ */ _)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2322 |
"op \<le> :: int => int => bool" ("(_ <=/ _)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2323 |
"op < :: int => int => bool" ("(_ </ _)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2324 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2325 |
quickcheck_params [default_type = int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2326 |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
2327 |
hide (open) const Pls Min Bit0 Bit1 succ pred |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2328 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2329 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2330 |
subsection {* Legacy theorems *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2331 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2332 |
lemmas zminus_zminus = minus_minus [of "z::int", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2333 |
lemmas zminus_0 = minus_zero [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2334 |
lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2335 |
lemmas zadd_commute = add_commute [of "z::int" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2336 |
lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2337 |
lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2338 |
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
2339 |
lemmas zmult_ac = mult_ac |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
2340 |
lemmas zadd_0 = add_0_left [of "z::int", standard] |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
2341 |
lemmas zadd_0_right = add_0_right [of "z::int", standard] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2342 |
lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2343 |
lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2344 |
lemmas zmult_commute = mult_commute [of "z::int" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2345 |
lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2346 |
lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2347 |
lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2348 |
lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2349 |
lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2350 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2351 |
lemmas zmult_1 = mult_1_left [of "z::int", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2352 |
lemmas zmult_1_right = mult_1_right [of "z::int", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2353 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2354 |
lemmas zle_refl = order_refl [of "w::int", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2355 |
lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard] |
33657 | 2356 |
lemmas zle_antisym = order_antisym [of "z::int" "w", standard] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2357 |
lemmas zle_linear = linorder_linear [of "z::int" "w", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2358 |
lemmas zless_linear = linorder_less_linear [where 'a = int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2359 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2360 |
lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2361 |
lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2362 |
lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2363 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2364 |
lemmas int_0_less_1 = zero_less_one [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2365 |
lemmas int_0_neq_1 = zero_neq_one [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2366 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2367 |
lemmas inj_int = inj_of_nat [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2368 |
lemmas zadd_int = of_nat_add [where 'a=int, symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2369 |
lemmas int_mult = of_nat_mult [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2370 |
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2371 |
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2372 |
lemmas zless_int = of_nat_less_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2373 |
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2374 |
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2375 |
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2376 |
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2377 |
lemmas int_0 = of_nat_0 [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2378 |
lemmas int_1 = of_nat_1 [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2379 |
lemmas int_Suc = of_nat_Suc [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2380 |
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2381 |
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2382 |
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2383 |
lemmas zless_le = less_int_def |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2384 |
lemmas int_eq_of_nat = TrueI |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2385 |
|
30960 | 2386 |
lemma zpower_zadd_distrib: |
2387 |
"x ^ (y + z) = ((x ^ y) * (x ^ z)::int)" |
|
2388 |
by (rule power_add) |
|
2389 |
||
2390 |
lemma zero_less_zpower_abs_iff: |
|
2391 |
"(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)" |
|
2392 |
by (rule zero_less_power_abs_iff) |
|
2393 |
||
2394 |
lemma zero_le_zpower_abs: "(0::int) \<le> abs x ^ n" |
|
2395 |
by (rule zero_le_power_abs) |
|
2396 |
||
31015 | 2397 |
lemma zpower_zpower: |
2398 |
"(x ^ y) ^ z = (x ^ (y * z)::int)" |
|
2399 |
by (rule power_mult [symmetric]) |
|
2400 |
||
2401 |
lemma int_power: |
|
2402 |
"int (m ^ n) = int m ^ n" |
|
2403 |
by (rule of_nat_power) |
|
2404 |
||
2405 |
lemmas zpower_int = int_power [symmetric] |
|
2406 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
2407 |
end |